HADAMARD AND FEJÉR-HADAMARD TYPE INTEGRAL INEQUALITIES FOR HARMONICALLY CONVEX FUNCTIONS VIA AN EXTENDED GENERALIZED MITTAG-LEFFLER FUNCTION

In this paper generalized form of some new inequalities of the Hadamard and the Fejér-Hadamard type have been established. Fractional integral operators due to an extended generalized Mittag-Leffler function via harmonically convex functions are utilized to obtain the new results.


Introduction
Convex functions are equivalently defined by the Hadamard inequality stated in the following theorem.
Theorem 1.1. Let I be an interval of real numbers and f : I → R be a convex function on I.
Then for a, b ∈ I, a < b the following inequality holds.
It is always in the focus of researchers especially working in the field of mathematical analysis. Now a days it is under consideration via a variety of fractional integral operators and fractional differential operators (see for example [1,2,4,6,7,11] and references there in). In this paper, we are interested to find the Hadamard and the Fejér-Hadamard and related fractional inequalities for harmonically convex functions via fractional integral operators due to an extended generalized Mittag-Leffler function. First we give the definition of harmonically convex function as follows. (1) holds for all a, b ∈ I and t ∈ [0, 1]. If inequality in (1) is reversed, then f is said to be harmonically concave function for more detail one can see [6].
Then the extended generalized Mittag-Leffler function E γ,δ ,k,c µ,ν,l (t; p) is defined by where β p is the generalized beta function defined by and (c) nk is the Pochhammer symbol defined as (c) nk = Γ(c+nk) Γ(c) . Remark 1.1. (2) is a generalization of the following functions.
A lot of authors of this age are working on inequalities involving fractional integral operators for example for Riemann-Liouville, Caputo, Hillfer, Canvati etc [3,5,11].
Kunt et al. in [7], produced the following result for harmonically convex functions.
, then the following inequalities for fractional integrals hold Chen and Wu in [2] presented the following Fejér-Hadamard inequality for harmonically convex functions.
harmonically symmetric about a+b 2ab , then the following integral inequalities hold In the next section first we prove the Hadamard type inequality for harmonically convex functions via generalized fractional integral operators defined in (3) and (4). Also we produce the Hadamard type inequalities given in [1,2,6,7]. Then we investigate the Fejér-Hadamard type inequalities via fractional integral operators defined in (3) and (4) and reproduce such results given in [1,2,6,7].
In this section we give our results.
f is a harmonically convex function on [a, b], then the following inequalities for generalized fractional integral operators hold Proof. dt.
If we put in above x = tb+(1−t)a ab and y = ta+(1−t)b ab , then we have the following inequality After simplification, we get By using Definition 1.4, we get Again by using that harmonically convexity of f for t ∈ [0, 1], one can have Inequalities (8) and (11) provide the required inequality (5).
Proof. Multiplying (6) by 2t ν−1 E γ,δ ,k,c µ,ν,l (ωt µ ; p) on both sides and integrating over [0, 1 2 ], we have Putting in above Since f is harmonically symmetric about a+b 2ab , we replace 1 a + 1 b − x by x in first term on R.H.S. of the above inequality and after simplification, we have By using Definition 1.4, we get µ,ν,l (ωt µ ; p)dt.

Putting in above
Since f is harmonically symmetric about a+b 2ab , therefore by replacing 1 a + 1 b − x with x in first term of L.H.S. of above inequality and after simple calculation, we have ε γ,δ ,k,c µ,ν,l,ω , a+b 2ab Inequalities (14) and (16)  (ii) If we put ω = p = 0, then we get Theorem 1.2.