The Nonlocal Fractal Integral Reverse Minkowski’s and Other Related Inequalities on Fractal Sets

Rahman, Gauhar; Nisar, Kottakkaran Sooppy; Golamankaneh, Alireza Khalili

doi:https://doi.org/10.1155/2021/4764891

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2021 | Article ID 4764891 | https://doi.org/10.1155/2021/4764891

The Nonlocal Fractal Integral Reverse Minkowski’s and Other Related Inequalities on Fractal Sets

Gauhar Rahman,¹Kottakkaran Sooppy Nisar,²and Alireza Khalili Golamankaneh³

Academic Editor: Giovanni Falsone

Received17 Aug 2020

Revised23 Jan 2021

Accepted28 Jan 2021

Published16 Feb 2021

Abstract

In this paper, we study the generalized Riemann–Liouville fractional integral for the functions with fractal support. The aim of this article is to investigate reverse Minkowski’s inequalities and certain other related inequalities by employing the generalized Riemann–Liouville fractional integral for the functions with fractal support.

1. Introduction

Fractional calculus involves integrals and derivatives of arbitrary order. The applications of fractional calculus have been found in the field of several sciences and engineering [1–8]. In [1, 2], the nonlocal fractional integrals and derivatives are utilized to model the processes with memory effect. In [9–11], the researchers used the nonlocal derivatives to model more appropriately the dynamics of the nonconservative systems in formulation of Hamilton and Lagrange.

Fractal analysis has been studied by many researchers by using measure theory, harmonic analysis, stochastic process, fractional spaces, and other techniques [12–35].

Recently, Parvate and Gangal [36–41] proposed the -calculus on the fractal subset of real line and fractal curves. The researchers have applied transport materials on disordered systems such as fractal sets and fractal curves [42–44]. In [45], the researchers established Schrödinger’s equation on a fractal curve.

Such new developments in fractional calculus encourage future research to investigate new innovative ideas to unify the fractional operators and establish inequalities involving new fractional operators. The fractional integral inequalities (FII in short) and their applications play an important role in the field of applied mathematics. A wide number of integral inequalities and their extensions were built in the sense of classical fractional integral and fractional derivative operators (see, e.g., [46–50]).

The inequalities, especially the Hölder, the reverse Minkowski, the arithmetic, and geometric inequalities, have played a key role in the field of both pure as well as applied mathematics. These and several other essential inequalities are now in common use and, therefore, it is not surprising that several studies associated to these areas have been made in order to accomplish a diversity of desired goals. In the past few decades, the theory of inequalities has established rapidly and unexpected results were investigated, along with simpler new proofs for existing results, and, accordingly, new direction for research has been opened up. Recently, the theory of inequalities has gained more considerable interest from many mathematicians, and a large number of new inequalities have been estimated in the literature. It is recognized that, in general, some specific inequalities provide a useful and important device in the development of different branches of mathematics. In [51], Dahmani has investigated the reverse Minkowski fractional integral inequalities. Sousa and Capelas de Oliveira [52] have investigated the reverse Minkowski inequalities and certain other related inequalities for Katugampola fractional Integral operators. In [53, 54], the authors have studied the reverse Minkowski inequalities by considering Hadamard fractional integral operators. In this present article, we study the said inequalities by considering the generalized Riemann–Liouville fractional integral for the functions with fractal support.

The structure of the paper is follows.

In Section 2, we have given some known results and basic definitions. In Section 3, the nonlocal reverse Minkowski inequalities are presented for nonlocal fractal integrals on fractal subset of real line. In Section 4, some other related inequalities for nonlocal fractal integrals on fractal subset of real line are presented.

2. Preliminaries

Some well-known basic definitions and results associated with classical fractional integrals and generalized fractional integrals are presented in this section. The reverse Minkowski’s integral inequalities can be found in the work of [27, 55]. The reverse Minkowski’s inequalities are the motivation of the work performed so far, involving the classical Riemann integrals which are presented by the following theorems.

Theorem 1. (see [55]). Let the two functions and be positive on and . If , , then the following inequality holds:

Theorem 2. (see [55]). Let the two functions and be positive on and . If , , then the following inequality holds:

Definition 1. (see [5, 6]). The well-known classical fractional integrals of order are, respectively, defined byandwhere with .

Dahmani [51] has investigated the following inequalities by using classical fractional integral.

Theorem 3. (see [51]). Let the two functions and be positive on such that, for all , , . If , , then the following inequality holds:, , .

Theorem 4. (see [51]). Let the two functions , be positive on such that, for all , , . If , , then the following inequality holds:, , .

In [1], it is shown that the geometry of fractal is the geometry of real world. In [36, 37, 39], Parvate and Gangal proposed the calculus on fractals which is related to Riemann integrals.

Definition 2. (see [36, 37, 39]). For the thin Cantor set, the following integral staircase function is defined bywhere is the -dimension of thin Cantor set.

Definition 3. (see [36, 37, 39]). The -derivative is defined bywhen the limit exists.

Definition 4. The Gamma function with the fractal support is defined bywhere .

Here, we review the following nonlocal fractal integral operators for the functions with fractal support [56, 57].

Definition 5. If (-order differentiable function on ) and , then the left- and right-sided Riemann–Liouville fractal integral operators of order are, respectively, defined by [56, 57]andwhere is defined in (7), is the staircase function, and is the fractal set with -dimension (see, e.g., [56, 57]).

Remark 1. If we consider in (10) and (11), then we get (3) and (4), respectively.

One can easily prove the following lemma [56, 57].

Lemma 1.

3. The Nonlocal Reverse Minkowski Inequalities on Fractal Sets

In this section, we present the nonlocal fractal reverse Minkowski integral inequalities in the fractal support by using the generalized nonlocal fractal integral operator. The nonlocal reverse Minkowski fractal integral inequalities in fractal support are presented in the following theorems.

Theorem 5. Let , , , and and , (-order differentiable functions on ) be two positive functions on such that, for all , and . If , , then the following inequality holds:

Proof. Under the given hypothesis of Theorem 5, , , , we haveConsider a function:We conclude that the function is positive for all , , as each term of defined in (15) is positive in view of hypothesis of Theorem 5.
Therefore, conducting product on both sides of (14) by and integrating the estimated inequality with respect to from to , we havewhich can be written asHence, it follows thatNow, utilizing the condition , we haveIt follows thatAgain, conducting product on both sides of (20) by and integrating the estimated inequality with respect to from to , we obtainThus, by adding inequalities (18) and (21) yields the desired inequality.

Theorem 6. Let , , and let and (-order differentiable functions on ) be two positive functions on such that, for all , and . If , , then the following inequality holds:

Proof. The multiplication of inequalities (18) and (21) yieldsBy utilizing the Minkowski inequality to the right-hand side of [17], we haveThus, from inequalities (23) and (24), we get the desired inequality (22).

This section is devoted to deriving certain related nonlocal fractal integral inequalities on the fractal set.

Theorem 7. Let , , and let and (-order differentiable functions on ) be two positive functions on such that and . If , , we have

Proof. Since , , therefore, we haveIt follows thatConducting multiplication on both sides of (27) by where is defined by (15) and integrating the estimated inequality with respect to from to , we haveIt follows thatConsequently, we haveOn the contrary, , ; therefore, we haveIt follows thatAgain, conducting multiplication on both sides of (32) by where is defined by (15) and integrating the estimated inequality with respect to from to , we haveHence, we can writeMultiplying (30) and (34), we get the desired inequality.

Theorem 8. Let , , and let and (-order differentiable functions on ) be two positive functions on such that and . If , , we have

Proof. Replacing and by and , in Theorem 7, and we get the desired inequality (35).

Theorem 9. Let , , and let and (-order differentiable functions on ) be two positive functions on such that and . If where , , then the following inequality for left nonlocal fractal integral with fractal support holds:

Proof. By the given hypothesis , we haveConducting multiplication on both sides of inequality (37) by where id defined by (15) and integrating the estimated inequality with respect to over , we obtainIt follows thatOn the contrary, using , , we haveAgain, conducting multiplication on both sides of inequality (40) by where is defined by (15) and integrating the estimated inequality with respect to over , we obtainNow, using Young’s inequality, we haveTaking product on both sides of inequality (40) by where id defined by (15) and integrating the resultant identity with respect to over to , we obtainWith the aid of (39) and (41), (43) can be written asNow, using the inequality , one can obtainandHence, the proof of (36) can be followed from (44)–(46).

Theorem 10. Let , , and let and (-order differentiable functions on ) be two positive functions on such that , . If where , , then we have

Proof. Under the given hypothesis , we haveIt can be written asAlso, we haveIt follows thatAlso, we haveIt follows thatConducting product on both sides of inequality (51) by where is defined by (15) and integrating the resultant identity with respect to over , we obtainIt follows thatAgain, conducting product on both sides of inequality (53) by where is defined by (15) and integrating the resultant identity with respect to over , we obtainHence, by adding inequalities (55) and (56), we get the desired inequality (47).

Theorem 11. Let , , and let and (-order differentiable functions on ) be two positive functions on such that , . If and for all , then we have

Proof. Under the given hypothesis, we haveThe product of inequality (58) with givesFrom (59), we obtainandNow, conducting product on both sides of (60) and (61), respectively, by where id is defined by (15) and integrating the estimated identity with respect to over , we obtainandHence, by adding (62) and (63), we get the desired proof.

Theorem 12. Let , , and let and (-order differentiable functions on ) be two positive functions on such that , . If where for all , then we have

Proof. Under the given hypothesis, , we haveAlso, we have , which givesThe multiplication of (65) and (66) yieldsNow, conducting multiplication on both sides of inequality (67) by where id is defined by (15) and integrating the resultant identity with respect to over , we haveIt follows thatwhich completes the desired proof.

Theorem 13. Let , , and let and (-order differentiable functions on ) be two positive functions on such that and . If where for all , then the following inequality for the left nonlocal fractal integral on fractal set holds:where .

Proof. Under the given hypothesis where , we haveandFrom (71) and (72), we havewhere . Also, from the given hypothesis , we haveandFrom (74) and (75), we obtainIt follows thatFrom (73) and (77), we can writeandNow, conducting multiplication on both sides of (78) and (79), respectively, by where id is defined by (15) and integrating the resultant identity with respect to over , we obtainIt follows thatSimilarly, from (79), we obtainHence, by summing (81) and (82), we obtain the required proof.

Remark 2. We note that all results lead to standard fractional calculus by setting , that is, .

5. Concluding Remarks

In this present investigation, we presented the nonlocal reverse Minkowski’s inequalities and some other inequalities for nonlocal fractal integral operator on fractal sets. The special cases of this work can be found in the work of [51, 58, 59].

Data Availability

No data were used in this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 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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Mathematical Problems in Engineering

The Nonlocal Fractal Integral Reverse Minkowski’s and Other Related Inequalities on Fractal Sets

Abstract

1. Introduction

2. Preliminaries

3. The Nonlocal Reverse Minkowski Inequalities on Fractal Sets

4. Certain Related Nonlocal Fractal Integral Inequalities on Fractal Sets

5. Concluding Remarks

Data Availability

Conflicts of Interest

References

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