HERMITE-HADAMARD TYPE INEQUALITIES FOR HARMONICALLY S-CONVEX FUNCTIONS

In this paper, the author give some new Hermite-Hadamard type inequality, which estimate the difference between the middle and the leftmost terms in the ordinary Hermite-Hadamard type inequality, for harmonically s- convex functions in the second sense by setting up an integral identity for differentiable functions.


Introduction
Many inequalities have been established for convex functions but the most famous is the Hermite-Hadamard's inequality, due to its rich geometrical significance and applications, which is stated as follows: Let f : I ⊆ R → R be Received: June 16, 2014 c 2015 Academic Publications, Ltd. url: www.acadpubl.eua convex function and a, b ∈ I with a < b.Then following double inequalities hold: Hermite-Hadamard's inequalities for convex, (α, m)-convex, GA-convex and geometric convex functions and have received renewed attention in recent years and a remarkable variety of refinements and generalizations have been found in [1,2,3,7,8,9, ?, 10, 12, 13] and references therein.
Let us recall some definitions of several kinds of convex functions: Definition 1.Let I be an interval in R. Then f : I → R is said to be convex on I if the inequality holds, for all x, y ∈ I and t ∈ [0, 1].Definition 2. Let I be an interval in R + = (0, ∞).A function f : I → R is said to be harmonically convex on I if the inequality holds, for all x, y ∈ I and t ∈ [0, 1].If the inequality in (2) is reversed, then f is said to be harmonically concave.
In [14], Zhang etal.defined the harmonically quasi-convex functions and supplied several properties of this kind of functions.Definition 3. Let I be an interval in R + = (0, ∞).A function f : I → R is said to be harmonically quasi-convex on I if the inequality holds, for all x, y ∈ I and t ∈ [0, 1].If the inequality in (3) is reversed, then f is said to be harmonically quasi-concave.
In [4], Imdat Işcan established the following result of the Hermite-Hadamard type for harmonically convex functions: Also, in [4,5,6], Imdat Işcan established some new Hermite-Hadamard type and Ostrowski type inequalities, which estimate the difference between the middle and the rightmost terms in (3), for harmonically convex functions: , then the following inequality holds: In [4,5,6], Imdat Işcan introduced the harmonically s-convex functions and established some new supplied Hermite-Hadamard type inequalities: Definition 4. Let I be an interval in R + = (0, ∞).A function f : I → R is said to be harmonically s-convex in the second sense on I if the inequality holds, for all x, y ∈ I, t ∈ [0, 1] and for some fixed s ∈ (0, 1].If the inequality in (4) is reversed, then f is said to be harmonically s-concave in the second sense.
In this article we consider the following special functions: Here (q) n is the Pochhammer symbol, which is defined by Definition 6.The beta function, also called the Euler integral of the first kind, is a special function defined by In this paper, we give some new Hermite-Hadamard type inequalities, which gives an upper bound for the approximation of the integral average for harmonically s-convex functions in the second sense by setting up an integral identity for differentiable functions.

Main Results
In order to find some new inequalities of Hermite-Hadamard-like type inequalities connected with the rightmost and and middle parts of (1) for functions whose derivatives are harmonically s-convex in the second sense, we need the following lemma [11]: Now we turn our attention to establish the Hermite-Hadamard type inequalities, which estimate the difference between the middle and the leftmost terms in (1), for harmonically s-convex functions in the second sense by using the above lemma.
then for all t ∈ [0, 1] the following inequality holds: where Proof.From Lemma 1, we have (b) By substituting ( 10) and ( 11) in ( 9), we get the desired result.
Therefore, we can deduce the following results: Theorem 2.2.Let f : I ⊆ R + = (0, ∞) → R be a differentiable function on I 0 , the interior of an interval I, such that f ′ ∈ L([a, b]), where a, b ∈ I with a < b.If |f ′ | q is harmonically s-convex in the second sense on [a, b] for q ≥ 1 with 1 p + 1 q = 1, then for all t ∈ [0, 1] the following inequality holds: where Proof.From Lemma 1, we have By substituting (15) and ( 16) in ( 14), we get the desired result.
Theorem 2.3.Let f : , then for all t ∈ [0, 1] the following inequality holds: where Proof.From Lemma 1, we have Since |f ′ | q is harmonically s-convex in the second sense on [a, b], we have (a) By substituting ( 18) and ( 19) in (17), we get the desired result (16).
Theorem 2.4.Let f : I ⊆ R + = (0, ∞) → R be a differentiable function on I 0 , the interior of an interval I, such that f ′ ∈ L([a, b]), where a, b ∈ I with a < b.If |f ′ | q is harmonically s-convex in the second sense on [a, b] for q > 1 with 1 p + 1 q = 1, then for all t ∈ [0, 1] the following inequality holds, where Proof.From Lemma 1, we have (24) Since |f ′ | q is harmonically s-convex in the second sense on [a, b], we have By substituting (25) in (24), we get the desired result (23).

Theorem 1 . 3 .
Let f : I ⊆ R + = (0, ∞) → R be a harmonically s-convex function in the second sense and a, b ∈ I with a < b.If f ∈ L[a, b], then following double inequalities hold: