Abstract

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). The aim of this paper is to establish bounds of this unified integral operator by means of -convex functions. The 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 -convex functions.

1. Introduction

To prove the mathematical inequalities, fractional integral operators play an important role in the field of different branches of mathematics and engineering. Many mathematicians have used fractional integrals and conformable fractional integrals to develop integral inequalities [1–14]. We start from definitions of fractional integral operators which are direct consequences of unified integral operators given in (8) and (9).

Definition 1. (see [15]). Let be an integrable function. Also, let be an increasing and positive function on , having a continuous derivative on . The left-sided and right-sided fractional integrals of a function with respect to another function on of order , where , are defined bywhere is the gamma function.
A -analogue of the above definition is defined as follows.

Definition 2. (see [16]). Let be an integrable function. Also, let be an increasing and positive function on , having a continuous derivative on . The left-sided and right-sided fractional integrals of a function with respect to another function on of order are defined bywhere is defined by [17]A well-known function named Mittag-Leffler function is defined by [18]where and .
One can see [19–22] to study the Mittag-Leffler function and its generalizations. Also, (2) and (3) produce many types of fractional integral operators (see [10], Remark 6).
A generalized fractional integral operator containing an extended generalized Mittag-Leffler function is defined as follows.

Definition 3. (see [1]). Let , , and with , , and . Let and . Then, the generalized fractional integral operators and are defined bywhereis the extended generalized Mittag-Leffler function.
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.

Definition 4. (see [23]). Let , , be the functions such that be positive and and be differentiable and strictly increasing. Also, let be an increasing function on and , , , , and . Then, for , the left and right integral operators are defined bywhere .
For suitable settings of function , , 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].
The objective of this paper is to obtain bounds of unified integral operators explicitly which are directly linked with various fractional and conformable integrals. The -convexity has been used for establishing these bounds. The notion of -convexity is defined by Mihesan in [24].

Definition 5. A function , is said to be -convex, where , ifholds for all .

Remark 1. (i)If we put  = , then (10) gives the definition of -convex function(ii)If we put  = , then (10) gives the definition of convex function(iii)If we put  = , then (10) gives the definition of star-shaped functionFor some recent citations and utilizations of -convex functions, one can see [9, 25–28] and references therein. In the upcoming section, bounds of unified integral operators are established by using -convexity. These 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, 29–31]. 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 -convexity of , a modulus inequality is obtained.

2. Main Results

Bounds of unified integral operators (8) and (9) using -convexity are studied in the following result:

Theorem 1. Let be a positive integrable -convex function with . Let be differentiable and strictly increasing function, and also, let be an increasing function on . If , , , , and , then for , we haveand hence,

Proof. Under the assumptions of and , one can write the following inequality:Multiplying with , we can obtainBy using , the following inequality is obtained:Using the definition of -convexity for , the following inequality is valid:Multiplying (15) with (16) and integrating over , one can obtainBy using (8) of Definition 4 and integrating by parts, the following inequality is obtained:Now, on the other side, for and , the following inequality holds true:Using -convexity of , we haveAdopting 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.

Remark 2. (i)If we consider  = (1,1) in (12), Theorem 8 in [10] is obtained(ii)If we consider for the left-hand integral and for the right-hand integral and in (12), then Theorem 1 in [9] can be obtained(iii)If we consider in the result of (ii), then Corollary 1 in [9] can be obtained(iv)If we consider , , and  = (1,1) in (12), Theorem 1 in [6] is obtained(v)If we consider in the result of (iv), Corollary 1 in [6] is obtained(vi)If we consider for the left-hand integral and for the right-hand integral, =(1,1), , and , then Theorem 1 in [4] can be obtained(vii)If we consider in the result of (vi), then Corollary 1 in [4] can be obtained(viii)If we consider for the left-hand integral and for the right-hand integral, , and and =(1,1) in (12), then Theorem 1 in [5] is obtained(ix)By setting in the result of (viii), Corollary 1 in [5] can be obtained(x)By setting and in the result of (ix), Corollary 2 in [5] can be obtained(xi)By setting and in the result of (ix), Corollary 3 in [5] can be obtainedTo prove the next result, we need the following lemma [9].

Lemma 1. Let be an -convex function with . If , then the following inequality holds:for all and .

The following result provides upper and lower bounds of the sum of operators (8) and (9) in the form of a Hadamard inequality.

Theorem 2. With the assumptions of Theorem 1 in addition, if , then we have