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Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex

Ali Barani1*, Amir G Ghazanfari1 and Sever S Dragomir23

Author Affiliations

1 Department of Mathematics, Lorestan University, P.O. Box 465, Khoramabad, Iran

2 School of Engineering and Science, Victoria University, P.O. Box 14428, Melbourne City, VIC, 8001, Australia

3 School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Johannesburg, Wits, 2050, South Africa

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Journal of Inequalities and Applications 2012, 2012:247  doi:10.1186/1029-242X-2012-247

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2012/1/247


Received:6 August 2011
Accepted:28 May 2012
Published:29 October 2012

© 2012 Barani 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 this article, we extend some estimates of the right-hand side of a Hermite-Hadamard-type inequality for preinvex functions. Then, a generalization to functions of several variables on invex subsets of is introduced.

Keywords:
Hermite-Hadamard inequality; invex sets; preinvex functions

1 Introduction and preliminary

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M2">View MathML</a> be an interval on the real line ℝ, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M3">View MathML</a> be a convex function and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M4">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M5">View MathML</a>. We consider the well-known Hadamard’s inequality

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M6">View MathML</a>

(1)

Both inequalities hold in the reversed direction if f is concave. We note that Hadamard’s inequality may be regarded as a refinement of the concept of convexity and it follows easily from Jensen’s inequality. Hadamard’s inequality for convex functions has received renewed attention in recent years and a remarkable variety of refinements and generalizations have been found (see, for example, [1-4]).

The classical Hermite-Hadamard inequality provides estimates of the mean value of a continuous convex function <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M7">View MathML</a>.

Dragomir and Agarwal [5] used the formula,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M8">View MathML</a>

(2)

to prove the following results.

Theorem 1.1Assume<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M9">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M10">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M7">View MathML</a>is a differentiable function on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M12">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M13">View MathML</a>is convex on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M14">View MathML</a>then the following inequality holds true

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M15">View MathML</a>

(3)

Theorem 1.2Assume<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M9">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M10">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M7">View MathML</a>is a differentiable function on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M12">View MathML</a>. Assume<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M20">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M21">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M22">View MathML</a>is convex on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M14">View MathML</a>then the following inequality holds true

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M24">View MathML</a>

(4)

Ion [6] presented some estimates of the right-hand side of a Hermite-Hadamard-type inequality in which some quasi-convex functions are involved.

In recent years, several extensions and generalizations have been considered for classical convexity. A significant generalization of convex functions is that of invex functions introduced by Hanson [7]. Weir and Mond [8] introduced the concept of preinvex functions and applied it to the establishment of the sufficient optimality conditions and duality in nonlinear programming. Aslam Noor [9,10] introduced the Hermite-Hadamard inequality for preinvex and log-preinvex functions.

In this article, we generalize the results in [6] for functions whose first derivatives absolute values are preinvex. Also some results for functions whose second derivatives absolute values are preinvex will be given. Now, we recall some notions in invexity analysis which will be used throughout the article (see [11,12] and references therein).

Definition 1.1 A set is said to be invex with respect to the map , if for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M28">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M29">View MathML</a>

(5)

It is obvious that every convex set is invex with respect to the map <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M30">View MathML</a>, but there exist invex sets which are not convex (see [11]). Let be an invex set with respect to . For every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27">View MathML</a> the η-path <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M34">View MathML</a> joining the points x and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M35">View MathML</a> is defined as follows

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M36">View MathML</a>

Definition 1.2 Let be an invex set with respect to . Then, the function is said to be preinvex with respect to η, if for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M28">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M42">View MathML</a>

(6)

Every convex function is a preinvex with respect to the map <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M30">View MathML</a> but the converse does not holds. For properties and applications of preinvex functions, see [12,13] and references therein.

The organization of the article is as follows: In Section 2, some generalizations of Hermite-Hadamard-type inequality for first-order differentiable functions are given. Section 3 is devoted to a generalization to several variable preinvex functions. Hermite-Hadamard-type inequality for second-order differentiable functions are studied in Section 23.

2 First-order differentiable functions

In this section, we introduce some generalizations of Hermite-Hadamard-type inequality for functions whose first derivatives absolute values are preinvex. We begin with the following lemma which is a generalization of Lemma 2.1 in [5] to invex setting.

Lemma 2.1Letbe an open invex subset with respect toand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M46">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M47">View MathML</a>. Suppose thatis a differentiable function. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M49">View MathML</a>is integrable on theθ-path<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M50">View MathML</a>then, the following equality holds

(7)

Proof Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M52">View MathML</a>. Since A is an invex set with respect to θ, for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M28">View MathML</a> we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M54">View MathML</a>. Integrating by parts implies that

(8)

which completes the proof. □

Theorem 2.1Letbe an open invex subset with respect to. Suppose thatis a differentiable function. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M59">View MathML</a>is preinvex onAthen, for every<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M52">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M47">View MathML</a>the following inequality holds

(9)

Proof Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M52">View MathML</a>. Since A is an invex set with respect to θ, for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M28">View MathML</a> we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M54">View MathML</a>. By preinvexity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M66">View MathML</a> and Lemma 2.1 we get

(10)

where,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M68">View MathML</a>

 □

Now, we give an example of an invex set with respect to an θ which is satisfies the conditions of Theorem 2.1.

Example 2.1 Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M69">View MathML</a> and the function is defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M71">View MathML</a>

Clearly K is an open invex set with respect to θ. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M72">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M73">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M74">View MathML</a> hence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M75">View MathML</a>. Now,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M76">View MathML</a>

and

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M77">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M78">View MathML</a>.

Another similar result is embodied in the following theorem.

Theorem 2.2Letbe an open invex subset with respect to. Suppose thatis a differentiable function. Assume thatwith<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M21">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M84">View MathML</a>is preinvex onAthen, for every<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M46">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M86">View MathML</a>the following inequality holds

(11)

Proof Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M52">View MathML</a>. By assumption, Hölder’s inequality and the proof of Theorem 4.1 we have

(12)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M90">View MathML</a>. □

Note that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M91">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M92">View MathML</a> for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M93">View MathML</a> then, we can deduce Theorems 1.1 and 1.2, from Theorems 2.1 and 2.2, respectively.

3 An extension to several variables functions

The aim of this section is to extend the Proposition 1 in [6] and Theorem 2.2 to functions of several variables defined on invex subsets of .

The mapping is said to be satisfies the condition C if for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M97">View MathML</a>,

Note that, in Example 2.1, θ satisfies the condition C.

For every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27">View MathML</a> and every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M100">View MathML</a> from condition C we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M101">View MathML</a>

(13)

see [12] for details.

Proposition 3.1Letbe an invex set with respect toandis a function. Suppose thatηsatisfies conditionConS. Then, for every<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27">View MathML</a>the functionfis preinvex with respect toηonη-path<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M34">View MathML</a>if and only if the functiondefined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M108">View MathML</a>

is convex on<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M109">View MathML</a>.

Proof Suppose that φ is convex on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M109">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M111">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M112">View MathML</a>. Fix <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M113">View MathML</a>. Since η satisfies condition C, by (13) we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M114">View MathML</a>

(14)

Hence, f is preinvex with respect to η on η-path <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M34">View MathML</a>.

Conversely, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27">View MathML</a> and the function f be preinvex with respect to η on η-path <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M34">View MathML</a>. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M118">View MathML</a>. Then, for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M113">View MathML</a> we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M120">View MathML</a>

(15)

Therefore, φ is quasi-convex on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M109">View MathML</a>. □

The following theorem is a generalization of Proposition 1 in [6].

Theorem 3.1Letbe an open invex set with respect to. Assume thatηsatisfies conditionC. Suppose that for every<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27">View MathML</a>the functionis preinvex with respect toηonη-path<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M34">View MathML</a>. Then, for every<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M127">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M5">View MathML</a>the following inequality holds,

(16)

Proof Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M27">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M127">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M5">View MathML</a>. Since f is preinvex with respect to η on η-path <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M34">View MathML</a> by Proposition 3.1 the function defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M135">View MathML</a>

is convex on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M109">View MathML</a>. Now, we define the function as follows

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M138">View MathML</a>

Obviously for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M139">View MathML</a> we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M140">View MathML</a>

hence, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M141">View MathML</a> . Applying Theorem 1.1 to the function ϕ implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M142">View MathML</a>

and we deduce that (16) holds. □

4 Second-order differentiable functions

In this section, we introduce some generalizations of Hermite-Hadamard-type inequality for functions whose second derivatives absolute values are preinvex. We begin with the following lemma (see Lemma 1 in [14] and Lemma 4 in [15]).

Lemma 4.1Letbe an open invex subset with respect toand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M46">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M47">View MathML</a>. Suppose thatis a differentiable function. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M148">View MathML</a>is integrable on theθ-path<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M149">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M78">View MathML</a>then, the following equality holds

(17)

Proof Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M52">View MathML</a>. Since A is an invex set with respect to θ, for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M28">View MathML</a> we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M54">View MathML</a>. Integrating by parts implies that

(18)

which completes the proof. □

Theorem 4.1Letbe an open invex subset with respect toand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M47">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M159">View MathML</a>. Suppose thatis a twice differentiable function onA. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M161">View MathML</a>is preinvex onAand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M162">View MathML</a>is integrable on theθ-path<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M149">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M78">View MathML</a>then, the following inequality holds

(19)

Proof Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M52">View MathML</a>. Since A is an invex set with respect to θ, for every <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M28">View MathML</a> we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M54">View MathML</a>. By preinvexity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M169">View MathML</a> and Lemma 4.1 we get

(20)

which completes the proof. □

The corresponding version for powers of the absolute value of the second derivative is incorporated in the following theorem.

Theorem 4.2Letbe an open invex subset with respect toand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M47">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M159">View MathML</a>. Suppose thatis a twice differentiable function onAand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M176">View MathML</a>is preinvex onA, for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M177">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M162">View MathML</a>is integrable on theθ-path<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M149">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M78">View MathML</a>then, the following inequality holds

(21)

Proof By preinvexity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M169">View MathML</a>, Lemma 4.1 and using the well-known Hölder integral inequality, we get

which completes the proof. □

A more general inequality is given using Lemma 4.1, as follows:

Theorem 4.3Letbe an open invex subset with respect toand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M47">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M159">View MathML</a>. Suppose thatis a twice differentiable function onAand<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M189">View MathML</a>is preinvex onA, for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M190">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M162">View MathML</a>is integrable on theθ-path<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M149">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M78">View MathML</a>then, the following inequality holds

(22)

Proof By preinvexity of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2012/1/247/mathml/M169">View MathML</a>, Lemma 4.1 and using the well-known weighted power mean inequality, we get

which completes the proof. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors conceived of the study, and participated in the design and coordination. All authors read and approved the final manuscript.

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