Journal of Applied Mathematics

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

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

## Hermite-Hadamard-Type Inequalities for *r*-Preinvex Functions

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.

#### References

- M. A. Hanson, “On sufficiency of the Kuhn-Tucker conditions,”
*Journal of Mathematical Analysis and Applications*, vol. 80, no. 2, pp. 545–550, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Noor, “Variational-like inequalities,”
*Optimization*, vol. 30, no. 4, pp. 323–330, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Noor, “Invex equilibrium problems,”
*Journal of Mathematical Analysis and Applications*, vol. 302, no. 2, pp. 463–475, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Weir and B. Mond, “Pre-invex functions in multiple objective optimization,”
*Journal of Mathematical Analysis and Applications*, vol. 136, no. 1, pp. 29–38, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Noor, “Hermite-Hadamard integral inequalities for log-preinvex functions,”
*Journal of Mathematical Analysis and Approximation Theory*, vol. 2, no. 2, pp. 126–131, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. G. Pachpatte,
*Mathematical Inequalities*, vol. 67 of*North-Holland Mathematical Library*, Elsevier, Amsterdam, The Netherlands, 2005. View at MathSciNet - C. E. M. Pearce, J. Pečarić, and V. Šimić, “Stolarsky means and Hadamard's inequality,”
*Journal of Mathematical Analysis and Applications*, vol. 220, no. 1, pp. 99–109, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. P. N. Ngoc, N. V. Vinh, and P. T. T. Hien, “Integral inequalities of Hadamard type for
*r*-convex functions,”*International Mathematical Forum. Journal for Theory and Applications*, vol. 4, no. 33-36, pp. 1723–1728, 2009. View at Google Scholar · View at MathSciNet - G.-S. Yang and D.-Y. Hwang, “Refinements of Hadamard's inequality for
*r*-convex functions,”*Indian Journal of Pure and Applied Mathematics*, vol. 32, no. 10, pp. 1571–1579, 2001. View at Google Scholar · View at MathSciNet - G. Zabandan, A. Bodaghi, and A. Kılıçman, “The Hermite-Hadamard inequality for
*r*-convex functions,”*Journal of Inequalities and Applications*, vol. 2012, article 215, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - M. Bessenyei, “Hermite-Hadamard-type inequalities for generalized 3-convex functions,”
*Publicationes Mathematicae Debrecen*, vol. 65, no. 1-2, pp. 223–232, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Antczak, “Mean value in invexity analysis,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 60, no. 8, pp. 1473–1484, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. M. Yang, X. Q. Yang, and K. L. Teo, “Generalized invexity and generalized invariant monotonicity,”
*Journal of Optimization Theory and Applications*, vol. 117, no. 3, pp. 607–625, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. R. Mohan and S. K. Neogy, “On invex sets and preinvex functions,”
*Journal of Mathematical Analysis and Applications*, vol. 189, no. 3, pp. 901–908, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Noor and K. I. Noor, “Some characterizations of strongly preinvex functions,”
*Journal of Mathematical Analysis and Applications*, vol. 316, no. 2, pp. 697–706, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Noor and K. I. Noor, “Hemiequilibrium-like problems,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 64, no. 12, pp. 2631–2642, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. G. Pachpatte, “A note on integral inequalities involving two log-convex functions,”
*Mathematical Inequalities & Applications*, vol. 7, no. 4, pp. 511–515, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Chudziak and J. Tabor, “Characterization of a condition related to a class of preinvex functions,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 74, no. 16, pp. 5572–5577, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C.-P. Liu, “Some characterizations and applications on strongly
*α*-preinvex and strongly*α*-invex functions,”*Journal of Industrial and Management Optimization*, vol. 4, no. 4, pp. 727–738, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - M. Soleimani-Damaneh, “Nonsmooth optimization using Mordukhovich's subdifferential,”
*SIAM Journal on Control and Optimization*, vol. 48, no. 5, pp. 3403–3432, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - M. Soleimani-Damaneh, “Generalized invexity in separable Hilbert spaces,”
*Topology*, vol. 48, no. 2-4, pp. 66–79, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Soleimani-Damaneh, “The gap function for optimization problems in Banach spaces,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 69, no. 2, pp. 716–723, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Soleimani-Damaneh and M. E. Sarabi, “Sufficient conditions for nonsmooth
*r*-invexity,”*Numerical Functional Analysis and Optimization*, vol. 29, no. 5-6, pp. 674–686, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - M. Soleimani-Damaneh, “Optimality for nonsmooth fractional multiple objective programming,”
*Nonlinear Analysis. Theory, Methods & Applications*, vol. 68, no. 10, pp. 2873–2878, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Yang and X. Yang, “Two new characterizations of preinvex functions,”
*Dynamics of Continuous, Discrete & Impulsive Systems B*, vol. 19, no. 3, pp. 405–410, 2012. View at Google Scholar · View at MathSciNet - T. Antczak, “
*r*-preinvexity and*r*-invexity in mathematical programming,”*Computers & Mathematics with Applications*, vol. 50, no. 3-4, pp. 551–566, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - T. Antczak, “A new method of solving nonlinear mathematical programming problems involving
*r*-invex functions,”*Journal of Mathematical Analysis and Applications*, vol. 311, no. 1, pp. 313–323, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - S. S. Dragomir and C. E. M. Pearce,
*Selected Topics on Hermite-Hadamard Type Inequalities*, RGMIA Monograph, Victoria University, Melbourne, Australia, 2000. - J. E. Pečarić, F. Proschan, and Y. L. Tong,
*Convex Functions, Partial Orderings and Statistical Applications*, vol. 187 of*Mathematics in Science and Engineering*, Academic Press, New York, NY, USA, 1992. View at MathSciNet - S. S. Dragomir and B. Mond, “Integral inequalities of Hadamard type for log-convex functions,”
*Demonstratio Mathematica*, vol. 31, no. 2, pp. 355–364, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. A. Noor, “On Hadamard integral inequalities involving two log-preinvex functions,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 8, no. 3, article 75, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet