Hadamard and Fejér–Hadamard inequalities for extended generalized fractional integrals involving special functions

In this paper we prove the Hadamard and the Fejér–Hadamard inequalities for the extended generalized fractional integral operator involving the extended generalized Mittag-Leffler function. The extended generalized Mittag-Leffler function includes many known special functions. We have several such inequalities corresponding to special cases of the extended generalized Mittag-Leffler function. Also there we note the known results that can be obtained.

If we denote by K m (b) the set of m-convex functions on [0, b] for which f (0) < 0, then we have whenever m ∈ (0, 1). Note that in the class K 1 (b) there are only convex functions f : [0, b] → R for which f (0) ≤ 0 (see [2]). An m-convex function need not be a convex function, as the following example shows.
For more results and inequalities related to m-convex functions one can consult for example [2,[4][5][6]. In the literature the integral inequality where f : I → R is a convex function on the interval I of real numbers and a, b ∈ I with a < b, is known as the Hadamard inequality. If f is concave, then the above inequality holds in the reverse direction. The Hadamard inequality has always retained the attention of mathematicians and a lot of results have been produced about it, for example see [6][7][8][9][10][11][12] and the references cited therein. In [13] Fejér gave a generalization of the Hadamard inequality as follows.
In the literature inequality (1.2) is known as the Fejér-Hadamard inequality. Nowadays the Hadamard and the Fejér-Hadamard inequalities via fractional calculus are in focus of researchers. Recently a lot of papers have been dedicated to this field (see [4,[14][15][16] and the references therein). Fractional calculus refers to integration or differentiation of fractional order, the origin of fractional calculus is as old as calculus. For a historical survey of this field the reader is referred to [17][18][19][20][21].
Fractional integral inequalities are useful in establishing the uniqueness of solutions for certain fractional partial differential equations. They also provide upper and lower bounds for the solutions of fractional boundary value problems. Many researchers have explored certain extensions and generalizations of integral inequalities by involving fractional calculus (see [14,16,22,23]).
We are going to give the Hadamard and the Fejér-Hadamard inequalities for the extended generalized fractional integral operator containing the extended generalized Mittag-Leffler function [24]. We give a two sided definition of the extended generalized fractional integral operator containing the extended generalized Mittag-Leffler function as follows: Definition 1.2 Let δ, α, β, τ , c ∈ C and R(δ), R(α), R(β), R(τ ), R(c) > 0, p ≥ 0 and q, r > 0. Then the extended generalized fractional integral operator ω,δ,q,r,c ·,α,β,τ containing the extended generalized Mittag-Leffler function E δ,r,q,c α,β,τ for a real-valued continuous function f is defined by and where the generalized beta function β p (x, y) is defined by For ω = 0 along with p = 0, the integral operator ω,δ,q,r,c ·,α,β,τ would correspond essentially to the two sided Riemann-Liouville fractional integral operator In [24][25][26][27][28][29] fractional boundary value problems and fractional differential equations are studied along with properties of Mittag-Leffler function. In the following results we see some properties of the Mittag-Leffler function [24].

Theorem 1.2 The series in (1.5) is absolutely convergent for all values of t provided that
We organize the paper so that in Sect.
Multiplying both sides of the above inequality with t β-1 E Again by using the fact that f is a convex function on [a, b] and for t ∈ [0, 1] we have  In the following remark we mention some published results.
In the following we mention some published results.

Hadamard and Fejér-Hadamard inequality for m-convex function via the extended generalized Mittag-Leffler function
In ; p Proof Since f is an m-convex function on [a, mb], for t ∈ [0, 1] we have Multiplying with t β-1 E δ,r,q,c α,β,τ (ωt α ; p) both sides of the above inequality we get Integrating with respect to t over [0, 1] we have 2f a + mb 2 Again by using that f is an m-convex function we have ; p From inequalities (3.3) and (3.5) we get the inequality in (3.1).
In the following remark we mention some published results.