Some Parameterized Quantum Simpson’s and Quantum Newton’s Integral Inequalities via Quantum Differentiable Convex Mappings

You, Xue Xiao; Ali, Muhammad Aamir; Budak, Hüseyin; Vivas-Cortez, Miguel; Qaisar, Shahid

doi:https://doi.org/10.1155/2021/5526726

Mathematical Problems in Engineering

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Abstract Introduction Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 5526726 | https://doi.org/10.1155/2021/5526726

Some Parameterized Quantum Simpson’s and Quantum Newton’s Integral Inequalities via Quantum Differentiable Convex Mappings

Xue Xiao You,¹Muhammad Aamir Ali,²Hüseyin Budak,³Miguel Vivas-Cortez,⁴and Shahid Qaisar⁵

Academic Editor: Nhon Nguyen-Thanh

Received22 Feb 2021

Accepted29 Mar 2021

Published26 Dec 2021

Abstract

In this work, two generalized quantum integral identities are proved by using some parameters. By utilizing these equalities, we present several parameterized quantum inequalities for convex mappings. These quantum inequalities generalize many of the important inequalities that exist in the literature, such as quantum trapezoid inequalities, quantum Simpson’s inequalities, and quantum Newton’s inequalities. We also give some new midpoint-type inequalities as special cases. The results in this work naturally generalize the results for the Riemann integral.

1. Introduction

Thomas Simpson has developed crucial methods for the numerical integration and estimation of definite integrals considered as Simpson’s rule during 1710–1761. Nevertheless, a similar approximation was used by J. Kepler almost one hundred years earlier, so it is also known as Kepler’s rule. Simpson’s rule includes the three-point Newton–Cotes quadrature rule, so estimation based on the three-step quadratic kernel is sometimes called Newton-type results.(1)Simpson’s quadrature formula (Simpson’s 1/3 rule):(2)Simpson’s second formula or Newton–Cotes quadrature formula (Simpson’s 3/8 rule):

There are a large number of estimations related to these quadrature rules in the literature; one of them is the following estimation known as Simpson’s inequality.

Theorem 1. Suppose that is a four-time continuously differentiable mapping on , and let . Then, one has the inequality

In recent years, many authors have focused on Simpson’s type inequality in various categories of mappings. Specifically, some mathematicians have worked on the results of Simpson’s and Newton’s type in obtaining a convex map because convexity theory is an effective and powerful way to solve a large number of problems from different branches of pure and applied mathematics. For example, Dragomir et al. [1] presented the new Simpson’s inequalities and their applications in quadrature formulas for numerical integration. In addition, some inequalities of Simpson’s type of s-convex functions were determined by Alomari et al. in [2]. Subsequently, Sarikaya et al. noted the variance of Simpson’s type inequality based on convexity in [3]. For the further studies of this area, one can consult [4–6].

On the contrary, many well-known integral inequalities have been studied in the setup of -calculus using the concept of classical convexity [7–28].

2. Preliminaries of -Calculus and Some Inequalities

In this section, we first present some known definitions and related inequalities in -calculus. Set the following notation (see [29]):

Jackson [30] defined the -integral of a given function from 0 to as follows:provided that the sum converges absolutely. Moreover, he defined the -integral of a given function over the interval as follows:

Definition 1 (see [31]). We consider the mapping . Then, the -derivative of at is defined by the following expression:If , we define if it exists and it is finite.

Definition 2 (see [32]). We consider the mapping . Then, the -derivative of at is defined byIf , we define if it exists and it is finite.

Definition 3 (see [31]). We consider the mapping . Then, the -definite integral on is defined byIn [11, 21], the authors proved quantum Hermite–Hadamard-type inequalities and their estimations by using the notions of the -derivative and -integral.
On the contrary, in [32], Bermudo et al. gave the following definition and obtained the related Hermite–Hadamard-type inequalities:

Definition 4 (see [32]). We consider the mapping . Then, the -definite integral on is defined by

Theorem 2 (see [32]). Let be a convex function on and . Then, -Hermite–Hadamard inequalities are given as follows:

In [33], Budak proved the left and right bounds of inequality (11).

The present paper aims to generalize the results proved in [13, 33, 34]. The key benefit of our paper is that it includes multiple inequalities at the same time, such as Simpson’s inequalities, Newton’s inequalities, midpoint inequalities, and trapezoidal inequities.

3. Crucial Identities

We deal with the three identities which are necessary to obtain our main results in this section.

Let us start with the following useful lemma.

Lemma 1. If is a -differentiable function on such that is continuous and integrable on , then we have the following identity:where .

Proof. By Definition 2, we see thatAfter applying the fundamental properties of quantum integrals, we deduce thatFrom Definition 4, we conclude thatandWe can obtain the required identity (12) by putting the computed integrals (15)–(17) in (14).

Remark 1. If we assume and in Lemma 1, then we obtain Lemma 2 of [13].

Remark 2. In Lemma 1, by taking the limit , we have Lemma 2.1 of [34] for .

Remark 3. In Lemma 1, if we choose , then we obtain the following identity:which is proved by Budak in Lemma 1 of [33].

Corollary 1. In Lemma 1, if we choose and , then we obtain the following new identity:

Lemma 2. If is a -differentiable function on such that is continuous and integrable on , then we have the following identity:where .

Proof. After applying the fundamental properties of quantum integrals, we deduce thatIf the same steps in the proof of Lemma 1 are applied for the rest of this proof, we can obtain the desired identity (20).

Remark 4. By assuming , , and in Lemma 2, we obtain Lemma 3 of [13].

Remark 5. If we take in Lemma 2, then we recapture identity (18).

Corollary 2. If we take the limit in Lemma 2, then we obtain the following new identity:

Now, we calculate the integrals in the following lemma which will be used in our next section.

Lemma 3. The subsequent quantum integrals are true:

Proof. Case I: let .
By the definition of the -integral, we haveCase II: let .
From the definition of the -integral, we getThis gives the proof of equality (28). In a similar way, we can prove the others.

4. Simpson’s Type Inequalities for Quantum Integrals

An extension of quantum Simpson’s inequalities for quantum differentiable convex functions using the quantum integrals is given in this section.

Theorem 3. We assume that the given conditions of Lemma 1 hold. If the mapping is convex on , then the following Simpson’s type inequality holds:where – are given in (28)–(30), respectively.

Proof. By Lemma 1 and the convexity of , we conclude thatwhich is the desired conclusion.

Remark 6. By taking the limit in Theorem 3, we have Theorem 2.1 of [34] for .

Remark 7. If we assume in Theorem 3, then we obtain the following trapezoidal type inequality:which is established by Budak in Theorem 3 of [33].

Corollary 3. In Theorem 3, if we choose and , then we obtain the following midpoint-type inequality:

Remark 8. In Theorem 3, if we assume and , then Theorem 3 reduces to Theorem 4 of [13].

Theorem 4. We assume that the given conditions of Lemma 1 hold. If the mapping , , is convex on , then the following Simpson’s type inequality holds:where , , and – are given in (23), (24), and (28)–(30), respectively.

Proof. From Lemma 1 and the power mean inequality, we obtainBy applying the convexity of , we obtainand the proof is completed.

Remark 9. If we take the limit in Theorem 4, then we have Theorem 2.3 of [34] for .

Remark 10. If we assume in Theorem 4, then we obtain the following trapezoidal type inequality:which is established by Budak in Theorem 4 of [33].

Remark 11. If we assume and in Theorem 4, then Theorem 4 reduces to Theorem 6 of [13].

Corollary 4. In Theorem 4, if we choose and , then we obtain the following midpoint-type inequality:

Theorem 5. Assume that the given conditions of Lemma 1 hold. If the mapping , , is convex on , then the following Simpson’s type inequality holds:where and

Proof. From Lemma 1 and the Hölder inequality, we haveBy using the convexity of , we obtainand the proof is completed.

Remark 12. If we take the limit in Theorem 5, then Theorem 5 becomes Theorem 2.2 of [34] for .

Remark 13. If we assume and in Theorem 5, then Theorem 5 becomes Theorem 5 of [13].

5. Newton’s Type Inequalities for Quantum Integrals

In this section, a new extension of quantum Newton’s inequalities for quantum differentiable convex functions is given.

Theorem 6. We assume that the given conditions of Lemma 2 hold. If the mapping is convex on , then the following Newton’s type inequality holds:where – are given in (31)–(36), respectively.

Proof. If we consider Lemma 2 and apply the same method that is used in the proof of Theorem 3, then we can obtain the desired inequality (53).

Corollary 5. If we take the limit in Theorem 6, then we obtain the following Newton’s type inequality:where

Remark 14. If we take , , and in Theorem 6, then Theorem 6 reduces to Theorem 7 of [13].

Remark 15. If we assume in Theorem 6, then we recapture inequality (42).

Theorem 7. We assume that the given conditions of Lemma 2 hold. If the mapping , , is convex on , then the following Newton’s type inequality holds:where – and – are given in (31)–(36) and (25)–(27), respectively.

Proof. If we apply the steps used in the proof of Theorem 4 and taking into account Lemma 2, we can obtain the required inequality (56).

Corollary 6. If we take the limit in Theorem 7, then we obtain the following Newton’s type inequality:where – are defined in Corollary 5 and

Remark 16. If we take , , and in Theorem 7, then Theorem 7 reduces to Theorem 9 of [13].

Remark 17. If we assume in Theorem 7, then we recapture inequality (47).

Theorem 8. We assume that the given conditions of Lemma 2 hold. If the mapping , , is convex on , then the following Newton’s type inequality holds:where and

Proof. If we apply the steps used in the proof of Theorem 5 and taking into account Lemma 2, we can obtain the required inequality (59).

Remark 18. If we take , , and in Theorem 8, then Theorem 8 becomes Theorem 8 of [13].

6. Conclusions

In this work, using quantum integrals, we developed a new extension of quantum trapezoid, quantum Simpson’s, and quantum Newton’s type estimations for quantum differentiable convex functions. It is also shown that the results presented here are generalizations of the findings presented in [13, 33, 34]. In the special cases of newly developed findings, we also obtained several new Simpson’s type, Newton’s type, midpoint-type, and trapezoidal type inequalities. It is an interesting and innovative problem that future researchers may investigate in order to achieve similar inequalities for convex and coordinated convex functions via different quantum integrals.

Data Availability

Data sharing is not applicable to this paper as no datasets were generated or analyzed during the current 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 and read and approved the final manuscript.

Acknowledgments

The authors want to thank the Dirección de Investigación from Pontificia Universidad Católica del Ecuador for the technical support to our research project entitled “Algunas desigualdades integrales para funciones convexas generalizadas y aplicaciones.”

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Copyright © 2021 Xue Xiao You 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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