On New Modifications Governed by Quantum Hahn’s Integral Operator Pertaining to Fractional Calculus

Abstract

In the article, we present several generalizations for the generalized Čebyšev type inequality in the frame of quantum fractional Hahn’s integral operator by using the quantum shift operator . As applications, we provide some associated variants to illustrate the efficiency of quantum Hahn’s integral operator and compare our obtained results and proposed technique with the previously known results and existing technique. Our ideas and approaches may lead to new directions in fractional quantum calculus theory.

1. Introduction

In 1882, Čebyšev discovered a fascinating and significantly valuable integral inequality as follows: if and are two integrable and synchronous functions on , where the functions and are said to be synchronous on if for all .

It is well known that the Čebyšev inequality (1) has wild applications in the fields of pure and applied mathematics [1–10]. Recently, the generalizations and variants for the Čebyšev inequality (1) have attracted the attention of many researchers [11–20].

Quantum difference operators are receiving an increase of interest due to their applications [21, 22]. Roughly speaking, quantum calculus can substitute the classical derivative by a difference operator, which allows to deal nondifferentiation functions.

Let , , be an interval such that , and be a real-valued function. Then, the Hahn difference operator [23] is defined by if is differentiable at .

The Hahn difference operator (3) unifies (in the limit) the Jackson -difference derivative [24] for and the forward difference for , which are defined by if exists for , and for .

The Hahn difference operator has been applied successfully in the construction of families of orthogonal polynomials as well as in approximation problems [25–28].

In [29], the authors introduced some concepts of fractional quantum calculus in terms of a -shifting operator .

Let be an interval. Then, the point of Hahn calculus on the interval generated by the quantum numbers and is given by

We state that for all consequences of our investigation; the quantum Hahn shifting operator is defined by and the iterated -times quantum shifting is given by with for .

Let us recall the basic knowledge of quantum Hahn calculus on an interval (see [30]).

Definition 1. Let be a function defined on . Then, the quantum Hahn difference operator is defined by if is differentiable at .

Definition 2. Let be a given function and . Then, the -quantum Hahn integral of from to is defined by where for provided that the series converge at and . The function is said to be -integrable on if (11) exists for all .

Before approaching the main definitions of fractional quantum Hahn calculus on , we present the -power function which is stated as

Precisely, if , then with for .

The -gamma function is defined as for .

Obviously, , where and is the quantum number.

Now, we introduce the concepts of fractional quantum Hahn derivative and integral of Riemann-Liouville type [31].

Definition 3 (see [31]). Suppose that and is a real-valued function. Then, the fractional quantum Hahn derivative of the Riemann-Liouville type of order is defined by where is the smallest integer greater than or equal to .

Definition 4 (see [31]). Let and be a real-valued function. Then, the fractional quantum Hahn integral of the Riemann-Liouville type of order is defined by with if the right-hand side exists.

Theorem 5 (see [31]). Let , , and . Then, one has

Fractional calculus is invariably important in almost all fields of mathematics and applied sciences. Also, the fractional differential equations can provide adequate models for many physical problems in areas such as heat equation, wave equation, Poisson equation and Laplace equation, fluid mechanics, biological populations, viscoelasticity, advection-diffusion, and signal processing [32, 33].

Inequality plays an irreplaceable role in the development of mathematics. Very recently, many new inequalities such as Hermite-Hadamard type inequality [34–38], Petrović type inequality [39], Pólya-Szegö type inequality [40], Ostrowski type inequality [41], reverse Minkowski inequality [42], Jensen type inequality [43, 44], Bessel function inequality [45], trigonometric and hyperbolic function inequalities [46], fractional integral inequality [47–51], complete and generalized elliptic integral inequalities [52–57], generalized convex function inequality [58–60], and mean value inequality [61–63] have been discovered by many researchers. In particular, the applications of integral inequalities have gained considerable importance among researchers for fixed-point theorems; the existence and uniqueness of solutions for differential equations [64–68] and numerous numerical and analytical methods have been recommended for the advancement of integral inequalities [69–75].

Asawasamrit et al. [76] expounded the concept of -derivative over the interval and derived several inequalities on quantum analogues, for example, -Cauchy-Schwarz inequality, -Grüss-Čebyšev integral inequality, -Grüss inequality, and other integral inequalities, by use of the convexity theory.

The main purpose of the article is to provide the novel versions of the generalized Čebyšev inequalities and present the associated variants via quantum Hahn’s fractional integral operator.

To end this section, we give the definition of the one-sided fractional quantum Hahn integral in the Riemann-Liouville sense.

Definition 6. Let and be a real-valued function. Then, the one-sided fractional quantum Hahn integral of Riemann-Liouville type of order is defined by

2. Certain Extended Weighted Čebyšev Fractional Quantum Hahn Integral Operator

In this section, we provide several new generalizations for the weighted extensions of Čebyšev functionals via a quantum Hahn integral operator.

Theorem 7. Let with , , , be a positive -integrable function defined on , and and be two -differentiable functions defined on such that and . Then, the inequalities hold for all .

Proof. Let and Then, can be written as Multiplying both sides of (21) by and then performing the -integration with respect to over , we have Inequality (22) can be rewritten as Multiplying both sides of (23) by and then performing the -integration with respect to over , we get Similarly, we have Taking into account the Hölder inequality, we have It follows from (26) and (27) that From (24) and (28), we obtain Making use of the Hölder inequality for bivariate integral, we have It follows from (30) and the inequalities that Therefore, we get the desired inequality

Let . Then, Theorem 7 leads to Corollary 8 which provide a new result for -fractional integral operator.hold for all

Corollary 8. Let with , , be a positive -integrable function defined on , and and be two -differentiable functions defined on such that and . Then, the inequalities hold for all .

Remark 9. If , then Theorem 7 reduces to the result for the Riemann-Liouville fractional integral operator given in [77]. Some results given in the literature [13, 78] can also be obtained from Theorem 7 immediately.

Theorem 10. Let with , , , and be the positive -integrable functions defined on , and and be the -differentiable functions defined on such that and . Then, the inequality holds for all .

Proof. Multiplying both sides of (23) by and performing -integration with respect to over , we have Taking modulus on both sides of (36), one has

Let for . Then, Theorem 10 leads to Corollary 11 which provide a new result for -fractional integral operator.

Corollary 11. Let with , , and be the positive -integrable functions defined on , and and be the -differentiable functions defined on such that and . Then, the inequality holds for all .

Remark 12. Let . Then, Theorem 10 becomes Theorem 14 of [77].

3. Some New Generalizations by Fractional Quantum Hahn Integral Operator

Theorem 13. Let with , and for , and and be the the positive -integral functions. Then the following inequalities
(A1)
(A2)
(A3)
(A4)
hold for all .

Proof. Considering the well-known Young inequality for all and with .
Substituting and into (39) gives Multiplying both sides of (40) by and then performing the -integration with respect to and over lead to the conclusion that Consequently, we have which implies (A₁). The remaining inequalities can be derived by adopting the similar argument and accompanying the selection of parameters in Young inequality as follows:
(A₂)
(A₃)
(A₄)

Theorem 14. Let with , and for , and and be the positive -integral functions defined on . Then, the inequalities.
(A₅)
(A₆)
(A₇)
(A₈)
hold for all .

Proof. Considering the well-known weighted inequality Substituting and yields Conducting product on both sides of (45) by and then performing the -integration with respect to and over , we obtain Consequently, we have which implies (A₅). The rest of variants can be derived by adopting the similar strategy and accompanying the selection of parameters in inequality as follows:
(A₆)
(A₇)
(A₈)

Theorem 15. Let , , and be the positive -integrable functions defined on , and Then, the inequalities
(A₉)
(A₁₀)
(A₁₁)
hold for all .

Proof. From (49) and (50), we clearly see that Conducting product on both sides of (51) by and then performing the -integration with respect to over yield It follows from and that From (52) and (54), we conclude that which implies (A₉). By making few changes in (A₉), we can get (A₁₀) and (A₁₁).

Theorem 16. Let , , and and be the positive -integrable functions defined on such that Then, the inequalities
(A₁₂)
(A₁₃)
(A₁₄)(A₁₄)
hold for all .

Proof. It follows from (56) and (57) that The proof can be derived by following Theorem 15.

Theorem 17. Let with , , , and be the positive -integrable functions defined on such that (56) and (57) hold. Then, the inequalities
(A₁₅)
(A₁₆)(A₁₆)
hold for all .

Proof. It follows from that Multiplying both sides of (59) by , we have Taking into account the well-known inequality for (60), we get which implies (A₁₅).
Replacing, respectively, and by and in (61) and (56), we attain the required inequality (A₁₆):

Theorem 18. Let , , and and be the -integrable functions defined on such that and (56) holds. Then, for all , we have
(A₁₇)(A₁₇)
(A₁₈)(A₁₈)
(A₁₉)(A₁₉)

Proof. We clearly see that Inequality (63) can be written as Conducting product on both sides of (64) by and then performing the -integration with respect to over yield Also, by Cauchy inequality, we get Multiplying both sides of the inequality (66) by , we get .
Alternately, it follows from and that From and (68), we get which conclude . Similarly, we can prove the inequality .

Theorem 19. Let , , and and be the -integrable functions defined on such that and for all . Then, for all , we have

Proof. Under the given assumption, we have which implies that Conducting product on both sides of (75) by and then performing the -integration with respect to and over , we get (71).
From (65), (71), and the Cauchy inequality we get which completes the proof of (72) and (73).

4. Conclusion

We have discovered several generalizations for the generalized Čebyšev type inequality via quantum fractional Hahn’s integral operator by using the quantum shift operator , provided some associated variants to show the efficiency of quantum Hahn’s integral operator, and compared our obtained results and proposed technique with the previously known results and existing technique. The outcome shows that the proposed plans are extremely important and computationally appealing to deal with several sorts of differential equations. As a future research course of this paper, the new techniques obtained in the present paper can be prolonged to attain analytical solutions of quantum mechanics introduced in different works distributed currently connected with high-dimensional fractional equations.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally to 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, 11971142, 11871202, 11301127, 11701176, 11626101, and 11601485).