SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

On the Hermite-Hadamard type inequalities

Chang-Jian Zhao1*, Wing-Sum Cheung2 and Xiao-Yan Li3

Author Affiliations

1 Department of Mathematics, China Jiliang University, Hangzhou, 310018, P.R. China

2 Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, P.R. China

3 Department of Mathematics, Hunan Normal University, Changsha, 410000, P.R. China

For all author emails, please log on.

Journal of Inequalities and Applications 2013, 2013:228  doi:10.1186/1029-242X-2013-228


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/228


Received:20 October 2012
Accepted:18 April 2013
Published:7 May 2013

© 2013 Zhao et al.; 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 the present paper, we establish some new Hermite-Hadamard type inequalities involving two functions. Our results in a special case yield recent results on Hermite-Hadamard type inequalities.

MSC: 26D15.

Keywords:
Hermite-Hadamard inequality; Barnes-Godunova-Levin inequality; Minkowski integral inequality; Hölder inequality

1 Introduction

The following inequality is well known in the literature as Hermite-Hadamard’s inequality [1].

Theorem 1.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M1">View MathML</a>be a convex function on an interval of real numbers. Then the following Hermite-Hadamard inequality for convex functions holds:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M2">View MathML</a>

(1.1)

If the functionfis concave, the inequality (1.1) can be written as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M3">View MathML</a>

(1.2)

Recently, many generalizations, extensions and variants of this inequality have appeared in the literature (see, e.g., [2-10]) and the references given therein. In particular, in 2010, Özdemir and Dragomir [11] established some new Hermite-Hadamard inequalities and other integral inequalities involving two functions in ℝ. Following this work, the main purpose of the present paper is to establish some dual Hermite-Hadamard type inequalities involving two functions in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M4">View MathML</a>. Our results provide some new estimates on such type of inequalities.

2 Preliminaries

A region <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M5">View MathML</a> is called convex if it contains the close line segment joining any two of its points, or equivalently, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M6">View MathML</a> whenever <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M7">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M8">View MathML</a>.

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M9">View MathML</a> be a duality function on the convex region <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M5">View MathML</a>. <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M9">View MathML</a> is called a duality convex function on the convex region D if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M12">View MathML</a>

(2.1)

whenever <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M13">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M8">View MathML</a>.

If the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15">View MathML</a> is concave, the inequality (2.1) can be written as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M16">View MathML</a>

(2.2)

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M17">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M18">View MathML</a> be two positive nm-tuples, and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M19">View MathML</a>. Then, on putting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M20">View MathML</a>, it easy follows that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M21">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M22">View MathML</a>

(2.3)

(also see, e.g., [[1], p.15]). Here, the rth power mean of x with weights p is the following: <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M23">View MathML</a> if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M24">View MathML</a>; <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M25">View MathML</a> if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M26">View MathML</a>; <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M27">View MathML</a> if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M28">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M29">View MathML</a> if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M30">View MathML</a>.

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M31">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M32">View MathML</a>. Now, we define the p-norm of the function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15">View MathML</a> on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M34">View MathML</a> as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M35">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M36">View MathML</a>

and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M37">View MathML</a> is the set of all functions <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M31">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M39">View MathML</a>.

Lemma 2.1 (see [12]) (Barnes-Godunova-Levin inequality)

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M41">View MathML</a>be nonnegative concave functions on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M34">View MathML</a>, then for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M43">View MathML</a>we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M44">View MathML</a>

(2.4)

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M45">View MathML</a>

Lemma 2.2 (see [1]) (Hermite-Hadamard inequality)

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M46">View MathML</a>be a convex function. Then the following dual Hermite-Hadamard inequality for convex functions holds:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M47">View MathML</a>

(2.5)

The inequality is reversed if the function<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15">View MathML</a>is concave.

Lemma 2.3 (see [13]) (A reversed Minkowski integral inequality)

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M41">View MathML</a>be positive functions satisfying

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M51">View MathML</a>

(2.6)

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M52">View MathML</a>

(2.7)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M53">View MathML</a>.

3 Main results

Our main results are established in the following theorems.

Theorem 3.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M54">View MathML</a>and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M55">View MathML</a>be nonnegative functions such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M56">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M57">View MathML</a>are concave on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M34">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M59">View MathML</a>

(3.1)

where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M60">View MathML</a>is the Barnes-Godunova-Levin constant given by (2.4).

Proof Observe that whenever <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M61">View MathML</a> is concave on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M34">View MathML</a>, the nonnegative function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15">View MathML</a> is also concave on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M34">View MathML</a>. Namely,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M65">View MathML</a>

that is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M66">View MathML</a>

and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M67">View MathML</a>, using the power-mean inequality (2.3), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M68">View MathML</a>

For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M69">View MathML</a>, similarly, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M70">View MathML</a> is concave on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M71">View MathML</a>, the nonnegative function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M41">View MathML</a> is concave on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M73">View MathML</a>.

In view that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M61">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M70">View MathML</a> are concave functions on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M34">View MathML</a>, from Lemma 2.2, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M77">View MathML</a>

(3.2)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M78">View MathML</a>

(3.3)

By multiplying the above inequalities, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M79">View MathML</a>

(3.4)

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M54">View MathML</a>, then it is easy to show that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M81">View MathML</a>

(3.5)

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M82">View MathML</a>

(3.6)

Thus, by applying Barnes-Godunova-Levin inequality to the right-hand side of (3.4) with (3.5), (3.6), we get (3.1).

The proof is complete. □

Remark 3.1 By multiplying inequalities (3.2), (3.3), we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M83">View MathML</a>

(3.7)

By applying the Hölder inequality to the left-hand side of (3.7) with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M84">View MathML</a>, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M85">View MathML</a>

(3.8)

Remark 3.2 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M41">View MathML</a> change to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M88">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M89">View MathML</a>, respectively, and with suitable changes in Theorem 3.1 and Remark 3.1, we have the following.

Corollary 3.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M54">View MathML</a>and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M91">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M92">View MathML</a>, be nonnegative functions such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M93">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M94">View MathML</a>are concave on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M95">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M96">View MathML</a>

and if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M84">View MathML</a>, then one has

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M98">View MathML</a>

This is just Theorem 2.1 established by Özdemir and Dragomir [11].

Theorem 3.2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M32">View MathML</a>and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M100">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M101">View MathML</a>, and let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M55">View MathML</a>be positive functions with

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M103">View MathML</a>

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M104">View MathML</a>

(3.9)

Proof Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M41">View MathML</a> are positive, as in the proof of Lemma 2.3 (see [[13], p.2]), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M107">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M108">View MathML</a>

By multiplying the above inequalities and in view of the Minkowski inequality, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M109">View MathML</a>

(3.10)

Hence

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M110">View MathML</a>

This proof is complete. □

Remark 3.3 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M41">View MathML</a> change to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M88">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M89">View MathML</a>, respectively, and with suitable changes in (3.9), (3.9) reduces to an inequality established by Özdemir and Dragomir [11].

Theorem 3.3If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M61">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M70">View MathML</a>are as in Theorem 3.1, then the following inequality holds:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M117">View MathML</a>

(3.11)

Proof If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M61">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M70">View MathML</a> are concave on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M71">View MathML</a>, then from Lemma 2.2, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M121">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M122">View MathML</a>

which imply that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M123">View MathML</a>

(3.12)

On the other hand, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M124">View MathML</a>, from (2.3) we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M125">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M126">View MathML</a>

which imply that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M127">View MathML</a>

(3.13)

Combining (3.12) and (3.13), we obtain the desired inequality as

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M128">View MathML</a>

This proof is complete. □

Remark 3.4 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M41">View MathML</a> change to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M88">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M89">View MathML</a>, respectively, and with suitable changes in (3.11), (3.11) reduces to an inequality established by Özdemir and Dragomir [11].

Theorem 3.4Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M133">View MathML</a>be functions such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M56">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M57">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M136">View MathML</a>are in<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M137">View MathML</a>, and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M138">View MathML</a>

Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M139">View MathML</a>

(3.14)

where

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M140">View MathML</a>

and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M84">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M67">View MathML</a>.

Proof Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M143">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M144">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M145">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M146">View MathML</a>

In view of the Young-type inequality and using the elementary inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M147">View MathML</a>

we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M148">View MathML</a>

This completes the proof. □

Remark 3.5 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M15">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M41">View MathML</a> change to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M88">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M89">View MathML</a>, respectively, and with suitable changes in (3.14), (3.14) reduces to an inequality established by Özdemir and Dragomir [11].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

C-JZ, W-SC and X-YL jointly contributed to the main results Theorems 3.1-3.4. All authors read and approved the final manuscript.

Acknowledgements

The first author’s research is supported by Natural Science Foundation of China (10971205). The second author’s research is partially supported by a HKU Seed Grant for Basic Research.

References

  1. Mitrinović, DS, Pecarič, JE, Fink, AM: Classical and New Inequalities in Analysis, Kluwer Academic, Dordrecht (1993)

  2. Kavurmaci, H, Avci, M, Özdemir, ME: New inequalities of Hermite-Hadamard type for convex functions with applications. J. Inequal. Appl.. 2011, Article ID 86 (2011)

  3. Özdemir, ME, Kavurmaci, H, Ocak Akdemir, A, Avci, M: Inequalities for convex and s-convex functions on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M153">View MathML</a>. J. Inequal. Appl.. 2012, Article ID 20 (2012)

  4. Dragomir, SS: A sequence of mappings associated with the Hermite-Hadamard inequalities and applications. Appl. Math.. 49, 123–140 (2004)

  5. Kotrys, D: Hermite-Hadamard inequality for convex stochastic processes. Aequ. Math.. 83, 143–151 (2012). Publisher Full Text OpenURL

  6. Makó, J, Páles, Z: Hermite-Hadamard inequalities for generalized convex functions. Aequ. Math.. 69, 32–40 (2005). Publisher Full Text OpenURL

  7. Ger, R, Pečarić, J: On vector Hermite-Hadamard differences controlled by their scalar counterparts. Int. Ser. Numer. Math.. 161, 165–173 (2010)

  8. Xi, B-Y, Bai, R-F, Qi, F: Hermite-Hadamard type inequalities for the m- and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/228/mathml/M155">View MathML</a>-geometrically convex functions. Filomat. 27, 1–7 (2013)

  9. Dragomir, SS, Hunt, E, Pearce, CEM: Interpolating maps, the modulus map and Hadamard’s inequality. Optimization, Part 1. 207–223 (2009). PubMed Abstract | Publisher Full Text OpenURL

  10. Niculescu, CP: The Hermite-Hadamard inequality for convex functions of a vector variable. Math. Inequal. Appl.. 5(4), 619–623 (2002)

  11. Set, E, Özdemir, ME, Dragomir, SS: On the Hermite-Hadamard inequality and other integral inequalities involving two functions. J. Inequal. Appl.. 2010, Article ID 148102 (2010)

  12. Pachpatte, BG: Inequalities for Differentiable and Integral Equations, Academic Press, Boston (1997)

  13. Bougoffa, L: On Minkowski and Hardy integral inequalities. J. Inequal. Pure Appl. Math.. 7(2), Article ID 60 (2006)