Some New Tempered Fractional Pólya-Szegö and Chebyshev-Type Inequalities with Respect to Another Function

Rahman, Gauhar; Nisar, Kottakkaran Sooppy; Abdeljawad, Thabet; Samraiz, Muhammad

doi:https://doi.org/10.1155/2020/9858671

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Fractional Calculus and Related Inequalities

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Volume 2020 | Article ID 9858671 | https://doi.org/10.1155/2020/9858671

Some New Tempered Fractional Pólya-Szegö and Chebyshev-Type Inequalities with Respect to Another Function

Gauhar Rahman,¹Kottakkaran Sooppy Nisar,²Thabet Abdeljawad,^3,4,5and Muhammad Samraiz⁶

Academic Editor: Tepper L Gill

Received18 Apr 2020

Accepted03 Aug 2020

Published02 Nov 2020

Abstract

In this present article, we establish certain new Pólya–Szegö-type tempered fractional integral inequalities by considering the generalized tempered fractional integral concerning another function in the kernel. We then prove certain new Chebyshev-type tempered fractional integral inequalities for the said operator with the help of newly established Pólya–Szegö-type tempered fractional integral inequalities. Also, some new particular cases in the sense of classical tempered fractional integrals are discussed. Additionally, examples of constructing bounded functions are considered. Furthermore, one can easily form new inequalities for Katugampola fractional integrals, generalized Riemann–Liouville fractional integral concerning another function in the kernel, and generalized fractional conformable integral by applying different conditions.

1. Introduction

The well-known Chebyshev functional [1] is defined bywhere the functions and are integrable on . If the functions and are synchronous on , i.e.,for any , then . Functional (1) has gained more recognition due to its diverse applications in the fields of transform theory, numerical quadrature, probability, and statistical problems. Additionally, the researchers have established a large number of integral inequalities by utilizing functional (1). The interesting readers may consult [2–5]. In [6], Tassaddiq et al. recently established certain inequalities via fractional conformable integrals by considering functional (1).

In [7], Grüss introduced the following inequality:where the functions and are integrable on such that and satisfy the inequalities and , for all and for some constant .

In [8], Pólya–Szegö presented the following inequality:

In [9], Dragomir and Diamond presented the following inequality with the help of Pólya–Szegö inequality:where the functions and are positive and integrable on such that and satisfy the inequalities and , for all and for some constant .

In the last few decades, the researchers have considered that fractional integral inequalities are the most powerful tools for the development of both applied and pure mathematics. In [10], the authors presented some Grüss-type integral inequalities by considering fractional integrals. Some new integral inequalities in sense of Riemann–Liouville fractional integrals can be found in the work of Dahmani [11].

In [12], Sarikaya et al. gave the idea of generalized -fractional integrals with applications. Set et al. [13] investigated some Grüss-type inequalities by considering generalized -fractional integrals.

Very recently, the idea of fractional conformable and proportional fractional integral operators was proposed by Jarad et al. [14, 15]. Later on, Huang et al. [16] presented generalized Hermite–Hadamard-type inequalities by considering generalized fractional conformable integrals. In [17], Qi et al. established Chebyshev-type inequalities for generalized fractional conformable integrals.

In [18], Ntouyas et al. investigated some new Pólya–Szegö- and Chebyshev-type inequalities by considering Riemann–Liouville fractional integrals. The tempered fractional integral was first studied by Buschman [19], but Li et al. [20] and Meerschaert et al. [21] have described the associated tempered fractional calculus more explicitly. Fernandez and Ustaoǧlu [22] investigated several analytic properties of the tempered fractional integral. In [23], Fahad et al. proposed the general form of the generalized tempered fractional integral concerning another function. In this paper, we investigate the said inequalities for the so-called tempered fractional integrals containing another function in the kernel.

The structure of the paper as follows.

In Section 2, some basic definitions are presented. Some new Pólya–Szegö-type for the so-called generalized tempered fractional integral in the sense of another function is presented in Section 3. In Section 4, we present some new generalized Chebyshev-type tempered fractional integral inequalities. In Section 5, certain new particular cases in terms of classical tempered fractional integrals are discussed. An example of constructing bounding functions is considered in Section 6. Finally, the concluding remarks are discussed in Section 7.

2. Preliminaries

In this section, we consider some well-known definitions and mathematical preliminaries.

Definition 1. (see [7]). Suppose that the functions are positive with and , for all , then the following inequality holds:where the constants , , and is the sharp of inequality (6).

Definition 2. (see [24, 25]). The function will be in the space ifIf we apply , then (7) gives

Definition 3. (see [26]). Suppose that the function and assume that the function is positive, monotone, and increasing on and having continuous derivative on with . Then, the Lebesgue real-valued measurable function defined on is said to be in the space , ifWhen , thenNote that, the space coincides with the space if for and similarly with the space if for .

Definition 4. (see [20–22]). The left-sided tempered fractional integral of order and with and is defined by

Remark 1. By setting in (11) yields the following Riemann–Liouville fractional integral, which is defined byThe following results for (11) hold:We define the following one-sided tempered fractional integral.

Definition 5. The one-sided tempered fractional integral of order , is defined by

Definition 6. (see [23]). Let the function be an integrable in the space and assume that the function is positive, monotone, and increasing on , and its derivative is continuous on with . Then, the left-sided generalized tempered fractional integral of the function concerning another function in the kernel is defined bywhere , with , and is the well-known gamma function.

Remark 2. The following results can be obtained:(i)Applying Definition 6 for , we get (11).(ii)Applying Definition 6 for , then it will reduce to the left-sided generalized Riemann–Liouville fractional integral operator [27].(iii)Applying Definition 6 for , then it will reduce to the following left-sided Hadamard tempered integral defined by [23](iv)Applying Definition 6 for , and , then it will reduce to the left-sided Katugampola [24] fractional integral.(v)Applying Definition 6 for and , the left Riemann–Liouville fractional integral (12) will be obtained(vi)Applying Definition 6 for and (where , and ), then it reduces to the left-sided generalized fractional conformable integral given by [28](vii)Applying Definition 6 for , , then it reduces to the fractional conformable integral defined by Jarad et al. [14].In this article, we consider the following one-sided GTF-integral.

Definition 7. Let the function be integrable in the space and assume that the function is positive, monotone, and increasing on , and its derivative is continuous on with . Then, the one-sided generalized tempered fractional integral of the function concerning another function in the kernel is defined by

Definition 8. For , we define the following subintegrals for the generalized tempered integralNote that

Remark 3. If we set and , then (18) will reduce to the subintegrals of Riemann–Liouville fractional integral defined by [18].

3. Pólya–Szegö-Type Tempered Fractional Integral Inequalities

In this section, we provide some new Pólya–Szegö-type tempered fractional integral inequalities for positive and integrable functions via tempered fractional integral (17) containing another function in the kernel.

Lemma 1. Let the functions and be positive and integrable on and assume that the function is positive, monotone, and increasing on , and its derivative is continuous on with . Suppose that and are four positive and integrable functions on such thatThen, for , , and , the following tempered fractional integral inequality holds:

Proof. From the given hypothesis, we haveSimilarly, we haveTaking product of (22) and (23), we getFrom (24), it can be written asNow, taking product of (25) with and integrating the resultant identity with respect to over , we haveWith the aid of Definition 8, we can writeBy applying AM-GM inequality, i.e., , we getIt follows thatwhich gives the desired assertion (21).

Corollary 1. Let the functions and be positive and integrable on and assume that the function is positive, monotone, and increasing on , and its derivative is continuous on with . Suppose that , and are four positive and integrable functions on such thatThen, for , , and , the following tempered fractional integral inequality holds:

Lemma 2. Let all the conditions of Lemma 1 hold. Then, for , , and , the following tempered fractional integral inequality holds:

Proof. From the hypothesis defined by (20), we havewhich follows thatTaking product on both sides of (18) by , we obtainTaking product of both sides of (36) with and integrating the resultant identity with respect to over , we havewhich by applying (8) becomesAgain, taking product of both sides of (38) with and integrating the resultant identity with respect to over and then applying (17), we getBy using AM-GM inequality, we getIt follows thatwhich completes the desired assertion (17).

Corollary 2. Let the functions and be positive and integrable on satisfying the hypothesis defined by (16) and assume that the function is positive, monotone, and increasing on , and its derivative is continuous on with . Then, for , , and , the following tempered fractional integral inequality holds:whereand is the well-known incomplete gamma function (see [22]).

Lemma 3. Suppose that all the conditions of Lemma 1 hold and assume that the function is positive, monotone, and increasing on , and its derivative is continuous on with . Then, for , , and , the following tempered fractional integral inequality holds:

Proof. From the hypothesis defined by (20), we havewhich in view of (17) yieldsSimilarly, one can obtainHence, the product of (46) and (47) yields the desired assertion (44).

Corollary 3. Let the functions and be positive and integrable on satisfying the hypothesis defined by (30) and assume that the function is positive, monotone, and increasing on , and its derivative is continuous on with . Then, for , , and , the following tempered fractional integral inequality holds:

4. Chebyshev-Type Tempered Fractional Integral Inequalities

In this section, certain Chebyshev-type inequalities via tempered fractional integral (20) are presented with the help of Pólya–Szegö integral inequality given by Lemma 1.

Theorem 1. Let the functions and be positive and integrable on and assume that the function is positive, monotone, and increasing on , and its derivative is continuous on with . Suppose that , and are four positive and integrable functions on satisfying the hypothesis defined by (10). Then, for , , and , the following tempered fractional integral inequality holds:where

Proof. By the given hypothesis, both the functions and are positive and integrable functions on . Therefore, for with , we define byMultiplying (52) by and double integrating the resultant identity with respect to and over and then using (17), we obtainBy applying Cauchy–Schwartz inequality for double integrals, we haveIn view of (17) and (43), we getApplying Lemma 1 for , we getIt follows thatSimilarly, one can getAgain applying Lemma 1 for , we getThus, by considering (53) to (60), we arrive at the desired assertion (49). This is the desired proof of Theorem 1.

Theorem 2. Suppose that all the conditions of Theorem 1 are satisfied. Then, for , , and , the following tempered fractional integral inequality holds:where

Proof. Applying Theorem 1 for , we get the desired result in (61).

Remark 4. If we consider , , , and , then we have

Corollary 4. Let the functions and be positive and integrable on and satisfying the hypothesis given by (30). Then, for , , and , the following tempered fractional integral inequality holds:

5. Particular Cases

The following new Pólya–Szegö- and Chebyshev-type inequalities for classical tempered fractional integral (14) can be easily established.

Lemma 4. Let the functions and be positive and integrable on . Suppose that , and are four positive and integrable functions on satisfying the hypothesis defined by (20). Then, for , , and , the following tempered fractional integral inequality holds:

Proof. Applying Lemma 1 for , we get Lemma 4.

Lemma 5. Let all the conditions of Lemma 4 are satisfied. Then, for , , and , the following tempered fractional integral inequality holds:

Proof. Applying Lemma 2 for , we get Lemma 5.

Similarly, we can derive the particular case of Lemma 3. The following theorem represents the particular case of Theorem 1 in terms of classical tempered fractional integral.

Theorem 3. Let the functions and be positive and integrable on . Suppose that , and are four positive and integrable functions on satisfying the hypothesis defined by (20). Then, for , , and , the following tempered fractional integral inequality holds:where

Proof. Applying Theorem 1 for , then we get the desired Theorem 3.

Similarly, we can derive particular result of Theorem 2.

6. Applications

In this section, we define a way for constructing four bounded functions and then utilize them to present certain estimates of Chebyshev-type tempered fractional integral inequalities of two unknown functions.

Let the unit function be defined byand let the Heaviside unit step function be defined by

Suppose that the function is piecewise continuous function on defined bywhere and . Similarly, we definewhere the constants . If there exists an integrable function on satisfying the hypothesis , then we have for each , .

Proposition 1. Let the functions and be two positive and integrable on . Assume that the functions , , , and are defined by (73)–(76), respectively, and satisfying the hypothesis defined by (30). Then, for , the following inequality for tempered fractional integral holds:

Proof. By applying the Definition 8, we haveHence, by applying Lemma 1, we get the desired assertion (77).

Proposition 2. By setting in Proposition 1, then we arrive to the following result in terms of classical tempered fractional integral:

Remark 5. Throughout in the paper, if we apply and , then all the newly presented inequalities will be reduced to the work derived earlier by Ntouyas et al. [18].

7. Concluding Remarks

Certain new Pólya–Szegö- and Chebyshev-type inequalities by utilizing tempered fractional integral are presented in this paper. These inequalities generalized the existing inequalities. We can easily get the said Pólya–Szegö- and Chebyshev-type inequalities for Katugampola, generalized Riemann–Liouville, classical Riemann–Liouville, generalized conformable, and conformable fractional integrals by applying different conditions on function given in Remark 2.

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 the authors contributed equally, and they have read and approved the final manuscript for publication.

Acknowledgments

Thabet Abdeljawad would like to thank Prince Sultan University for the support through the research group: Nonlinear Analysis Methods in Applied Mathematics (NAMAM) (group number RG-DES-2017-01-17).

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Copyright © 2020 Gauhar Rahman 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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