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.