Journal of Applied Mathematics

Volume 2013 (2013), Article ID 978493, 5 pages

http://dx.doi.org/10.1155/2013/978493

Research Article

Refinements on the Hermite-Hadamard Inequalities for -Convex Functions

Feixiang Chen and Xuefei Liu

School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, Chongqing 404000, China

Received 10 April 2013; Accepted 24 August 2013

Academic Editor: Kazutake Komori

Copyright © 2013 Feixiang Chen and Xuefei 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.

We give some new generalizations of the well-known Hermite-Hadamard inequality for -convex functions.

1. Introduction

Let be a convex function on the interval ; then for any , with , we have the following double inequality:

This remarkable result is well known in the literature as the Hermite-Hadamard inequality. Note that some of the classical inequalities for means can be derived from (1) for appropriate particular selections of the mapping . Both inequalities hold in the reversed direction if is concave. Some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors (e.g., [1–9]). The Hermite-Hadamard inequality was generalized in [10] to an -convex positive function which is defined on an interval []. A positive function is called -convex on [], if for each , and

It is obvious that 0-convex functions are simply log-convex functions and 1-convex functions are ordinary convex functions. If is a positive -concave function, then inequality (2) is reversed. We note that if and are convex and is increasing, then is convex; moreover, since , it follows that a log-convex function is convex. This follows directly from (2) because, by the arithmetic-geometric mean inequality, we have

In [6], Gill et al. proved the following two theorems.

Theorem 1. Suppose that is a positive -convex function on . Then

If is a positive -concave function, then the inequality is reversed.

Theorem 2. Suppose that is a positive log-convex function on . Then

If is a positive log-concave function, then the inequality is reversed.

In [11], Sulaiman obtained the following result for -convex functions.

Theorem 3. Let be a positive -convex function on , . Set

Then, the following inequality holds:

In [12], Sulaiman obtained the following result for log-convex functions.

Theorem 4. Assume that is an increasing log-convex function. Then for all , one has

where

For recent results and generalizations concerning the Hermite-Hadamard inequality, see [7, 8] and the references given therein.

In this note, we establish some generalizations of the above results for the class of -convex functions.

2. Lemmas

Lemma 5. If , , then

Remark 6. Applying Lemma 5 and (2), for an -convex , let ; then is log-convex. Now we let in Theorem 1; then we get Theorem 2 from

Lemma 7. If , then

Lemma 8. If and , then

Lemma 9. If , , then the following inequality holds:

Lemma 10 (see [11, Theorem  3.1]). Let be -convex on and ; then

3. Main Results

Theorem 11. Let be -convex and non-decreasing on and ; for , , , and arbitrary , the following inequality holds:

where

Proof. Observing that , and Jensen’s inequality for , we have

where the second inequality follows from replacing and by and , respectively, and (1) for . Therefore

where the third inequality follows from replacing and by and , respectively, and (4). The proof is completed.

Remark 12. Applying Theorem 11 for , we get Theorem 3.

Corollary 13. With the above notations, if is -convex and nondecreasing on and , one has the following inequality:

where and are defined in Theorem 11.

Theorem 14. Let be log-convex and nondecreasing on ; for , , , and arbitrary , the following inequality holds:

where

Proof. Let be a positive -convex function on , , by Theorem 3: then

Applying Lemma 5, let ; then is log-convex and

Observing that ,

where the first inequality follows from replacing and by and , respectively, and (5). The proof is completed.

Remark 15. Applying Theorem 14 for and , = , we get Theorem 4.

Corollary 16. With the above notations, suppose that is log-convex and nondecreasing on ; one has the following inequality:

where and are defined in Theorem 14.

Acknowledgments

This work is supported by the Natural Science Foundation Project of “CQ CSTC” (no. cstc2012jjA00013) and Youth Project of Chongqing Three Gorges University of China.

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