The Scientific World Journal

Volume 2014 (2014), Article ID 279158, 7 pages

http://dx.doi.org/10.1155/2014/279158

Research Article

Some Hermite-Hadamard Type Inequalities for Harmonically s-Convex Functions

Feixiang Chen¹ and Shanhe Wu²

¹School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou, Chongqing 404000, China

²Department of Mathematics and Computer Science, Longyan University, Longyan, Fujian 364012, China

Received 3 June 2014; Accepted 17 July 2014; Published 24 July 2014

Academic Editor: Wangmeng Zuo

Copyright © 2014 Feixiang Chen and Shanhe Wu. 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 establish some estimates of the right-hand side of Hermite-Hadamard type inequalities for functions whose derivatives absolute values are harmonically s-convex. Several Hermite-Hadamard type inequalities for products of two harmonically s-convex functions are also considered.

1. Introduction

Let be a convex function and with ; then

Inequality (1) is known as the Hermite-Hadamard inequality.

In [1], Hudzik and Maligranda considered the class of functions which are -convex in the second sense. This class of functions is defined as follows.

A function is said to be -convex in the second sense if the inequality

holds for all , and for some fixed .

It can be easily seen that, for , -convexity reduces to ordinary convexity of functions defined on .

In [2], Dragomir and Fitzpatrick established a variant of Hermite-Hadamard inequality which holds for the -convex functions in the second sense.

Theorem 1 (see [2]). Suppose that is an -convex function in the second sense, where and let , . If , then the following inequalities hold:

Some generalizations, improvements, and extensions of inequalities (1) and (3) can be found in the recent papers [2–18].

In [16], İşcan investigated the Hermite-Hadamard type inequalities for harmonically convex functions.

Definition 2 (see [16]). Let be a real interval. A function is said to be harmonically convex, if

for all and . If the inequality in (4) is reversed, then is said to be harmonically concave.

Theorem 3 (see [16]). Let be a harmonically convex function and with . If , then one has

Theorem 4 (see [16]). Let be a differentiable function on ( is the interior of ), with , and ; then

In [19], İşcan investigated the Hermite-Hadamard type inequalities for harmonically -convex functions.

Definition 5 (see [19]). Let be a real interval. A function is said to be harmonically -convex, if

for all , and for some fixed . If the inequality in (7) is reversed, then is said to be harmonically -concave.

Theorem 6 (see [19]). Let be a harmonically -convex function and with . If , then one has

In [20], Pachpatte established two new Hermite-Hadamard type inequalities for products of convex functions asserted by Theorem 7.

Theorem 7 (see [20]). Let and be real-valued, nonnegative, and convex functions on . Then

where and .

For more results concerning the Hermite-Hadamard inequality, we refer the reader to [21–25] and the references cited therein.

In this paper, we establish some estimates of the right-hand side of Hermite-Hadamard type inequalities for functions whose derivatives absolute values are harmonically -convex. Moreover, we provide several Hermite-Hadamard type inequalities for products of two harmonically -convex functions.

2. Inequalities for Harmonically -Convex Functions

We recall the following special functions.

The gamma function is as follows:

the beta function is as follows:

the hypergeometric function is as follows:

Our main results are given in the following theorems.

Theorem 8. Let be a differentiable function on such that , where with . If is harmonically -convex on for some fixed , , then

where

Proof. Let . Using Theorem 4, the power mean inequality, and the harmonically -convexity of , we have

where

Calculating , , and , we find

Similarly, we get

This completes the proof of Theorem 8.

Theorem 9. Let be a differentiable function on such that , where with . If is harmonically -convex on for some fixed , , then

where .

Proof. Let . Utilizing Theorem 4, the Hölder inequality, and the harmonically -convexity of , we have

where

The proof of Theorem 9 is completed.

3. Inequalities for Products of Harmonically -Convex Functions

Theorem 10. Let , , , be functions such that . If is harmonically -convex and is harmonically -convex on for some fixed , then

where and .

Proof. Since is harmonically -convex and is harmonically -convex on , then for we get

From (24), we get

Integrating both sides of the above inequality with respect to over , we obtain

The proof of Theorem 10 is completed.

Remark 11. Taking in Theorem 10, we obtain

Remark 12. Choosing and in Theorem 10 gives

which is the right-hand side inequality of (5).

Theorem 13. Let , , , be functions such that . If is harmonically -convex and is harmonically -convex on for some fixed , then

where and .

Proof. Using the harmonically -convexity of and , we have for all

Choosing and , we have

Integrating the resulting inequality with respect to over , we get

That is,

From

we get

This completes the proof of Theorem 13.

Remark 14. Putting in Theorem 13 gives

Remark 15. If we take and in Theorem 13, then we obtain

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The present investigation was supported, in part, by the Youth Project of Chongqing Three Gorges University of China (no. 13QN11) and, in part, by the Foundation of Scientific Research Project of Fujian Province Education Department of China (no. JK2012049).

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