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Fractional Calculus and Related Inequalities

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Volume 2020 |Article ID 4189036 | https://doi.org/10.1155/2020/4189036

Yu-Ming Chu, Muhammad Uzair Awan, Muhammad Zakria Javad, Awais Gul Khan, "Bounds for the Remainder in Simpson’s Inequality via -Polynomial Convex Functions of Higher Order Using Katugampola Fractional Integrals", Journal of Mathematics, vol. 2020, Article ID 4189036, 10 pages, 2020. https://doi.org/10.1155/2020/4189036

Bounds for the Remainder in Simpson’s Inequality via -Polynomial Convex Functions of Higher Order Using Katugampola Fractional Integrals

Academic Editor: Efthymios G. Tsionas
Received12 May 2020
Revised24 Jul 2020
Accepted24 Jul 2020
Published24 Aug 2020

Abstract

The goal of this paper is to derive some new variants of Simpson’s inequality using the class of -polynomial convex functions of higher order. To obtain the main results of the paper, we first derive a new generalized fractional integral identity utilizing the concepts of Katugampola fractional integrals. This new fractional integral identity will serve as an auxiliary result in the development of the main results of this paper.

1. Introduction and Preliminaries

The following inequality known in the literature as Simpson’s inequality [1].where is a four times continuously differentiable function on and

Simpson’s inequality plays a significant role in analysis [24]. Over the years, it has been extended and generalized in different directions using novel and innovative approaches. The survey by Dragomir et al. [5] is very informative regarding the developments of Simpson’s inequality and its applications.

In recent years, the fractional calculus [610] is often known as noninteger calculus which has become a powerful tool in mathematics because it provides a good tool to describe physical memory. Fractional calculus has wide applications in real life through its help in solving different physical problems [1120]. The classic definition of Riemann–Liouville fractional integrals is one of the most basic concepts in fractional calculus which is defined as:

Definition 1. Let . Then Riemann–Liouville integrals and of order with are defined bywhereis a well known gamma function.
In past few decades, several successful attempts have been made in generalizing the classical concepts of fractional calculus. Erdelyi–Kober operator is a significant generalization of fractional integrals introduced and was studied by Arthur Erdelyi and Hermann Kober. But there is a drawback that one cannot get the Hadamard version of the derivatives and integrals from Erdelyi–Kober operators. Katugampola [21] gave a well-defined concept of fractional integrals as:

Definition 2. Let be a finite interval. Then the left and right sides of Katugampola fractional integrals of order of are defined by if the integral exists.
If we take , then we can recapture Riemann–Liouville fractional integrals from the Katugampola fractional integrals. It worth to mention here that Erdelyi–Kober operators and Katugampola fractional integrals are not equivalent to each other.
Sarikaya et al. [22] are the first authors to utilize the concepts of Riemann–Liouville fractional integrals in obtaining the fractional analogues of Hermite–Hadamard’s inequality. This idea inspired several inequalities expert, and resultantly huge number of articles have been written on the fractional analogues of classical inequalities. For example, Hu et al. [23] obtained some new fractional analogues of integral inequalities using Katugampola fractional integrals. Nie et al. [24] obtained -fractional analogues of Simpson’s inequality. Peng et al. [25] also obtained some new fractional analogues of Simpson’s inequality. Set [26] obtained fractional analogues of Ostrowski’s inequality. Wu et al. [27] obtained fractional analogues of inequalities using -th order differentiable functions. Kermausuor [28] obtained new Simpson type inequalities involving Katugampola fractional integrals essentially using the class of Breckner type -convex function.
Recently, Toplu et al. [29] introduced the notion of -polynomial convex functions as follows.

Definition 3. Let . Then a nonnegative function is said to be a -polynomial convex function if the inequalityholds for every and .
Many researchers have also derived several new Hermite–Hadamard’s like inequalities [3036] using the concept of -polynomial convex functions. We would like to point out here that the class of -polynomial convex functions generalize the class of convex functions if we take , then we have the class of 1-polynomial convex functions which is just the classical convex functions. Also we can get other type of convexities: for example, for , we have 2-polynomial convexity. Another point of pondering here is that every -polynomial convex function is an -convex function with the function . So more generally, every nonnegative convex function is also an -polynomial convex function.
The idea behind the study of this article is to extend the notion of -polynomial convex functions with the introduction of higher order -polynomial convex functions. We derive a new fractional integral identity using the concepts of Katugampola fractional integrals. This new identity will serve as an auxiliary result in the development of some new fractional analogues of Simpson’s inequalities using the concept of -polynomial convex functions of higher order.
Before we move to our main results, we would like to introduce the notion of -polynomial convex functions of higher order.

Definition 4. Let . Then a nonnegative function is said to be a higher order -polynomial convex function if the inequalityholds for every , , and .

Remark 1. Note that if in (7), then the class of -polynomial convex functions of higher order reduces to the class of -polynomial convex functions. If , then we have a new class of strongly -polynomial convex functions. If , then we have the class of higher-order convex functions [37]. And along with , if we have , then the class of -polynomial convex functions of higher order reduces to the class of strongly convex functions [38]. From this, it is evident that the class of -polynomial convex functions of higher order is quite unifying one as it relates several other unrelated classes of convexity [3945].

2. Main Results

In this section, we discuss our main results.

2.1. A Key Lemma

We now derive the main auxiliary result of the paper.

Lemma 1. Let , and let be a differentiable function on , with such that , then

Proof. ConsiderIntegrating by parts , we haveSimilarly,By substituting the values of and in , we get the required result.
This completes the proof.

2.2. Results and Discussions

We now derive the main results using Lemma 1.

Theorem 1. Let and let be a differentiable function on , with such that . If is higher-order -polynomial convex function of order , then

Proof. By using Lemma 1 and because is higher-order -polynomial convex function of order , we haveNote thatSimilarly,where we have used the fact that for all . Using and in (13), we get the required result.

Theorem 2. Let , and let be a differentiable function on , with such that . If is higher-order -polynomial convex function of order , where and , then

Proof. Using Lemma 1, Holder’s inequality and because is a higher-order -polynomial convex function of order , we haveNote thatSimilarly,where for all . Using and in (17), we get the required result.

Theorem 3. Let , and let be a differentiable function on , with such that . If is a higher-order -polynomial convex function of order where , then

Proof. Using Lemma 1, Power mean inequality, and because is a higher-order -polynomial convex function of order , we have Note thatSimilarly,where we have used the fact that for all . Using and in (21), we get the required result.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgments

This work was supported by the Natural Science Foundation of China (Grant nos. 61673169, 11701176, 11626101, and 11601485).

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Copyright © 2020 Yu-Ming Chu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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