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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 725987, 11 pages

http://dx.doi.org/10.1155/2014/725987
Research Article

New Hermite-Hadamard Type Inequalities for -Times Differentiable and -Logarithmically Preinvex Functions

1School of Mathematical Sciences, Dalian University of Technology, Dalian 11602, China

2College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China

Received 6 March 2014; Revised 27 June 2014; Accepted 29 June 2014; Published 20 July 2014

Academic Editor: Cristina Pignotti

Copyright © 2014 Shuhong Wang and Ximin Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The concept of s-logarithmically preinvex function is introduced, and by creating an integral identity involving an n-times differentiable function, some new Hermite-Hadamard type inequalities for s-logarithmically preinvex functions are established.

1. Introduction

Throughout this paper, let , , denote the set of all positive integers, denote the interval in , and denote the set in .

Let us recall some definitions of various convex functions.

Definition 1. A function is said to be convex if holds for all and . If inequality (1) reverses, then is said to be concave on .

Definition 2 (see [1]). A set is said to be invex with respect to the map , if for every and The invex set is also called a -connected set.

It is obvious that every convex set is invex with respect to the map , but there exist invex sets which are not convex (see [1], e.g.).

Definition 3 (see [1]). Let be an invex set with respect to . For every , the -path joining the points and is defined by

Definition 4 (see [2]). Let be an invex set with respect to . A function is said to be preinvex with respect to , if for every and The function is said to be preincave if and only if is preinvex.

Every convex function is preinvex with respect to the map , but not conversely (see [2], e.g.).

Definition 5 (see [3]). Let be an invex set with respect to . The function on the set is said to be logarithmically preinvex with respect to , if for every and

For properties and applications of preinvex and logarithmically preinvex functions, please refer to [18] and closely related references therein.

The most important inequality in the theory of convex functions, the well known Hermite-Hadamard’s integral inequality, may be stated as follows. Let and with . If is a convex function on , then If is concave on , then inequality (6) is reversed.

Inequality (6) has been generalized by many mathematicians. Some of them may be recited as follows.

Theorem 6 (see [9, Theorem 2.2]). Let be a differentiable mapping on and with . If is convex on , then

Theorem 7 (see [10, Theorem 1]). Let and with . If is differentiable on such that is a convex function on for , then

Theorem 8 (see [11, Theorem 2.3]). Let be differentiable on , with and . If is convex on , then

Theorem 9 (see [6]). Let be an open invex set with respect to and with for all . If is a preinvex function on , then the following inequality holds:

Theorem 10 (see [4, Theorem 4.3]). Let be an open invex set with respect to and with for all . Suppose that is a twice differentiable function on and is preinvex on . If and is integrable on the -path     for , then

Theorem 11 (see [12, Theorem 3.1]). For and , let be an open invex set with respect to and with for all . Suppose that is an -times differentiable function on and is integrable on the -path for . If is preinvex on for , then

Theorem 12 (see [13, Theorem 5]). For and , let be an open invex set with respect to and with for all . Suppose that is a function such that exits on and is integrable on . If is logarithmically preinvex on     for , then we have the inequality where

Recently, some related inequalities for preinvex functions were also obtained in [14, 15].

In the paper, the concept of -logarithmically preinvex function is introduced, and by creating an integral identity involving an -times differentiable function, some new Hermite-Hadamard type inequalities for -logarithmically preinvex functions are established which generalize some known results.

2. New Definition and Lemmas

Now we introduce concepts of -logarithmically preinvex functions.

Definition 13. Let be an invex set with respect to . The function on the set is said to be -logarithmically preinvex with respect to , if for every , and some

Clearly, when taking in (15), then becomes the standard logarithmically convex function on .

In order to obtain our main results, we need the following lemmas.

Lemma 14. For , let be an open invex set with respect to and with   for all . If is an -times differentiable function on and is integrable on the -path   for , then where and the above summation is zero for .

Proof. Since and is an invex set with respect to , for every , we have . When , by integrating by part in the right-hand side of (16), one gives Hence, the identity (16) holds for .

When and , suppose that the identity (16) is valid.

When , by the hypothesis, we have

Therefore, when , the identity (16) holds. By induction, the proof of Lemma 14 is complete.

Remark 15. Under the conditions of Lemma 14, we have

Proof. These are special cases of Lemma 14 for ,   , , respectively.

Remark 16. Adding the identities (19) and (21) and then dividing by 2 result in Lemma 14 from [12].

Lemma 17. Let and . Then for , .

Proof. When , the proof is straightforward.

When , for , we have which coincides with the right-hand side of (22) for .

For , we get which coincides with the right-hand side of (22) for .

Suppose that (22) is true for ,   , then, for , it follows that Therefore, when , the identity (22) holds. By induction, the proof of Lemma 17 is complete.

Lemma 18 (see [16]). Let , , and . Then where .

By Lemmas 17 and 18, a straightforward computation gives the following lemmas.

Lemma 19. Let and . Then for , .

Lemma 20. Let , , and . Then where .

Lemma 21 (see [17]). Let and , . Then

3. Hermite-Hadamard Type Inequalities

Now we start out to establish some new Hermite-Hadamard type inequalities for -times differentiable and -logarithmically preinvex functions.

Theorem 22. For and , let be an open invex set with respect to and with for all . Suppose that is an -times differentiable function on and is integrable on the -path for . If   is -logarithmically preinvex on for , then for and some where , is defined in Lemma 19, and

Proof. Since and is an invex set with respect to , for every , we have . Using Lemma 14, Hölder’s inequality, and -logarithmically preinvexity of , it yields that By Lemmas 17 and 21, for , we give For , we get For , we obtain For , we have Using Lemma 19 and substituting (33) to (39) into (32), we get inequality (30).

Theorem 22 is thus proved.

Corollary 23. Under the assumptions of Theorem 22,

(1) if , then

(2) if , then

Theorem 24. For and , let be an open invex set with respect to and with for all . Suppose that is an -times differentiable function on and is integrable on the -path for . If is -logarithmically preinvex on for , then for and some where , and are given in Theorem 22 and is defined by (26).

Proof. Since and is an invex set with respect to , for every , we have . Using Lemma 14, Hölder’s inequality, and -logarithmically preinvexity of , it follows that

The rest is the same as the proof of Theorem 22.

Corollary 25. Under the assumptions of Theorem 24, if , then

Theorem 26. For and , let be an open invex set with respect to and with for all . Suppose that is an -times differentiable function on and is integrable on the -path for . If is -logarithmically preinvex on for , then for