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Fractional Calculus and Related Inequalities

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Volume 2020 |Article ID 2456463 | https://doi.org/10.1155/2020/2456463

Hong Ye, Ghulam Farid, Babar Khan Bangash, Lulu Cai, "Derivation of Bounds of an Integral Operator via Exponentially Convex Functions", Journal of Mathematics, vol. 2020, Article ID 2456463, 9 pages, 2020. https://doi.org/10.1155/2020/2456463

Derivation of Bounds of an Integral Operator via Exponentially Convex Functions

Academic Editor: Serkan Araci
Received17 Mar 2020
Accepted21 May 2020
Published04 Jul 2020

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 [13]. 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 [514] 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, 1825].

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 :

Corollary 4. If we take and , then (11) and (12) produce the fractional integral operators defined in [18] as follows:

From (16), they satisfy the following bound:

Corollary 5. If we take , and , then (11) and (12) produce the fractional integral operators defined as follows:From (16), they satisfy the following bound:

Next, we prove boundedness and continuity of integral operators.

Theorem 2. Let the assumptions of Theorem 1 are satisfied. If , then integral operators defined in (11) and (12) are continuous.

Proof. From (20), we haveIt is given that is increasing on :Further is increasing; therefore, we haveTherefore, (39) givesIf , then is decreasing on and we getIf , then is increasing on and we get from (42)Hence, is bounded and it is linear, and therefore, is continuous.
Similarly, continuity of can be proved.
For a differentiable function , as is exponentially convex, the following result holds:

Theorem 3. Let be a differentiable function. If is exponentially convex and is a differentiable and strictly increasing function. Also, let be an increasing function on , then for , , the following inequalities for integral operators holds:where

Proof. An exponentially convex function satisfies the following inequality:From which, we can writeInequalities (17) and (49) lead to the following integral inequality:while (50) givesFrom (48), we can writeAdopting the same method as we did for (49), the following integral inequality holds:From (51) and (53), (45) can be achieved.
An exponentially convex function satisfies the following inequality:From which, we can writeInequalities (21) and (55) lead the following integral inequality:while (56) givesFrom (54), we can writeAdopting the same method as we did for (55), the following inequality holds:From (57) and (59), (46) can be achieved.

3. Hadamard Type Inequalities for Exponentially Convex Function

In this section, we prove the Hadamard type inequality for an exponentially convex function. In order to prove this inequality result, we need the following lemma.

Lemma 1. (see [30]). Let be an exponentially convex function. If is exponentially symmetric, then the following inequality holds:

Theorem 4. Let be positive, exponentially convex, and symmetric about and be a differentiable and strictly increasing function. Also, let be an increasing function on . Then, for , , the following estimations of Hadamard type are valid.where for and for .

Proof. For the kernel of integral operator (11), we haveAn exponentially convex function satisfies the following inequality:Inequalities (62) and (63) lead the following integral inequality:while (64) givesOn the contrary, for the kernel of integral operator (12), we haveInequalities (63) and (66) lead the following integral inequality:while the abovementioned inequality givesFrom (65) and (68), the following inequality can be obtained:Now, using Lemma 1 and multiplying (60) with , then integrating over , we haveFrom which, we haveAgain using Lemma 1 and multiplying (60) with , then integrating over , we haveFrom which, we haveFrom (71) and (73), the following inequality can be achieved:From (69) and (74), (61) can be achieved.

Remark 3. For , in (61), Theorem 3 in [28] can be achieved.

Corollary 6. If we put , then the inequality (61) produces the following Hadamard type inequality:

Corollary 7. If we put , then the inequality (61) produces the following Hadamard type inequality:

Corollary 8. If we put and as identity function, then the inequality (61) produces the following Hadamard type inequality:

Corollary 9. If we put and as identity function, then the inequality (61) produces the following Hadamard type inequality:

4. Concluding Remarks

We have studied an integral operator for exponentially convex functions; this operator has direct consequences to several fractional and conformable integral operators. We have obtained bounds of the integral operator in different forms. In Theorem 1, upper bounds of this operator are studied for an exponentially convex function and several special cases have been presented in the form of propositions and corollaries. The boundedness is studied in Theorem 2. In Theorem 3, we have obtained results for differentiable function such that is exponentially convex. A version of the Hadamard inequality is proved in Theorem 4 which leads to its several variants for fractional and conformable integral operators.

Data Availability

No data were used to support this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the Basic Public Welfare Research Projects in Zhejiang Province (LGG 19F050003).

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