Study of fractional integral inequalities involving Mittag-Leffler functions via convexity

*Correspondence: saddiqa.maryam@hotmail.com 3Department of Mathematics, Air University, Islamabad, Pakistan Full list of author information is available at the end of the article Abstract This paper studies fractional integral inequalities for fractional integral operators containing extended Mittag-Leffler (ML) functions. These inequalities provide upper bounds of leftand right-sided fractional integrals for (α,h –m) convex functions. A generalized fractional Hadamard inequality is established. All the results hold for h-convex, (h,m)-convex, (α,m)-convex, (s,m)-convex, and associated functions.


Introduction
Convexity was introduced at the beginning of the twentieth century. Due to having many fascinating and important properties, a convex function plays a vital role in almost all areas of mathematical analysis, probability theory, optimization theory, graph theory, etc. It has been defined in different convenient ways, for example, graph of a convex function always lies below the chord joining any two points lying on its graph, the derivative of a differentiable convex function is increasing and vice versa, a convex function has line of support at each point of the interior of its domain, and many others. In the theory of inequalities it is frequently defined in the form of an inequality which can be interpreted very nicely in the plane. A function f : I ⊆ R → R satisfying the inequality f (ta + (1t)b) ≤ tf (a) + (1t)f (b), where I is an interval, t ∈ [0, 1], and a, b ∈ I, is called convex.
This analytic form of presentation of a convex function motivated the authors to define other types of convex functions for example m-convex, s-convex, (s, m)-convex, h-convex, (h, m)-convex, (α, m)-convex, exponentially convex, etc. In this age convex functions lead to the theory of convex analysis, theory of inequalities, a lot of research articles and books are dedicated to the literature which has been developed due to convex function, see [1,3,4,20,22,25,31].
The goal of this paper is to study the bounds of fractional integral operators involving Mittag-Leffler (ML) functions in their kernels by utilizing a generalized form of convex functions, namely (α, hm)-convex functions which unify h-convex, (h, m)-convex, (α, m)-convex, and (s, m)-convex functions. Therefore the results of this paper will simultaneously hold for all these kinds of convex functions.
In 2007, Varošance introduced the h-convex function.
Definition 1 ([30]) A function f : I → R is said to be h-convex if the following inequality holds: where h is a nonnegative function defined on J, a, b ∈ I, t ∈ [0, 1], I and J are real intervals such that (0, 1) ⊂ J.
Next we give the definition of Mittag-Leffler functions and associated definitions of fractional integral operators.
Mittag-Leffler function E ξ (·) for one parameter is defined as follows [19]: where t, ξ ∈ C, (ξ ) > 0, and (·) is the gamma function. It is a natural extension of exponential, hyperbolic, and trigonometric functions. This function and its extensions are useful in solving fractional integral/differential equations. It is also studied extensively in various fields of sciences; for details, see [2,7,10,16,17,24,26,27].
A derivative formula of the extended Mittag-Leffler function is given in the following lemma.
Next, we give the definition of fractional integral operators containing the extended Mittag-Leffler function (1.3).
Then the generalized fractional integral operators containing Mittag-Leffler function are defined by Remark 3 Operators (1.5) and (1.6) produce in particular several kinds of known fractional integral operators, see [29,Remark 1.4].
The classical Riemann-Liouville fractional integral operator is defined as follows.
Then Riemann-Liouville fractional integral operators of order ξ > 0 are defined by . From fractional integral operators (1.5) and (1.6) we can write In the upcoming section the extended Mittag-Leffler (ML) function (1.3) and the corresponding generalized fractional integral operators are used to evaluate the bounds of sum of left-and right-sided operators by using (α, hm)-convexity. Their particular cases are also discussed. Furthermore, the lower and upper bounds of sum of these operators are presented in the form of a Hadamard inequality for (α, hm)-convex functions. Also the presented results are connected with several already known results.
Some particular results are stated in the following corollaries.
1] 2 , m = 0, then for ξ , η ≥ 1, the following fractional integral inequality for generalized fractional integral operators (1.5) and (1.6) holds: (2.14) Proof Let x ∈ [x 0 , y 0 ] and t ∈ [x 0 , x). Then, by using the definition of (α, hm)-convexity of |ϕ |, we have From (2.15), we can write Multiplication of (2.2) with (2.16) gives the following inequality: Now, integrating over [x 0 , x], we get The left-hand side of (2.18) is computed as follows: Substituting xt = r, using the derivative property (1.4) of Mittag-Leffler function, we have Now, for xr = t in the second term of the right-hand side of the above equation and then using (1.5), we get

Therefore (2.18) becomes
Again from (2.15) we have Similar as we did for (2.16), we can obtain Now, for x ∈ [x 0 , y 0 ] and t ∈ (x, y 0 ], again by using the (α, hm)-convexity of |ϕ |, we have Proceeding along similar lines as we did to get (2.23), we can obtain the following inequality: From inequalities (2.23) and (2.25), triangular inequality (2.14) can be obtained.
It is easy to prove the next lemma which will be helpful to establish estimations in the form of a Hadamard-type inequality.
from which we can get the following inequality: