Some Hermite-Hadamard-Fejer type inequalities for Harmonically convex functions via Fractional Integral

In this paper, it is a fuction that is a harmonically-convex differentiable for a new identity. As a result of this identity some new and general inequalities for differentiable harmonically-convex functions are obtained.


Introduction
The classical or the usual convexity is defined as follows, holds for all x, y ∈ I and t ∈ [0, 1].
A number of papers have been written on inequalities using the classical convexty and one of the most captivating inequalities in mathematical analysis is stated as follows, where f : I ⊆ R −→ be a convex mapping and a, b ∈ I with a ≤ b . Both the inequalities hold in reversed direction if f is concave. The inequalities stated in (1) are known as Hermite-Hadamard inequalities.
The usual notion of convex function have been generalized in diverse manners. One of them is the so called harmonically s-convex functions and is stated in the definition below.
It can be easily seen that for s = 1 in Defination 2 reduces to following Defination 3, holds for all x, y ∈ I and t ∈ [0, 1] . If the inequality is reversed, then f is said to be harmonically concave. For the properties of harmonically-convex functions and harmonically-s-convex function, we refer the reader to [1,5,6,7,8,10,11] and the reference there in.
Most recently, a number of findings have been seen on Hermite-Hadamard type integral inequalities for harmonically-convex and for harmonically-s-convex functions.
In [6],İşcan gave defination of harmonically convex functions and established following Hermite-Hadamard type inequality for harmonically convex functions as follows.
[15] Let f : I ⊂ R\{0}→ R be a harmonically convex function and a, b ∈ I with a < b . If f ∈ L [a, b] then the following inequalities hold: In [11], Iscan with α > 0 and h(x) = 1/x.
holds for all x ∈ [a, b].
Theorem 4. In [1] Chan and Wu represented Hermite-Hadamard-Fejer inequality for harmonically convex functions as follows: In [10]İşcan and Kunt represented Hermite-Hadamard-Fejer type inequality for harmonically convex functions in fractional integral forms and established following identity as follows: nonnegative, integrable and harmonically symmetric with respect to 2ab/a + b, then the following inequalities for fractional integrals hold:

Definition 5. Let f ∈ L[a, b]. The right-hand side and left-hand side Hadamard fractional integrals J
In [4] D. Y. Hwang found out a new identity and by using this identity, established a new inequalities. Then in [12]İ. Işcan and S. Turhan used this identity for GA-convex functions and obtain generalized new inequalities. In this paper, we established a new inequality similar to inequality in [12] and then we obtained some new and general integral inequalities for differentiable harmonically-convex functions using this lemma. The following sections, let the notion, ]then the following inequality holds: Proof. By the integration by parts, we have Therefore where ζ 1 (a, b) , ζ 2 (a, b) , ζ 3 (a, b) are defined in Lemma 3.
by (15) and Lemma 2, this proof is complete.
continuous positive mapping and symmetric to 2ab a+b in Teorem 7, we obtain: Specially in (16) and using Lemma 1, for 0 < α ≤ 1 we have: Proof. By left side of inequality (15) in Teorem 7, when we write h(t) = On the other hand, right side of inequality (15), with Since g(x) is symmetric to x = 2ab a+b , we have and for all t ∈ [0, 1]. By (18)-(20), we have In the last inequality, By Lemma 1, we have A combination of (21) and (22), we have (16). This complete is proof.

Corollary 2. In Corollary 1,
(i) If α = 1 is in corollary, we obtain following Hermite-Hadamard-Fejer Type inequality for harmonically-convex function which is related the left-hand side of (17): where for a, b, H > 0, we have (16):

is in corollary, we obtain following Hermite-Hadamard-Fejer Type inequality for harmonically-convex function which is related the left-hand side of
] .
(iii) If g(x) = 1 and α = 1 is in corollary, we obtain following Hermite-Hadamard-Fejer Type inequality for harmonically-convex function which is related the left-hand side of (17): ] . (25) , then the following inequality holds: Proof. Continuing from (14) in Theorem 7, we use Hölder Inequality and we use that | f ′ | q is harmonically-convex. Thus this proof is complete.