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

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.


Introduction
Let : ⊆ R → R 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.
In [2], Dragomir and Fitzpatrick established a variant of Hermite-Hadamard inequality which holds for the -convex functions in the second sense.
Definition 2 (see [16]). Let ⊆ R \ {0} be a real interval. A function : → R is said to be harmonically convex, if for all , ∈ and ∈ [0, 1]. If the inequality in (4) is reversed, then is said to be harmonically concave.
In this paper, we establish some estimates of the righthand side of Hermite-Hadamard type inequalities for functions whose derivatives absolute values are harmonicallyconvex. Moreover, we provide several Hermite-Hadamard type inequalities for products of two harmonically -convex functions.

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.

Inequalities for Products of Harmonically -Convex Functions
The Scientific World Journal 5 From (24), we get ( + (1 − ) ) ( + (1 − ) ) Integrating both sides of the above inequality with respect to over [0, 1], we obtain The proof of Theorem 10 is completed.
which is the right-hand side inequality of (5).
Proof. Using the harmonically -convexity of and , we have for all , ∈ [ , ] Choosing = /( + (1 − ) ) and = /( + (1 − ) ), we have The Scientific World Journal Integrating the resulting inequality with respect to over [0, 1], we get That is, we get This completes the proof of Theorem 13.