Journal of Applied Mathematics

Volume 2013 (2013), Article ID 126457, 5 pages

http://dx.doi.org/10.1155/2013/126457

Research Article

Hermite-Hadamard-Type Inequalities for r-Preinvex Functions

Wasim Ul-Haq and Javed Iqbal

Department of Mathematics, Abdul Wali Khan University Mardan, Mardan 23200, Pakistan

Received 7 May 2013; Revised 24 August 2013; Accepted 24 August 2013

Academic Editor: Zhongxiao Jia

Copyright © 2013 Wasim Ul-Haq and Javed Iqbal. 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.

We aim to fi…nd Hermite-Hadamard inequality for r-preinvex functions. Also, it is investigated for the product of an r-preinvex function and s-preinvex function.

1. Introduction

Several interesting generalizations and extensions of classical convexity have been studied and investigated in recent years. Hanson [1] introduced the invex functions as a generalization of convex functions. Later, subsequent works inspired from Hanson's result have greatly found the role and applications of invexity in nonlinear optimization and other branches of pure and applied sciences. The basic properties and role of preinvex functions in optimization, equilibrium problems and variational inequalities were studied by Noor [2, 3] and Weir and Mond [4]. The well known Hermite-Hadamard inequality has been extensively investigated for the convex functions and their variant forms. Noor [5] and Pachpatte [6] studied these inequalities and related types for preinvex and log-preinvex functions.

The Hermite-Hadamard inequality was investigated in [7] to an -convex positive function which is defined on an interval . Certain refinements of the Hadamard inequality for -convex functions were studied in [8, 9]. Recently, Zabandan et al. [10] extended and refined the work of [8]. Bessenyei [11] studied Hermite-Hadamard-type inequalities for generalized -convex functions. In this paper, we introduce the -preinvex functions and establish Hermite-Hadamard inequality for such functions by using the method of [10].

2. Preliminaries

Let and be continuous functions, where is a nonempty closed set. We use the notations, and , for inner product and norm, respectively. We require the following well known concepts and results which are essential in our investigations.

Definition 1 (see [2, 4]). Let . Then, the set is said to be invex at with respect to if

is said to be invex set with respect to if it is invex at every . The invex set is also called a connected set.

Geometrically, Definition 1 says that there is a path starting from the point which is contained in . The point should not be one of the end points of the path in general, see [12]. This observation plays a key role in our study. If we require that should be an end point of the path for every pair of points , then , and consequently, invexity reduces to convexity. Thus, every convex set is also an invex set with respect to , but the converse is not necessarily true; see [4, 13] and the references therein. For the sake of simplicity, we assume that , unless otherwise specified.

Definition 2 (see [4]). The function on the invex set is said to be preinvex with respect to if

Note that every convex function is a preinvex function, but the converse is not true. For example, the function is not a convex function, but it is a preinvex function with respect to , where

The concepts of the invex and preinvex functions have played very important roles in the development of generalized convex programming, see [14–17]. For more characterizations and applications of invex and preinvex functions, we refer to [3, 15, 18–25].

Antczak [26, 27] introduced and studied the concept of -invex and -preinvex functions. Here, we define the following.

Definition 3. A positive function on the invex set is said to be -preinvex with respect to if, for each ,  :

Note that 0-preinvex functions are logarithmic preinvex and -preinvex functions are classical preinvex functions. It should be noted that if is -preinvex function, then is preinvex function ().

The well known Hermite-Hadamard inequality for a convex function defined on the interval is given by

see [6, 28, 29].

Hermite-Hadamard inequalities for log-convex functions were proved by Dragomir and Mond [30]. Pachpatte [6, 17] also gave some other refinements of these inequalities related to differentiable log-convex functions. Noor [5] proved the following Hermite-Hadamard inequalities for the preinvex and log-preinvex functions, respectively.

(i)Let be a preinvex function on the interval of real numbers (interior of ) and with . Then, the following inequalities hold:

(ii)Let be a log-preinvex function defined on the interval . Then,

where is the logarithmic mean of positive real numbers , .

3. Main Results

Theorem 4. Let be an -preinvex function on the interval of real numbers (interior of ) and with . Then, the following inequalities hold:

Proof. From Jensen’s inequality, we obtain

Since is preinvex, then, using Hermite-Hadamard inequality for preinvex functions (see [5]), we have

Hence,

This completes the proof.

Corollary 5. Let be a -preinvex function on the interval of real numbers (interior of ) and with . Then,

Theorem 6. Let be an -preinvex  function (with on the interval of real numbers (interior of ) and with . Then, the following inequalities hold:

Proof. For , Noor [5] proved this result. We proceed for the case . Since is -preinvex function, for all , we have

Therefore,

Putting , we have

which completes the proof.

Note that for , in Theorem 6, we have the same inequality again as in Corollary 5.

Theorem 7. Let be an -preinvex function with on the interval of real numbers (interior of ) and with . Then, is -preinvex function.

Proof. To prove this, we need the following inequality:

(see [10]) where ,   . Since is -preinvex, by inequality (17), for all ,  , we obtain

Hence, is -preinvex function.

As a special case of Theorem 7, we deduce the following result.

Corollary 8. Let be an -preinvex function with on the interval of real numbers (interior of ) and with . Then, is preinvex function.

Theorem 9. Let be an -preinvex function with on the interval of real numbers (interior of ) and with . Then, the following inequalities hold:

Proof. The left side of these inequalities is clear from Theorem 7. For the right hand side, we have from inequality (17)

Integrating both sides with respect to , we obtain

Again, using the other part of (17) and integrating, one can easily deduce the second part of the required inequalities.

Corollary 10. Let be an -preinvex function with on the interval of real numbers (interior of ) and with . Then, the following inequalities hold:

Theorem 11. Let be -preinvex and -preinvex functions, respectively, with on the interval of real numbers (interior of ) and with . Then, the following inequalities hold:

Proof. Since is -preinvex and is -preinvex, for all , we have

Therefore,

Now, applying Cauchy’s inequality, we obtain

which leads us to the required result.

Note. By putting ,  , in Theorem 11, we have the following inequality:

Theorem 11, for , was proved by Noor [31].

Theorem 12. Let be -preinvex () and -preinvex functions, respectively, on the interval of real numbers (interior of ) and with . Then, the following inequalities hold:

Proof. Since is -preinvex and is -preinvex, for all , we have

Therefore,

Now, applying Cauchy’s inequality, we obtain

which is the required result.

Acknowledgment

The authors would like to thank referees of this paper for their insightful comments which greatly improved the entire presentation of the paper. They would also like to acknowledge Prof. Dr. Ehsan Ali, VC AWKUM, for the financial support on the publication of this article.

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