Study of Fractional Integral Operators Containing Mittag-Leffler Functions via Strongly -Convex Functions

Abstract

The main aim of this paper is to give refinement of bounds of fractional integral operators involving extended generalized Mittag-Leffler functions. A new definition, namely, strongly -convex function is introduced to obtain improvements of bounds of fractional integral operators for convex, -convex, and -convex functions. The results of this paper will provide simultaneous generalizations as well as refinements of various published results.

1. Introduction

Convexity is one of the most important and key concept in mathematics, and many researchers have extended, generalized, and refined it in different ways. Numerous generalizations and extensions have been produced in recent past, for example, in generalization and extension point of views, m-convexity, -convexity, -convexity, -convexity, -convexity, and -convexity are remarkable, and in refinement point of view, the strongly convexity is the tremendous notion. In this paper, we have introduced the notion of strongly -convex function. By utilizing this refined form of convex function, we obtain refinements of the bounds of fractional integral operators involving Mittag-Leffler functions in their kernels. Therefore, the results of this paper are refinements of all the results proved in [1]. First, we give definitions of convex, strongly convex, and -convex functions.

Definition 1. (see [2]). Let be an interval on real line. A function is said to be convex function if the following inequality holds:for all and .

Definition 2. (see [3]). Let be an interval on real line. A real-valued function is said to be strongly convex with modulus on if, for each and , we have

Definition 3. (see [4]). A function is said to be -convex function, where and if, for every and , we haveThe well-known Mittag-Leffler function is denoted by and defined as follows (see [5]):where , and is the gamma function. It is a natural extension of exponential, hyperbolic, and trigonometric functions. This function and its extensions appear as solution of fractional integral equations and fractional differential equations. For a detailed study of Mittag-Leffler function and its extensions, see [6–10]. The following extended generalized Mittag-Leffler function is introduced by Andrić et al.

Definition 4. (see [11]). Let , , and with , , and . Then, the Mittag-Leffler function is defined bywhere is defined byand..
A derivative formula of the extended generalized Mittag-Leffler function is given in the following lemma.

Lemma 1. (see [11]). If with , and , then

Remark 1. The extended generalized Mittag-Leffler function (5) produces the related functions defined in [8–10, 12, 13], see Remark 1.3 in [14].
Next, we have the definition of the generalized fractional integral operator containing the extended generalized Mittag-Leffler function (5).

Definition 5. (see [11]). Let , , and with , , and . Let and . Then, the generalized fractional integral operators containing Mittag-Leffler function are defined byand

Remark 2. The operators defined in (8) and (9) produce several kinds of known fractional integral operators, see Remark 1.4 in [14].
The classical Riemann–Liouville fractional integral operator is defined as follows.

Definition 6. (see [13]). Let . Then, Riemann–Liouville fractional integral operators of order are defined byIt can be noted that and . From fractional integral operators (8) and (9), we can haveIn view of wide applications of Riemann–Liouville fractional integrals and derivatives, the problems which involve this integral operator are studied extensively by many authors. The aim of this paper is to provide fractional integral inequalities which are generalizations of Riemann–Liouville fractional integral inequalities. These inequalities also give associated inequalities for fractional integral operators containing Mittag-Leffler functions with different parameters.
The bounds of fractional integrals (8) and (9) for -convex functions are given in the following theorems.

Theorem 1. (see [1]). Let be a real-valued function. If is positive -convex, then for , the following fractional integral inequality for generalized integral operators (8) and (9) holds:

Theorem 2. (see [1]). Let be a real-valued function. If is differentiable and is -convex, then, for , the following fractional integral inequality for generalized integral operators (8) and (9) holds:

Theorem 3. (see [1]). Let , , be a real-valued function. If is positive, -convex, and , then, for , the following fractional integral inequality for generalized integral operators (8) and (9) holds:In Section 2, by using definition of strongly -convex function, we establish new refinements of the bounds of generalized fractional integral operators. Also, the refinements of bounds of these operators are presented in the form of Hadamard-like inequality by using strongly -convex functions. The results of this paper are connected with several well-known inequalities.

2. Main Results

First, we define strongly -convex function.

Definition 7. A function is said to be strongly -convex function, with modulus , for , ifholds, for all and .

Remark 3. (i)By setting in (17), strongly -convex function can be obtained [15](ii)By setting in (17), -convex function can be obtained(iii)By setting and in (17), convex function can be obtained(iv)By setting and in (17), -convex function can be obtained(v)By setting in (17), strongly convex function can be obtainedIn the following, by using strongly -convex functions the refinement of already proved results are given.

Theorem 4. Let be a real-valued function. If is positive and strongly -convex, then, for , the following fractional integral inequality for generalized integral operators (8) and (9) holds:

Proof. Let . Then, for and , one can have the following inequality:The function is strongly -convex; therefore, one can obtainBy multiplying (19) and (20) and then integrating over , we obtainFrom which we have that the left integral operator satisfies the following inequality:Now, on the contrary, for and , one can have the following inequality:Again from strongly -convexity of , we haveBy multiplying (23) and (24) and then integrating over , we haveThe right integral operator satisfies the following inequality:By adding (22) and (26), the required inequality (18) is established.

Remark 4. (i)Inequality (18) provides the refinement of Theorem 2.1 in [1](ii)If in (18), then result for strongly -convex will be obtained(iii)If and in (18), then Corollary 1 [16] is obtained(iv)If , , and in (18), then Theorem 1 in [17] is obtained(v)If and in (18), then Theorem 3 in [18] is obtained

Corollary 1. If we set in (18), then the following inequality is obtained:

Remark 5. Inequality (27) provides the refinement of Corollary 2.1 in [1].

Corollary 2. Along with assumptions of Theorem 1, if , then the following inequality is obtained:

Corollary 3. For in (28), we get the following result:

Theorem 5. Let be a real-valued function. If is differentiable and is strongly -convex, then for , the following fractional integral inequality for generalized integral operators (8) and (9) holds:

Proof. As and , by using strongly -convexity of , we haveFrom (31), one can haveThe product of (19) and (32) gives the following inequality:After integrating above inequality over , we obtainThe left-hand side of (34) is calculated as follows:Put , that is, ; also using the derivative property (7) of Mittag-Leffler function, we haveNow, put in the second term of the right-hand side of the above equation, and then using (8), we obtainTherefore, (34) takes the following form:Also, from (31), one can haveFollowing the same procedure as we did for (32), one can obtainFrom (38) and (40), we obtainNow, we let and . Then, by using strongly -convexity of , we haveOn the same lines as we have done for (19), (32), and (39), one can get, from (23) and (12), the following inequality:From inequalities (41) and (43), (30) is obtained.

Remark 6. (i)Inequality (30) provides the refinement of Theorem 2.2 [1](ii)If in (30), then result for strongly -convex will be obtained(iii)If and in (30), then Corollary 2 [16] is obtained(iv)If , , and in (30), then Theorem 2 [17] is obtained(v)If and in (30), then Theorem 5 in [18] is obtained

Corollary 4. If we put in (30), then the following inequality is obtained:

Remark 7. Inequality (44) provides the refinement of Corollary 2.2 in [8].
To prove our next result, we consider following lemma.

Lemma 2. Let , be strongly -convex function. If and ; then, the following inequality holds:

Proof. As is strongly -convex function, we haveLet , and we have .Hence, by using , inequality (45) can be obtained.

Theorem 6. Let , , be a real-valued function. If is positive, strongly -convex, and , then, for , the following fractional integral inequality for generalized integral operators (8) and (9) holds: where

Proof. For , we haveAs is strongly -convex, so, for , we haveBy multiplying (50) and (51) and then integrating over , we obtainFrom which we havethat is,Now, on the contrary, for , we haveBy multiplying (51) and (55) and then integrating over , we obtainFrom which we havethat is,Adding (54) and (58), we get the second inequality of (48). Multiplying (45) with and integrating over , we obtainBy using (9) and (12), we obtainThe integral operator appearing in the last term is computed as follows:Now, inequality (43) becomesBy multiplying (45) with and integrating over , also using (8) and (12), we obtainThe integral operator appearing in the last term is computed as follows:Now, inequality (63) becomesBy adding (62) and (65), first inequality of (48) can be obtained.

Remark 8. (i)Inequality (48) provides the refinement of Theorem 2.3 in [1](ii)If in (48), then result for strongly -convex will be obtained(iii)If and in (48), then Corollary 2 in [16] is obtained(iv)If , , and in (48), then Theorem 2 in [17] is obtained(v)If and in (48), then Theorem 7 in [18] is obtained

Corollary 5. If we put in (48), then the following inequality is obtained:

Remark 9. Inequality (66) provides the refinement of Corollary 2.3 in [1].

3. Concluding Remarks

The presented results are the refinements of the bounds of generalized fractional integral operators given in (8) and (9) for strongly -convex functions. From the presented results, one can obtain already proved results for convex, -convex, and -convex functions. Moreover, the refinements of some known fractional versions of the Hadamard inequality are also given.

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.