Abstract

In this paper, bounds of fractional and conformable integral operators are established in a compact form. By using exponentially convex functions, certain bounds of these operators are derived and further used to prove their boundedness and continuity. A modulus inequality is established for a differentiable function whose derivative in absolute value is exponentially convex. Upper and lower bounds of these operators are obtained in the form of a Hadamard inequality. Some particular cases of main results are also studied.

1. Introduction

We start with the definition of convex function.

Definition 1. (see [1]). A function is said to be convex ifholds for all and . If inequality (1) is reversed, then the function will be concave on .
Convex functions are very useful in many areas of mathematics and other subjects due to their fascinating properties and characterizations. Their geometric and analytic interpretations provide straightforward proofs of many mathematical inequalities including Hadamard, Jensen, Hölder, and Minkowski [1–3]. Theoretically, convex functions have been generalized and extended as -convex, -convex, -convex, -convex, -convex, -convex, etc. Awan et al. [4] defined the function named exponentially convex function as follows:

Definition 2. A function , where is an interval, is said to be an exponentially convex function ifholds for all , and . If the inequality in (2) is reversed, then is called exponentially concave.
If , then (2) gives inequality (1). For some recent citations and utilization of exponentially convex functions, one can see [5–14] and references therein. Our goal in this paper is to prove generalized integral inequalities for exponentially convex functions by using integral operators given in Definition 7. In the following, we give definitions of Riemann–Liouville fractional integrals:

Definition 3. Let . Then, the left-sided and right-sided Riemann–Liouville fractional integral operators of order are defined byA -fractional analogue is given as follows:

Definition 4. (see [15]). Let . Then, for , the -fractional integral operators of of order are defined byA more general definition of the Riemann–Liouville fractional integral operators is given as follows:

Definition 5. (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 the gamma function.

Definition 6. 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 integral operators, , of a function with respect to another function on of order are defined bywhere is the -gamma function.
A compact form of integral operators defined above is given as follows:

Definition 7. (see [17]). Let be the functions such that be positive and   and be differentiable and strictly increasing. Also, let be an increasing function on . Then, for , the left- and right-sided integral operators are defined bywhere .
Integral operators defined in (11) and (12) produce several fractional and conformable integral operators defined in [16, 18–25].

Remark 1. Integral operators given in (11) and (12) produce fractional and conformable integral operators as follows:(i)If we consider , then (11) and (12) integral operators coincide with (9) and (10) fractional integral operators.(ii)If we consider , , then (11) and (12) integral operators coincide with (7) and (8) fractional integral operators.(iii)If we consider and as identity function (11) and (12), integral operators coincide with (5 and 6) fractional integral operators.(iv)If we consider , , and along with as identity function (11) and (12), integral operators coincide with (3 and 4) fractional integral operators.(v)If we consider , , and , then (11) and (12) produce Katugampola fractional integral operators defined by Chen and Katugampola in [18].(vi)If we consider , , and , then (11) and (12) produce generalized conformable integral operators defined by Khan and Khan in [22].(vii)If we consider and , in (11) and and  = , in (12), respectively, then conformable -fractional integrals are achieved as defined by Habib et al. in [20].(viii)If we consider and , then (11) and (12) produce conformable fractional integrals defined by Sarikaya et al. in [24].(ix)If we consider and , in (11) and and , in (12) with , respectively, then conformable fractional integrals are achieved as defined by Jarad et al. in [21].(x)If we consider , then (11) and (12) produce generalized -fractional integral operators defined by Tunc et al. in [25].(xi)If we consider then following generalized fractional integral operators with exponential kernel are obtained [19]:(xii)If we consider and , then Hadamard fractional integral operators will be obtained [16, 23].(xiii)If we consider and , then Harmonic fractional integral operators given in [16] will be obtained and given as follows:(xiv)If we consider , then left- and right-sided logarithmic fractional integrals are obtained in [19] and given as follows:In the upcoming section, we will derive bounds of sum of the left- and right-sided integral operators defined in (11) and (12) for exponentially convex functions. These bounds lead to produce results associated to several kinds of well-known operators for exponentially convex functions, some of the results are presented in particular cases. Further in Section 3, bounds are presented in the form of a Hadamard inequality; several Hadamard type inequalities are obtained.

2. Bounds of Integral Operators and Their Consequences

Theorem 1. Let be a positive and exponentially convex function and be a differentiable and strictly increasing function. Also, let be an increasing function on . Then, for the following inequality for integral operators (11) and (12) holds

Proof. For the kernel of integral operator (11), we haveAn exponentially convex function satisfies the following inequality:Inequalities (17) and (18) lead to the following integral inequality:while (19) givesAgain, for the kernel of integral operator (12), we haveAn exponentially convex function satisfies the following inequality:Inequalities (21) and (22) lead to the following integral inequality:while (23) givesBy adding (20) and (24), (16) can be achieved.
The following remark connected the abovementioned theorem with already known results.

Remark 2. (1)For , and in (16), Corollary 1 in [26] can be achieved.(2)For , , and in (16), Corollary 1 in [27] can be achieved.(3)For , in (16), Theorem 1 in [28] can be achieved.Next results indicate upper bounds of several known fractional and conformable integral operators.

Proposition 1. Let . Then, (11) and (12) produce the fractional integral operators (7) and (8) as follows:Further they satisfy the following bound for :

Proposition 2. Let . Then, (11) and (12) produce integral operators defined in [29] as follows:Further they satisfy the following bound:

Corollary 1. If we take , then (11) and (12) produce the fractional integral operators (9) and (10) as follows:

From (16), the following bound holds for :

Corollary 2. If we take , , and , then (11) and (12) produce left- and right-sided Riemann–Liouville fractional integral operators (3) and (4) as follows:From (16), the following bound holds for :

Corollary 3. If we take and , then (11) and (12) produce the fractional integral operators (5) and (6) as follows:From (16), the following bound holds for :