Inequalities for a Unified Integral Operator via (α, m)-Convex Functions

Recently, a unified integral operator has been introduced by Farid, 2020, which produces several kinds of known fractional and conformable integral operators defined in recent decades (Kwun, 2019, Remarks 6 and 7). +e aim of this paper is to establish bounds of this unified integral operator by means of (α, m)-convex functions. +e resulting inequalities provide the bounds of all associated fractional and conformable integral operators in a compact form. Also, the results of this paper hold for different kinds of convex functions connected with (α, m)-convex functions.

Definition 1 (see [15]). Let η 1 : [a, b] ⟶ R be an integrable function. Also, let η 2 be an increasing and positive function on (a, b], having a continuous derivative η 2 ′ on (a, b). e left-sided and right-sided fractional integrals of a function η 1 with respect to another function η 2 on [a, b] of order μ, where R(μ) > 0, are defined by where Γ(·) is the gamma function.
A k-analogue of the above definition is defined as follows.
Definition 2 (see [16]). Let η 1 : [a, b] ⟶ R be an integrable function. Also, let η 2 be an increasing and positive function on (a, b], having a continuous derivative η 2 ′ on (a, b). e left-sided and right-sided fractional integrals of a function η 1 with respect to another function η 2 on [a, b] of order μ; R(μ), k > 0 are defined by where Γ k (·) is defined by [17] A well-known function named Mittag-Leffler function is defined by [18] where α, z ∈ C and R(α) > 0.
A generalized fractional integral operator containing an extended generalized Mittag-Leffler function is defined as follows.
Recently, Farid defined a unified integral operator which unifies several kinds of fractional and conformable integrals in a compact formula which is defined as follows.
For suitable settings of function ϕ, η 2 , and certain values of parameters included in Mittag-Leffler function (7), very interesting consequences are obtained which are comprised in Remarks 6 and 7 of [10]. e objective of this paper is to obtain bounds of unified integral operators explicitly which are directly linked with various fractional and conformable integrals. e (α, m)-convexity has been used for establishing these bounds.
ese bounds provide general formulas to obtain bounds of fractional and conformable integral operators described in Remarks 6 and 7 of [10]. Among the well-known inequalities which are related to the integral mean of a convex function, the Hadamard inequality is of great importance. Many mathematicians worked on new types of Hadamard inequalities using convex functions, see [8,[29][30][31]. We also established the general Hadamard-type inequality by applying Lemma 1 which further produces various inequalities of Hadamard type for fractional and conformable integrals. At the end, by using (α, m)-convexity of |η 1 ′ |, a modulus inequality is obtained.

Main Results
Bounds of unified integral operators (8) and (9) using (α, m)-convexity are studied in the following result: be differentiable and strictly increasing function, and also, let ϕ/x be an increasing function on [a, b].
and hence, Proof. Under the assumptions of ϕ and η 2 , one can write the following inequality: Multiplying with E c,δ,k,c By using E , the following inequality is obtained: Using the definition of (α, m)-convexity for η 1 , the following inequality is valid: Multiplying (15) with (16) and integrating over [a, x], one can obtain By using (8) of Definition 4 and integrating by parts, the following inequality is obtained:

Journal of Mathematics 3
Now, on the other side, for t ∈ (x, b] and x ∈ (a, b), the following inequality holds true: Using (α, m)-convexity of η 1 , we have Adopting the same procedure as we did for (15) and (16), the following inequality from (19) and (20) can be obtained: By adding (18) and (21), (12) can be obtained.
, then the following inequality holds: for all x ∈ [a, b] and m ∈ (0, 1]. e following result provides upper and lower bounds of the sum of operators (8) and (9) in the form of a Hadamard inequality.