GENERALIZATIONS OF THE HERMITE-HADAMARD TYPE INEQUALITIES FOR HARMONICALLY UASI-CONVEX FUNCTIONS

In this paper, some new results related to the right-hand side of the Hermite-Hadamard type inequality for the class of functions whose derivatives at certain powers are harmonically quasi-convex functions are obtained.


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 have received renewed attention in recent years and a remarkable variety of refinements and generalizations for them has been found in [1,2,3,7,8,9,10,11,13,14] 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 [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: where In [15], Zhang et.al defined the harmonically quasi-convex function 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 (2) is reversed, then f is said to be harmonically quasi-concave.
In this article we consider the following special functions: Here (q) n is the Pochhammer symbol, which is defined by , that is, estimate the difference between the middle and the rightmost terms in (1), 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 [12]: Proof By the integration by parts, we have which implies that the identity (6) holds.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 quasi-convex functions in the second sense by using the above lemma.
then for all t ∈ [0, 1] the following inequality holds: Proof From Lemma 1, we have where we have used the facts that (i) Therefore, we can deduce the following results: (1) If r = 1 in (7), then the following inequality holds: (2) If r = 0 in (7), then the following inequality holds: 21 (a, b, r, q) + r where Proof From Lemma 1, we have By the harmonically quasi-convexity of |f ′ | q and using the Hölder integral inequality, we have 21 (a, b, r, q) + r Corollary 2.2.In the inequality (8) in Theorem 2.2, additionally, if then the following inequality hold: 21 (a, b, r, q) + r 22 (a, b, r, q) .Theorem 2.3.Let f : I ⊆ R + = (0, ∞) → R be a differentiable function on I 0 , the interior of an interval I, such that then for all x ∈ [a, b] the following inequality holds: Proof From Lemma 1, we have By the harmonically quasi-convexity of |f ′ | q and using the Hölder integral inequality, we have which completes the proof.
Theorem 2.4.Let f : where µ 41 (a, b, r, q Proof From Lemma 1 and the the Hölder integral inequality, we have By the harmonically quasi-convexity of |f ′ | q and using the Hölder integral inequality, we have 41 (a, b, r, q) + r where we have used the fact that , Theorem 2.5.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 quasi-convex on [a, b] for q > 1 with 1 p + 1 q = 1, then for all x ∈ [a, b] the following inequality holds: where µ 4i (i = 1, 2) are defined in Theorem 2.4.
Proof From Lemma 1 and the the Hölder integral inequality, we have By the harmonically quasi-convexity of |f ′ | q and using the Hölder integral inequality, we have Theorem 2.6.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 quasi-convex on [a, b] for q ≥ 1 with 1 p + 1 q = 1, then for all x ∈ [a, b] the following inequality holds: × sup f ′ (a) q , f ′ (b) Proof From Lemma 1, Hölder integral inequality and the harmonically quasi-convexity of |f ′ | q , we have 21 (a, b, r, p) q , f ′ (b) 22 (a, b, r, p) r r + 1 q , f ′ (b) × sup f ′ (a) q , f ′ (b) paper, we give some new Hermite-Hadamard type inequalities, which gives an upper bound for the approximation of the integral average 1 b−a b a f (u)du by the value f (a)+f (b) 2 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.Assume |f ′ | is harmonically quasi-convex on [a, b].
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 quasi-convex on [a, b] for q ≥ 1, then fot all x ∈ [a, b] the following inequality holds: rf (a) + f (b)