Study of Fractional Integral Operators Containing Mittag-Leffler Functions via Strongly 
 
 
 α
 ,
 m
 
 
 -Convex Functions

The main aim of this paper is to give refinement of bounds of fractional integral operators involving extended generalized Mittag-Leffler functions. A new definition, namely, strongly 
 
 
 
 α
 ,
 m
 
 
 
 -convex function is introduced to obtain improvements of bounds of fractional integral operators for convex, 
 
 m
 
 -convex, and 
 
 
 
 α
 ,
 m
 
 
 
 -convex functions. The results of this paper will provide simultaneous generalizations as well as refinements of various published results.


Introduction
Convexity is one of the most important and key concept in mathematics, and many researchers have extended, generalized, and refined it in different ways. Numerous generalizations and extensions have been produced in recent past, for example, in generalization and extension point of views, m-convexity, (α, m)-convexity, s-convexity, (s, m)-convexity, h-convexity, and (h, m)-convexity are remarkable, and in refinement point of view, the strongly convexity is the tremendous notion. In this paper, we have introduced the notion of strongly (α, m)-convex function. By utilizing this refined form of convex function, we obtain refinements of the bounds of fractional integral operators involving Mittag-Leffler functions in their kernels. erefore, the results of this paper are refinements of all the results proved in [1]. First, we give definitions of convex, strongly convex, and (α, m)-convex functions.
Definition 1 (see [2]). Let I be an interval on real line. A function f: I ⟶ R is said to be convex function if the following inequality holds: for all u 1 , u 2 ∈ I and t ∈ [0, 1].
Definition 2 (see [3]). Let I be an interval on real line. A realvalued function f is said to be strongly convex with modulus λ ≥ 0 on I if, for each u 1 , u 2 ∈ I and t ∈ [0, 1], we have Definition 3 (see [4]). A function f: e well-known Mittag-Leffler function is denoted by E ξ (.) and defined as follows (see [5]): where t, ξ ∈ C, R(ξ) > 0, and Γ(.) is the gamma function. It is a natural extension of exponential, hyperbolic, and trigonometric functions. is function and its extensions appear as solution of fractional integral equations and fractional differential equations. For a detailed study of Mittag-Leffler function and its extensions, see [6][7][8][9][10]. e following extended generalized Mittag-Leffler function is introduced by Andrić et al.
A derivative formula of the extended generalized Mittag-Leffler function is given in the following lemma.
Next, we have the definition of the generalized fractional integral operator containing the extended generalized Mittag-Leffler function (5).
Let b]. en, the generalized fractional integral operators containing Mittag-Leffler function are defined by (8) and ϵ c,δ,k,c e operators defined in (8) and (9) produce several kinds of known fractional integral operators, see Remark 1.4 in [14]. e classical Riemann-Liouville fractional integral operator is defined as follows.
en, Riemann-Liouville fractional integral operators of order ξ > 0 are defined by It can be noted that (ϵ . From fractional integral operators (8) and (9), we can have In view of wide applications of Riemann-Liouville fractional integrals and derivatives, the problems which involve this integral operator are studied extensively by many authors. e aim of this paper is to provide fractional integral inequalities which are generalizations of Riemann-Liouville fractional integral inequalities. ese inequalities also give associated inequalities for fractional integral operators containing Mittag-Leffler functions with different parameters. e bounds of fractional integrals (8) and (9) for (α, m)-convex functions are given in the following theorems.
, then, for ξ, η > 0, the following fractional integral inequality for generalized integral operators (8) and (9) holds: In Section 2, by using definition of strongly (α, m)-convex function, we establish new refinements of the bounds of generalized fractional integral operators. Also, the refinements of bounds of these operators are presented in the form of Hadamard-like inequality by using strongly (α, m)-convex functions. e results of this paper are connected with several well-known inequalities.