Generalization of Some Fractional Integral Operator Inequalities for Convex Functions via Uniﬁed Mittag–Lefﬂer Function

: This paper aims to obtain the bounds of a class of integral operators containing Mittag– Lefﬂer functions in their kernels. A recently deﬁned uniﬁed Mittag–Lefﬂer function plays a vital role in connecting the results of this paper with the well-known bounds of fractional integral operators published in the recent past. The symmetry of a function about a line is a fascinating property that plays an important role in mathematical inequalities. A variant of the Hermite–Hadamard inequality is established using the closely symmetric property for ( α , m ) -convex functions.

The above function is the natural extension of many exponential, hyperbolic, and trigonometric functions.Due to its adverse use in various branches of mathematics, many scholars have published several generalizations and extensions of this function.Moreover, they have also proved several integral transforms of Mittag-Leffler functions and expressed them in the form of some famous special functions [2,3].The integral operators containing Mittag-Leffler functions are frequently used to prove several eminent inequalities such as the Hadamard, Ostrowski, Minkowski, Opial, and Chebyshev inequalities.
Motivated and inspired by the ongoing research in the field of integral inequalities, this paper aims to establish the boundedness of fractional integral operators containing an extended Mittag-Leffler function [13] called the unified Mittag-Leffler function.A Hadamard-type integral inequality is established using a symmetry-like function.For this purpose, we have utilized a well-known convexity named the (α, m)-convexity of a function.The results of this article are the generalizations of some fractional inequalities already proven in different papers.The classical Riemann-Liouville fractional integrals are defined as follows: . Then, left-sided and right-sided Riemann-Liouville fractional integrals of a function f of order µ where (µ) > 0 are defined as follows: and I where (µ) is real value of µ and Γ(µ) = ∞ 0 e −z z µ−1 dz.
The k-analogue of the Riemann-Liouville fractional integral is defined as follows: Then, k-fractional Riemann-Liouville integrals of order µ, where (µ) > 0, k > 0, are defined as: where Γ k (.) is defined as follows: The definition of generalized Riemann-Liouville fractional integrals by a monotonically increasing function is defined as follows: Definition 3 ([15]).Let ϕ : [σ 1 , σ 2 ] → R be an integrable function.Let ψ be an increasing and positive function on (σ 1 , σ 2 ], having a continuous derivative ψ on (σ 1 , σ 2 ).The left-sided and right-sided fractional integrals of a function ϕ with respect to another function ψ on [σ 1 , σ 2 ] of order µ where (µ) > 0 are defined by The k-analogue of a generalized Riemann-Liouville fractional integral is defined as follows: The left-sided and right-sided generalized fractional integral operators containing an extended generalized Mittag-Leffler function is defined as follows: and where is the extended generalized Mittag-Leffler function.
Recently, Gao et al. [20] gave the further generalization and extension of the above integral operator as follows: where 21) and ( 22), the unified integral operator given in Definition6 is obtained; 21) and (22), the operator given in Definition 5 is obtained; (iv) If Υ(ϑ) = ϑ α for α > 0, in (21), the operator given in [20] is obtained.Definition 10.By setting a i = l, s = 0 and (ρ) > 0 in ( 21) and ( 22), we get the fractional integral operator associated with the generalized Q function: where Convexity is an essential notion often used in mathematics, mathematical statistics, graph theory, etc.In particular, a convex function contributes a lot to the formulation of new inequalities which behave as generalizations of classical inequalities.The Hermite-Hadamard inequality is a straight sequel of convex functions.It describes the lower as well as upper bound of an integral mean of a convex function over an interval [σ 1 , σ 2 ].The Hermite-Hadamard inequality was first generalized by Fejér [21] with the help of the symmetric function about the midpoint of the interval [σ 1 , σ 2 ] on which the convex function is defined.In [22], Farid introduced an inequality of Hermite-Hadamard type by using symmetric convex functions.Presently, the Hermite-Hadamard inequality is generalized by defining new classes of convex functions which are clearly related to convex functions.This paper gives the Hermite-Hadamard-type inequality by using convex functions close to symmetric functions.The definition of (α, m)-convexity is defined as follows: holds for all ϑ, y ∈ [0, σ 2 ] and τ ∈ [0, 1].

Example 1.
(i) A (1, 1)-convex function is an example of convex function; (ii) A (α, 1)-convex function is an example of an α-convex function; (iii) A (1, m)-convex function is an example of an m-convex function.
In the upcoming section, we have investigated the bounds of integral operators given in (21) and (22) with the help of (α, m)-convexity.Moreover, we have also provided the Hermite-Hadamard-type inequality by using the condition close to symmetry about the interval's midpoint.The established results give many fractional and conformable integral inequalities.From here onward, we consider the parameters of Mittag-Leffler functions as real numbers.
Example 2. If α = 1, then the following inequality holds for an m-convex function: Example 3. If (α, m) = (1, 1), then the following inequality holds for a convex function: The following lemma is required to establish the next result.
then the following inequality holds: for all ϑ ∈ [σ 1 , σ 2 ] and m ∈ (0, 1]. The following result provides the upper and lower bounds of the sum of operators ( 21) and (22) in the form of a Hadamard inequality.

Theorem 2. Under the assumptions of Theorem 1, in addition
Proof.Under the assumptions of Υ and ψ, we have Multiplying with we can obtain from (41) the following inequality: By using the following inequality is obtained: Using the (α, m)-convexity of ϕ for ϑ ∈ [σ 1 , σ 2 ], we have Multiplying ( 43) and ( 44) and integrating the resulting inequality over [σ 1 , σ 2 ], one can obtain By using Definition 9 and integrating by parts, the following inequality is obtained: Now, for ϑ ∈ [σ 1 , σ 2 ], the following inequality holds true: Using the same technique that we did for ( 43) and ( 44), the following inequality can be observed from (44) and (46): By adding (45) and (47), the following inequality can be obtained: Multiplying both sides of (39) by , and integrating over [σ 1 , σ 2 ], we have From Definition 9, the following inequality is obtained: Similarly, multiplying both sides of (39) by By adding (49) and (50), the following inequality is obtained: Combining ( 48) and (51), inequality (40) can be achieved.

Conclusions
We have investigated the bounds of fractional integral operators containing the Mittag-Leffler function in their kernels.The established bounds are compact formulas that generate fractional integral inequalities for various well-known integral operators.All the presented and deduced results hold for convex, m-convex and star-shaped functions.This work can be further extended for different kinds of classes of functions that already exist in the literature.For example, classes of strongly convex and refined convex functions can be applied to improve such bounds of fractional integral operators.