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

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).