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Three families of two-parameter means constructed by trigonometric functions

Zhen-Hang Yang

Author Affiliations

Power Supply Service Center, ZPEPC Electric Power Research Institute, Jianguozhong Road 219, Hangzhou, Zhejiang, 310009, China

Journal of Inequalities and Applications 2013, 2013:541  doi:10.1186/1029-242X-2013-541

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/541


Received:16 April 2013
Accepted:10 October 2013
Published:19 November 2013

© 2013 Yang; licensee Springer.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we establish three families of trigonometric functions with two parameters and prove their monotonicity and bivariate log-convexity. Based on them, three two-parameter families of means involving trigonometric functions, which include Schwab-Borchardt mean, the first and second Seiffert means, Sándor’s mean and many other new means, are defined. Their properties are given and some new inequalities for these means are proved. Lastly, two families of two-parameter hyperbolic means, which similarly contain many new means, are also presented without proofs.

MSC: 26E60, 26D05, 33B10, 26A48.

Keywords:
trigonometric function; hyperbolic function; mean; inequality

1 Introduction

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M1">View MathML</a> denote the set of positive real numbers and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M2">View MathML</a>. A two-variable continuous function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M3">View MathML</a> is called a mean on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M1">View MathML</a> if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M5">View MathML</a>

holds. For convenience, however, we assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M6">View MathML</a> in what follows unless otherwise stated.

There exist many elementary means. They can be divided into three classes according to main categories of basic elementary functions by their composition. The first class is mainly constructed by power functions, like the Stolarsky means [1] defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M7">View MathML</a>

(1.1)

and Gini means [2] defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M8">View MathML</a>

(1.2)

It is well known that the Stolarsky and Gini means are very important, they contain many famous means, for instance, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M9">View MathML</a> - the logarithmic mean, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M10">View MathML</a> - the identric (exponential) mean, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M11">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M12">View MathML</a> - the p-order power mean, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M13">View MathML</a> - the p-order logarithmic mean, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M14">View MathML</a> - the p-order identric (exponential) mean; <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M15">View MathML</a> - the quadratic mean, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M16">View MathML</a> - the power-exponential mean, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M17">View MathML</a> - the p-order power-exponential mean, etc. The second class is mainly made up of exponential and logarithmic functions, such as the second part of Schwab-Borchardt mean (see [3], [[4], Section 3, equation (2.3)], [5]) defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M18">View MathML</a>

(1.3)

the logarithmic mean <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M19">View MathML</a>, the exponential mean defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M20">View MathML</a>

given in [6] (also see [7,8]) by Sándor and Toader, and the Neuman-Sándor mean defined in [5] by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M21">View MathML</a>

(1.4)

It should be noted that NS is actually a Schwab-Borchardt mean since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M22">View MathML</a> mentioned by Neuman and Sándor in [5].

The third class is mainly composed of trigonometric functions and their inverses, for example, the first part of Schwab-Borchardt mean defined by (1.3), the first and second Seiffert means [9,10] defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M23">View MathML</a>

(1.5)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M24">View MathML</a>

(1.6)

respectively, and the new mean presented recently by Sándor in [11,12] defined as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M25">View MathML</a>

(1.7)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M26">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M27">View MathML</a>, P is defined by (1.5). As Neuman and Sándor pointed out in [5], the first and second Seiffert means are generated by the Schwab-Borchardt mean, because <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M28">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M29">View MathML</a>.

From the published literature, the first and second classes have been focused on and investigated, and there are a lot of references (see [1,13-24]). While the third class is relatively little known.

The aim of this paper is to define three families of two-parameter means constructed by trigonometric functions, which include the Schwab-Borchardt mean SB, the first and second Seiffert means P, T, and Sándor’s mean X.

The paper is organized as follows. In Section 2, some useful lemmas are given. Three families of trigonometric functions and means with two parameters and their properties are presented in Sections 3-5. In Section 6, we establish some new inequalities for two-parameter trigonometric means. In the last section, two families of two-parameter hyperbolic means are also presented in the same way without proofs.

2 Lemmas

For later use, we give the following lemmas.

Lemma 2.1 [[25], p.26]

Letfbe a differentiable function defined on an intervalI. Then the divided differences functionFdefined on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M30">View MathML</a>by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M31">View MathML</a>

(2.1)

is increasing (decreasing) in both variables if and only iffis convex (concave).

Lemma 2.2 [[26], Theorem 1]

Letfbe a differentiable function defined on an intervalI, and letFbe defined on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M30">View MathML</a>by (2.1). Then the following statements are equivalent:

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M33">View MathML</a>is convex (concave) onI,

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M34">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M35">View MathML</a>,

(iii) Fis bivariate convex (concave) on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M30">View MathML</a>.

Lemma 2.3If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M37">View MathML</a>is a differentiable even function such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M33">View MathML</a>is convex in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M39">View MathML</a>, then the function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M40">View MathML</a>defined by (2.1) increases for positivexif<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M41">View MathML</a>and decreases if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M42">View MathML</a>provided<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M43">View MathML</a>.

Proof Differentiation yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M44">View MathML</a>

Since f is an even and differentiable function, it is easy to verify that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M45">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M46">View MathML</a>. From this we only need to prove that for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M41">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M48">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M49">View MathML</a> provided <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M43">View MathML</a> if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M33">View MathML</a> is convex on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M52">View MathML</a>.

To this end, we first show two facts. Firstly, application of Lemma 2.2 leads to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M53">View MathML</a>

The second one states that if h is a continuous and odd function on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M54">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M55">View MathML</a>) and is convex on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M56">View MathML</a>, then, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M57">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M58">View MathML</a>, the inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M59">View MathML</a>

holds. Indeed, using the fact <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M60">View MathML</a> and the property of a convex function, the second one easily follows.

Now we can prove the desired result. When <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M49">View MathML</a>, application of the two previous facts and notice that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M33">View MathML</a> is odd on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M63">View MathML</a> lead to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M64">View MathML</a>

Hence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M65">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M66">View MathML</a>.

This completes the proof. □

Lemma 2.4The following inequalities are true:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M67">View MathML</a>

(2.2)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M68">View MathML</a>

(2.3)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M69">View MathML</a>

(2.4)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M70">View MathML</a>

(2.5)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M71">View MathML</a>

(2.6)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M72">View MathML</a>

(2.7)

Proof Inequalities (2.2)-(2.5) easily follow by the elementary differential method, and we omit all details here. Inequality (2.6) can be derived from a well-known inequality given in [[27], p.238]) by Adamović and Mitrinović for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M73">View MathML</a>, while it is obviously true for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M74">View MathML</a>. Inequality (2.7) can be found in [[28], Problem 5.11, 5.12]. This lemma is proved. □

Lemma 2.5 [[29], pp.227-229]

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M75">View MathML</a>. Then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M76">View MathML</a>

(2.8)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M77">View MathML</a>is the Bernoulli number.

3 Two-parameter sine means

3.1 Two-parameter sine functions

We begin with the form of hyperbolic functions of Stolarsky means defined by (1.1) to introduce the two-parameter sine functions. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M78">View MathML</a>. Then the Stolarsky means can be expressed in hyperbolic functions as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M79">View MathML</a>

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M80">View MathML</a>

(3.1)

We call <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M81">View MathML</a> two-parameter hyperbolic sine functions. Accordingly, for suitable p, q, t, we can give the definition of sine versions of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M82">View MathML</a> as follows.

Definition 3.1 The function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M83">View MathML</a> is called a sine function with parameters if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M83">View MathML</a> is defined on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M85">View MathML</a> by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M86">View MathML</a>

(3.2)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M83">View MathML</a> is said to be a two-parameter sine function for short.

Now let us observe its properties.

Proposition 3.1Let the two-parameter sine function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M83">View MathML</a>be defined by (3.2). Then

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M83">View MathML</a>is decreasing inp, qon<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M90">View MathML</a>, and is log-concave in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M91">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M92">View MathML</a>and log-convex for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M93">View MathML</a>;

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M83">View MathML</a>is decreasing and log-concave inton<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M95">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M96">View MathML</a>, and is increasing and log-convex for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M97">View MathML</a>.

Proof We have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M98">View MathML</a>

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M99">View MathML</a>

(3.3)

(i) We prove that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M83">View MathML</a> is decreasing in p, q on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M90">View MathML</a>, and is log-concave in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M102">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M103">View MathML</a> and log-convex for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M104">View MathML</a>. By Lemmas 2.1 and 2.2, it suffices to check that f is concave in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M105">View MathML</a> and that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M33">View MathML</a> is concave for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M107">View MathML</a> and convex for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M108">View MathML</a>.

Differentiation and employing (2.2), (2.6) yield that for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M110">View MathML</a>

(3.4)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M111">View MathML</a>

(3.5)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M112">View MathML</a>

(3.6)

which prove part one.

(ii) Now we show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M83">View MathML</a> is decreasing and log-concave in t on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M114">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M96">View MathML</a>, and increasing and log-convex for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M97">View MathML</a>. It is easy to verify that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M33">View MathML</a> is an odd function on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M118">View MathML</a>, and so <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M119">View MathML</a> can be written in the form of integral as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M120">View MathML</a>

(3.7)

Differentiation and application of (2.2) and (2.3) give

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M121">View MathML</a>

It follows from (3.7) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M122">View MathML</a>

which proves part two and, consequently, the proof is completed. □

From the proof of Proposition 3.1, we see that f defined by (3.3) is an even function and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M123">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M124">View MathML</a> given by (3.6). Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M125">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M126">View MathML</a>. Then by Lemma 2.3 we immediately obtain the following.

Proposition 3.2For fixed<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M127">View MathML</a>, let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M128">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M129">View MathML</a>, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M130">View MathML</a>be defined by (3.2). Then the function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M131">View MathML</a>is decreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M132">View MathML</a>and increasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M133">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M134">View MathML</a>, and is increasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M135">View MathML</a>and decreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M136">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M137">View MathML</a>.

By Propositions 3.1 and 3.2 we can obtain some new inequalities for trigonometric functions.

Corollary 3.1For<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M139">View MathML</a>

(3.8)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M140">View MathML</a>

(3.9)

Proof (i) By Proposition 3.2, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M141">View MathML</a> is increasing in p on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M142">View MathML</a> and decreasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M143">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M144">View MathML</a>

Due to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M145">View MathML</a> is decreasing in q on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M90">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M147">View MathML</a>

Simplifying leads to (3.8).

(ii) Similarly, since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M148">View MathML</a> is increasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M149">View MathML</a> and decreasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M150">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M151">View MathML</a>

while <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M152">View MathML</a> follows by the monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M153">View MathML</a> in q on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M154">View MathML</a>. Simplifying yields inequalities (3.9). □

3.2 Definition of two-parameter sine means and examples

Being equipped with Propositions 3.1, 3.2, we can easily establish a family of means generated by (3.2). To this end, we have to prove the following statement.

Theorem 3.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M155">View MathML</a>, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M156">View MathML</a>be defined by (3.2). Then, for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158">View MathML</a>defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M159">View MathML</a>

(3.10)

is a mean ofaandbif and only if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M160">View MathML</a>.

Proof Without lost of generality, we assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M161">View MathML</a>. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162">View MathML</a>. Then the statement in question is equivalent to that the inequalities

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M163">View MathML</a>

(3.11)

hold for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109">View MathML</a> if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M160">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M130">View MathML</a> is defined by (3.2).

Necessity. We prove that the condition <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M160">View MathML</a> is necessary. If (3.11) holds, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M168">View MathML</a>

Using power series extension gives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M169">View MathML</a>

Hence we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M170">View MathML</a>

which implies that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M160">View MathML</a>.

Sufficiency. We show that the condition <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M160">View MathML</a> is sufficient. Clearly, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M173">View MathML</a>. Now we distinguish two cases to prove (3.11).

Case 1: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M174">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M175">View MathML</a>. This case can be divided into two subcases. In the first subcase of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M176">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M177">View MathML</a>, by the monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M130">View MathML</a> in p, q on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M179">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M180">View MathML</a>

In the second subcase of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M181">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M175">View MathML</a>, it is derived that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M183">View MathML</a>

From Proposition 3.2 it is seen that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M184">View MathML</a> is increasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M185">View MathML</a> and decreasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M186">View MathML</a>, which reveals that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M187">View MathML</a>, that is, the desired result.

Case 2: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M190">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M191">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M175">View MathML</a>. Because of the symmetry of p and q, we assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M193">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189">View MathML</a>. Due to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M196">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M175">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M198">View MathML</a>. Using the monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M130">View MathML</a> in p, q on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M179">View MathML</a> again leads us to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M201">View MathML</a>

On the other hand, from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M202">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M203">View MathML</a>, it is acquired that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M204">View MathML</a>

which proves Case 2 and the sufficiency is complete. □

Now we can give the definition of the two-parameter sine means as follows.

Definition 3.2 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M155">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M160">View MathML</a>, and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M130">View MathML</a> be defined by (3.2). Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158">View MathML</a> defined by (3.10) is called a two-parameter sine mean of a and b.

As a family of means, the two-parameter sine means contain many known and new means.

Example 3.1 Clearly, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>, all the following

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M211">View MathML</a>

(3.12)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M212">View MathML</a>

(3.13)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M213">View MathML</a>

(3.14)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M214">View MathML</a>

(3.15)

are means of a and b, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M215">View MathML</a> is the Schwab-Borchardt mean defined by (1.3).

To generate more means involving a two-parameter sine function, we need to note a simple fact: If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M216">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M217">View MathML</a>, M are means of distinct positive numbers x and y with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M218">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M219">View MathML</a> is also a mean and satisfies inequalities

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M220">View MathML</a>

Applying the fact to Definition 3.2, we can obtain more means involving a two-parameter sine function, in which, as mentioned in Section 1, G, A and Q denote the geometric, arithmetic and quadratic means, respectively, and we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M221">View MathML</a>.

Example 3.2 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M222">View MathML</a>. Then both the following

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M223">View MathML</a>

are means of a and b, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M224">View MathML</a> is the first Seiffert mean defined by (1.5) and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M225">View MathML</a> is Sándor’s mean defined by (1.7). Also, they lie between G and A.

Example 3.3 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M226">View MathML</a>. Then both the following

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M227">View MathML</a>

(3.16)

are means of a and b, and between G and Q.

It is interesting that the new mean <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M228">View MathML</a> is somewhat similar to the second Seiffert mean <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M229">View MathML</a>.

Example 3.4 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M230">View MathML</a>. Then both the following

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M231">View MathML</a>

are means of a and b, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M232">View MathML</a> is the second Seiffert mean defined by (1.6). Moreover, they are between A and Q.

3.3 Properties of two-parameter sine means

From Propositions 3.1, 3.2 and Theorem 3.1, we easily obtain the properties of two-parameter sine means.

Property 3.1 The two-parameter sine means <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158">View MathML</a> are symmetric with respect to parameters p and q.

Property 3.2 The two-parameter sine means <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158">View MathML</a> are decreasing in p and q.

Property 3.3 The two-parameter sine means <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158">View MathML</a> are log-concave in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M102">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M237">View MathML</a>.

Property 3.4 The two-parameter sine means <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158">View MathML</a> are homogeneous and symmetric with respect to a and b.

Now we prove the monotonicity of two-parameter sine means in a and b.

Property 3.5 Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>. Then, for fixed <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M240">View MathML</a>, the two-parameter sine means <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158">View MathML</a> are increasing in a on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M242">View MathML</a>. For fixed <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M243">View MathML</a>, they are increasing in b on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M244">View MathML</a>.

Proof (i) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M246">View MathML</a>. Differentiation yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M247">View MathML</a>

which, by part two of Proposition 3.1, reveals that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M248">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158">View MathML</a> is increasing in a on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M250">View MathML</a>.

(ii) Now we prove the monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158">View MathML</a> in b. We have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M252">View MathML</a>. Differentiation leads to

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M253">View MathML</a>

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M254">View MathML</a>

(3.17)

here

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M255">View MathML</a>

is an even function on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M154">View MathML</a>. Hence, to prove <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M257">View MathML</a>, it suffices to prove that for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M105">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M259">View MathML</a>, the inequality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M260">View MathML</a> is valid for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M129">View MathML</a>. Differentiation again gives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M262">View MathML</a>

It follows by Lemmas 2.1 and 2.3 that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M263">View MathML</a> is decreasing in p and q on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M154">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M265">View MathML</a> is increasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M266">View MathML</a> and decreasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M267">View MathML</a>.

Next we distinguish two cases to prove <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M268">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M105">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M160">View MathML</a>.

Case 1: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M174">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M175">View MathML</a>. This case can be divided into two subcases. In the first subcase of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M176">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M177">View MathML</a>, by the monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M275">View MathML</a> in p, q on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M179">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M277">View MathML</a>

In the second subcase of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M181">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M175">View MathML</a>, it is derived from the monotonicities of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M280">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M281">View MathML</a> that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M282">View MathML</a>

Case 2: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M190">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M191">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M175">View MathML</a>. Because of the symmetry of p and q, we assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M193">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189">View MathML</a>. This together with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M105">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M175">View MathML</a> gives <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M293">View MathML</a>. Therefore, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M294">View MathML</a>

which proves the monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M158">View MathML</a> in b on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M296">View MathML</a> and the proof is complete. □

Remark 3.1 Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>. Then, by the monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M298">View MathML</a> in a and b, we see that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M299">View MathML</a>

which implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M300">View MathML</a>

(3.18)

Similarly, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M301">View MathML</a>

4 Two-parameter cosine means

4.1 Two-parameter cosine functions

In the same way, the Gini means defined by (1.2) can be expressed in hyperbolic functions by letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M78">View MathML</a>:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M303">View MathML</a>

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M304">View MathML</a>

(4.1)

We call <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M305">View MathML</a> two-parameter hyperbolic cosine functions. Analogously, we can define the two-parameter cosine functions as follows.

Definition 4.1 The function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M306">View MathML</a> is called a two-parameter cosine function if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M306">View MathML</a> is defined on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M308">View MathML</a> by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M309">View MathML</a>

(4.2)

Similar to the proofs of Propositions 3.1 and 3.2, we give the following assertions without proofs.

Proposition 4.1Let the two-parameter cosine function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M306">View MathML</a>be defined by (4.2). Then

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M306">View MathML</a>is decreasing inp, qon<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M312">View MathML</a>, and is log-concave in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M102">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M314">View MathML</a>and log-convex for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M315">View MathML</a>;

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M306">View MathML</a>is decreasing and log-concave inton<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M114">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M96">View MathML</a>, and is increasing and log-convex for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M97">View MathML</a>.

Proposition 4.2For fixed<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M320">View MathML</a>, let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M321">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M129">View MathML</a>, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M323">View MathML</a>be defined by (4.2). Then the function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M324">View MathML</a>is decreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M325">View MathML</a>and increasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M326">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M327">View MathML</a>, and is increasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M328">View MathML</a>and decreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M329">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M330">View MathML</a>.

Propositions 4.1 and 4.2 also contain some new inequalities for trigonometric functions, as shown in the following corollary.

Corollary 4.1For<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M332">View MathML</a>

(4.3)

Proof By Propositions 4.1 and 4.2, we see that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M333">View MathML</a> is decreasing in q on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M334">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M335">View MathML</a> is decreasing in p on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M336">View MathML</a>. It is obtained that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M337">View MathML</a>

which by some simplifications yields the desired inequalities. □

4.2 Definition of two-parameter cosine means and examples

Similarly, by Propositions 4.1, 4.2, we can easily present a family of means generated by (4.2). Of course, we need to prove the following theorem.

Theorem 4.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M338">View MathML</a>, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M339">View MathML</a>be defined by (4.2). Then, for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M341">View MathML</a>defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M342">View MathML</a>

(4.4)

is a mean ofaandbif and only if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343">View MathML</a>.

Proof We assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M161">View MathML</a> and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162">View MathML</a>. Then the desired assertion is equivalent to the inequalities

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M346">View MathML</a>

(4.5)

hold for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109">View MathML</a> if and only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M323">View MathML</a> is defined by (4.2).

Necessity. If (4.5) holds, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M350">View MathML</a>

Using power series extension gives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M351">View MathML</a>

Hence we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M352">View MathML</a>

which yields <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343">View MathML</a>.

Sufficiency. We show that the condition <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343">View MathML</a> is sufficient. Clearly, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M173">View MathML</a>. Now we distinguish two cases to prove (4.5).

Case 1: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M174">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M357">View MathML</a>. By Proposition 4.1 it is obtained that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M358">View MathML</a>

From Proposition 4.2 it is seen that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M359">View MathML</a> is increasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M360">View MathML</a> and decreasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M361">View MathML</a>, which yields <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M362">View MathML</a>, which proves Case 1.

Case 2: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M190">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M191">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M357">View MathML</a>. We assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M193">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189">View MathML</a>. Due to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M371">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M357">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M373">View MathML</a>. Using the monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M323">View MathML</a> in p, q on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M375">View MathML</a> gives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M376">View MathML</a>

At the same time, since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M202">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M203">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M379">View MathML</a>

which proves Case 2 and the sufficiency is complete. □

Thus the two-parameter cosine means can be defined as follows.

Definition 4.2 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M338">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343">View MathML</a>, and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M323">View MathML</a> be defined by (4.2). Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M341">View MathML</a> defined by (4.4) is called a two-parameter cosine mean of a and b.

The two-parameter cosine means similarly include many new means, for example, when <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M386">View MathML</a>

(4.6)

is a mean, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M215">View MathML</a> is the Schwab-Borchardt mean defined by (1.3).

Additionally, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M388">View MathML</a>. Then all the following

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M389">View MathML</a>

are means of a and b, where P, T are the first and second Seiffert mean defined by (1.5) and (1.6), U is defined by (3.16), and they lie between G and A, G and Q, A and Q, respectively.

4.3 Properties of two-parameter cosine means

From Propositions 4.1, 4.2 and Theorem 4.1, we can deduce the properties of two-parameter cosine means as follows.

Property 4.1<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M341">View MathML</a> are symmetric with respect to parameters p and q.

Property 4.2<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M341">View MathML</a> are decreasing in p and q.

Property 4.3<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M341">View MathML</a> are log-concave in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M102">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M237">View MathML</a>.

Property 4.4<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M341">View MathML</a> are homogeneous and symmetric with respect to a and b.

Property 4.5 Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M396">View MathML</a>. Then, for fixed <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M240">View MathML</a>, the two-parameter cosine means <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M341">View MathML</a> are increasing in a on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M399">View MathML</a>. For fixed <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M243">View MathML</a>, they are increasing in b on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M401">View MathML</a>.

The proof of Property 4.5 is similar to that of Property 3.5, which is left to readers.

Remark 4.1 Assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>. Then employing the monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M341">View MathML</a> in a and b, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M404">View MathML</a>

5 Two-parameter tangent means

5.1 Two-parameter tangent functions

Now we define the two-parameter tangent function and prove its properties, proofs of which are also the same as those of Propositions 3.1 and 3.2.

Definition 5.1 The function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M405">View MathML</a> is called a two-parameter tangent function if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M405">View MathML</a> is defined on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M407">View MathML</a> by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M408">View MathML</a>

(5.1)

Proposition 5.1Let the two-parameter tangent function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M405">View MathML</a>be defined by (5.1). Then

(i) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M405">View MathML</a>is increasing inp, qon<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M312">View MathML</a>, and is log-convex in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M102">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M237">View MathML</a>and log-convex for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M414">View MathML</a>;

(ii) <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M405">View MathML</a>is increasing and log-convex intfor<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M96">View MathML</a>, and is decreasing and log-concave for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M97">View MathML</a>.

Proof We have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M418">View MathML</a>

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M419">View MathML</a>

(5.2)

(i) To prove part one, by Lemmas 2.1 and 2.2 it suffices to check that g is convex on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M312">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M421">View MathML</a> is convex on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M422">View MathML</a>. In fact, differentiation and application of (2.8) yield

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M423">View MathML</a>

(5.3)

Differentiation again gives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M424">View MathML</a>

Thus part one is proved.

(ii) For proving part two, we have to check that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M425">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M426">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M427">View MathML</a>. Differentiating <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M428">View MathML</a> given in (5.3) for t, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M429">View MathML</a>

In the same method as the proof of part two in Proposition 3.1, part two in this proposition easily follows.

This completes the proof. □

The following proposition is a consequence of Lemma 2.3, the proof of which is also the same as that of Proposition 3.2 and is left to readers.

Proposition 5.2For fixed<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M320">View MathML</a>, let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M321">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M129">View MathML</a>, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M433">View MathML</a>be defined by (5.1). Then the function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M434">View MathML</a>is increasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M325">View MathML</a>and decreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M436">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M327">View MathML</a>, and is decreasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M328">View MathML</a>and increasing on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M329">View MathML</a>for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M440">View MathML</a>.

As an application of Propositions 5.1 and 5.2, we give the following corollary.

Corollary 5.1For<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M442">View MathML</a>

(5.4)

Proof Propositions 5.1 and 5.2 indicate that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M443">View MathML</a> is increasing in q on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M444">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M445">View MathML</a> is decreasing in p on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M336">View MathML</a> and increasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M447">View MathML</a>. It follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M448">View MathML</a>

which, by some simplifications, yields the required inequalities. □

5.2 Definition of two-parameter tangent means and examples

Before giving the definition of two-parameter tangent means, we firstly prove the following statement.

Theorem 5.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M338">View MathML</a>, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M450">View MathML</a>be defined by (5.1). Then, for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452">View MathML</a>defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M453">View MathML</a>

(5.5)

is a mean ofaandbif<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343">View MathML</a>.

Proof We assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M161">View MathML</a> and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452">View MathML</a> is a mean of a and b if and only if the inequalities

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M458">View MathML</a>

hold for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M460">View MathML</a> is defined by (5.1). Similarly, it can be divided into two cases.

Case 1: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M174">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M357">View MathML</a>. From the monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452">View MathML</a> in p, q on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M375">View MathML</a>, it is deduced that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M465">View MathML</a>

By Proposition 5.2 we can see that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M466">View MathML</a> is decreasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M360">View MathML</a> and increasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M468">View MathML</a>, which yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M469">View MathML</a>

that is, the desired result.

Case 2: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M190">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M191">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M357">View MathML</a>. We assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M193">View MathML</a>. Analogously, there must be <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M476">View MathML</a>. Using the monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M433">View MathML</a> in p, q on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M375">View MathML</a> gives

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M479">View MathML</a>

Noticing that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M202">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M203">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M482">View MathML</a>

which proves Case 2 and the proof is finished. □

We are now in a position to define the two-parameter tangent means by (5.1).

Definition 5.2 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M338">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343">View MathML</a>, and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M433">View MathML</a> be defined by (5.1). Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452">View MathML</a> defined by (5.5) is called a two-parameter tangent mean of a and b.

Here are some examples of two-parameter tangent means.

Example 5.1 For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>, both the following

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M489">View MathML</a>

(5.6)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M490">View MathML</a>

(5.7)

are means of a and b, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M215">View MathML</a> is the Schwab-Borchardt mean defined by (1.3).

Example 5.2 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M492">View MathML</a>. Then all the following

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M493">View MathML</a>

are means of a and b, where P, T are the first and second Seiffert mean defined by (1.5) and (1.6), U is defined by (3.16). Also, they lie between G and A, G and Q, A and Q, respectively.

5.3 Properties of two-parameter tangent means

From Propositions 5.1 and 5.2 and Theorem 5.1, we see that the properties of two-parameter tangent means are similar to those of sine ones.

Property 5.1<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452">View MathML</a> is symmetric with respect to parameters p and q.

Property 5.2<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452">View MathML</a> is increasing in p and q.

Property 5.3<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452">View MathML</a> is log-convex in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M102">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M237">View MathML</a>.

Property 5.4<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452">View MathML</a> is homogeneous and symmetric with respect to a and b.

Now we prove the monotonicity of two-parameter trigonometric means in a and b.

Property 5.5 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>. Then, for fixed <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M243">View MathML</a>, the two-parameter tangent mean <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452">View MathML</a> is increasing in b on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M296">View MathML</a>. For fixed <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M240">View MathML</a>, the two-parameter tangent mean <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M505">View MathML</a> is increasing in a on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M250">View MathML</a>.

Proof (i) Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M508">View MathML</a>. Differentiation yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M509">View MathML</a>

Application of Proposition 5.1 yields <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M510">View MathML</a>, which proves part one.

(ii) Now we prove the monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452">View MathML</a> in a. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M512">View MathML</a> can be written as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M513">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M514">View MathML</a>

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M515">View MathML</a>

(5.8)

here

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M516">View MathML</a>

is even on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M312">View MathML</a>. Thus, to prove <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M518">View MathML</a>, it suffices to prove that for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M519">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343">View MathML</a>, the inequality <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M521">View MathML</a> holds for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109">View MathML</a>.

Utilizing (2.8) and differentiating again give

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M523">View MathML</a>

By Lemmas 2.1 and 2.3 we see that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M524">View MathML</a> is increasing in p and q on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M312">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M526">View MathML</a> is decreasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M336">View MathML</a> and increasing on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M447">View MathML</a>.

Now we distinguish two cases to prove <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M521">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M519">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343">View MathML</a>.

Case 1: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M174">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M357">View MathML</a>. By the monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M534">View MathML</a> in p, q on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M375">View MathML</a> and of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M536">View MathML</a> in p on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M422">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M538">View MathML</a>

Case 2: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M190">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M191">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M357">View MathML</a>. We assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M193">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M188">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M189">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M476">View MathML</a>. Therefore, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M548">View MathML</a>

which proves the monotonicity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452">View MathML</a> in a on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M250">View MathML</a>. □

Remark 5.1 Utilizing the monotonicity property, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M551">View MathML</a>

which indicates that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M552">View MathML</a>

Additionally, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M452">View MathML</a> has a unique property which shows the relation among two-parameter sine, cosine and tangent means.

Property 5.6 For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M556">View MathML</a>. In particular, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M557">View MathML</a>.

6 Inequalities for two-parameter trigonometric means

As shown in the previous sections-, by using Propositions 3.1-5.2 we can establish a series of new inequalities for trigonometric functions and reprove some known ones. However, we are more interested in how to establish new inequalities for two-parameter trigonometric means from these ones derived by using Propositions 3.1-5.2, as obtaining an inequality for bivariate mans from the corresponding one for hyperbolic functions (see [22-24,30]). In fact, Neuman also offered some successful examples (see [31]).

The inequalities involving Schwab-Borchardt mean SB are mainly due to Neuman and Sándor (see [31-33]), and Witkowski [34] also has some contributions to them. More often, however, inequalities for means constructed by trigonometric functions seem to be related to the first and second Seiffert means, see [11,33,35-55]. In this section, we establish some new inequalities for two-parameter trigonometric means by using their monotonicity and log-convexity. Our steps are as follows.

Step 1: Obtaining an inequality (I1) for trigonometric functions sint, cost and tant by using the monotonicity and log-convexity of two-parameter trigonometric functions and simplifying them.

Step 2: For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>, letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162">View MathML</a> in inequalities (I1) obtained in Step 1 and next multiplying both sides by b or a and simplifying yield an inequality (I2) for means involving trigonometric functions.

Step 3: Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M560">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M561">View MathML</a> be two means of a and b with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M562">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157">View MathML</a>. Making a change of variables <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M564">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M565">View MathML</a> leads to another inequality (I3) for means involving trigonometric functions.

Now we illustrate these steps.

Example 6.1

Step 1: For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109">View MathML</a>, we have (3.8).

Step 2: For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>, letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162">View MathML</a> and next multiplying each side of (3.8) by b and simplifying yield

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M569">View MathML</a>

Step 3: With <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M222">View MathML</a> yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M571">View MathML</a>

With <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M226">View MathML</a> yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M573">View MathML</a>

With <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M230">View MathML</a> yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M575">View MathML</a>

Example 6.2

Step 1: For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109">View MathML</a>, from (3.9) it is derived that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M577">View MathML</a>

(6.1)

Step 2: For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>, letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162">View MathML</a> and next multiplying each side of (6.1) by b and simplifying yield

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M580">View MathML</a>

(6.2)

Step 3: With <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M222">View MathML</a> yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M582">View MathML</a>

With <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M226">View MathML</a> yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M584">View MathML</a>

With <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M230">View MathML</a> yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M586">View MathML</a>

Example 6.3

Step 1: For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109">View MathML</a>, we have (4.3).

Step 2: For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>, letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162">View MathML</a> and next multiplying each side of (4.3) by b and simplifying yield

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M590">View MathML</a>

(6.3)

Step 3: With <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M222">View MathML</a> yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M592">View MathML</a>

(6.4)

With <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M593">View MathML</a> can yield corresponding inequalities.

Remark 6.1 From inequalities (6.3) it is derived that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M594">View MathML</a>

(6.5)

hold for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M596">View MathML</a> is the logarithmic mean of positive numbers x and y. The first inequality of (6.5) follows from the relation between the second and fourth terms, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M597">View MathML</a>, while the second one is obtained by the first one in (6.3).

Example 6.4

Step 1: For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109">View MathML</a>, we have (5.4).

Step 2: For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>, letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162">View MathML</a> and next multiplying each side of (5.4) by a and simplifying give

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M601">View MathML</a>

(6.6)

Step 3: With <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M222">View MathML</a> yields

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M603">View MathML</a>

(6.7)

With <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M593">View MathML</a>, we can derive corresponding inequalities.

Remark 6.2 Applying our method in establishing inequalities for means to certain known ones involving trigonometric functions, we can obtain corresponding inequalities which are possibly related to means. For example, the Wilker inequality states that for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M109">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M606','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M606">View MathML</a>

If for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>, put <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162">View MathML</a>, then we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M609">View MathML</a>

In a similar way, the following inequalities

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M610">View MathML</a>

(6.8)

can be changed into

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M611">View MathML</a>

by letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M162">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>, where the left inequality in (6.8) is due to Neuman and Sándor [[56], (2.5)] (also see [57-59]) and the right one is known as Cusa’s inequality.

Remark 6.3 The third inequality in (6.1) is clearly superior to Cusa’s inequality (the right one of (6.8)) because

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M614">View MathML</a>

While the second one in (6.1) is weaker than the first one in (6.8) since

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M615">View MathML</a>

7 Families of two-parameter hyperbolic means

After three families of two-parameter trigonometric means have been successfully constructed, we are encouraged to establish further two-parameter means of a hyperbolic version. They are included in the following theorems.

Theorem 7.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M616">View MathML</a>, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M82">View MathML</a>be defined by (3.1). Then, for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M619">View MathML</a>defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M620">View MathML</a>

(7.1)

is a mean ofaandbif and only if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M621">View MathML</a>

Theorem 7.2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M616">View MathML</a>, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M623">View MathML</a>be defined by (4.1). Then, for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M157">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M625">View MathML</a>defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M626">View MathML</a>

(7.2)

is a mean ofaandbif and only if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M343">View MathML</a>.

To prove the above theorems, it suffices to use comparison theorems given in [15,16] by Páles because both <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M619">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M629">View MathML</a> are means if and only if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M630">View MathML</a>

respectively. Here we omit further details.

The monotonicities and log-convexities of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M619">View MathML</a> and of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M629">View MathML</a> in parameters p and q are clearly the same as those of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M82">View MathML</a> and of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M623">View MathML</a>, which are in turn equivalent to those of Stolarsky means defined by (1.1) and of Gini means defined by (1.2), respectively. These properties can be found in [13,14,17-19,21].

The above theorems indicate that for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>, all the following

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M636">View MathML</a>

are means of a and b, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M637">View MathML</a> is the Schwab-Borchardt mean defined by (1.3).

It is easy to verify that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M638">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M639">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M640">View MathML</a> are logarithmic and identric means, respectively, while <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M641">View MathML</a> is the p-order power-exponential mean. Also, all the following

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M642','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M642">View MathML</a>

are means lying in G and Q. Likewise, all the following

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M643">View MathML</a>

are also means between A and Q, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M644">View MathML</a> is the Neuman-Sándor mean defined by (1.4).

It should be noted that the new mean <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M645">View MathML</a> is similar to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M644">View MathML</a>.

Similar to (5.1), for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M616">View MathML</a>, we can define the two-parameter hyperbolic tangent function as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M648">View MathML</a>

(7.3)

By some verifications, however, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M649">View MathML</a> does not have good properties like monotonicity in parameters p and q, and therefore, we fail to define a family of two-parameter hyperbolic tangent means. However, for certain p, q and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>, it is showed that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M651">View MathML</a> is a mean of a and b, for example,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M652">View MathML</a>

is clearly a mean of a and b. It is also proved that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M653">View MathML</a>

is also a mean of a and b. For this reason, we pose an open problem as the end of this paper.

Problem 7.1 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M210">View MathML</a>, and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M649">View MathML</a> be defined by (7.3). Finding p, q such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M656','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/541/mathml/M656">View MathML</a> is a mean of a and b.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author would like to thank Ms. Jiang Yiping for her help. The author also wishes to thank the reviewer(s) who gave some important and valuable advice.

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