New Post Quantum Analogues of Hermite–Hadamard Type Inequalities for Interval-Valued Convex Functions

Kalsoom, Humaira; Ali, Muhammad Aamir; Idrees, Muhammad; Agarwal, Praveen; Arif, Muhammad

doi:https://doi.org/10.1155/2021/5529650

Mathematical Problems in Engineering

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

Research Article | Open Access

Volume 2021 | Article ID 5529650 | https://doi.org/10.1155/2021/5529650

New Post Quantum Analogues of Hermite–Hadamard Type Inequalities for Interval-Valued Convex Functions

Humaira Kalsoom,¹Muhammad Aamir Ali,²Muhammad Idrees,³Praveen Agarwal,⁴and Muhammad Arif⁵

Academic Editor: Adrian Neagu

Received26 Feb 2021

Accepted21 May 2021

Published30 Jun 2021

Abstract

The main objective of this paper is to introduce -derivative and -integral for interval-valued functions and discuss their key properties. Also, we prove the -Hermite–Hadamard inequalities for interval-valued functions is the development of -Hermite–Hadamard inequalities by using new defined -integral. Moreover, we prove some results for midpoint- and trapezoidal-type inequalities by using the concept of Pompeiu–Hausdorff distance between the intervals. It is also shown that the results presented in this paper are extensions of some of the results already shown in earlier works. The proposed studies produce variants that would be useful for performing in-depth investigations on fractal theory, optimization, and research problems in different applied fields, such as computer science, quantum mechanics, and quantum physics.

1. Introduction

In mathematics, the quantum calculus is equivalent to usual infinitesimal calculus without the concept of limits or the investigation of calculus without limits (quantum is from the Latin word “quantus” and literally it means how much and in Swedish it is “Kvant”). Euler and Jacobi can be credited with establishing the basis of the modern understanding of quantum calculus, but these developments were recently applied in the field, bringing about tremendous development. This could be due to the fact that it acts as a connection between mathematics and physics. In 2002, the book [1] by Kac and Cheung presented some in-depth details of -calculus. Later on, a few scholars have continued to establish the idea of -calculus in a different direction of mathematics and physics. Jackson [2] created the concept of quantum-definite integrals in quantum calculus in the twentieth century. This inspired many quantum calculus analysts, and several papers have been published in this field as a consequence. Ernst [3] developed the history of -calculus and a new method for finding quantum calculus. Gauchman [4] derived integral inequalities in -Calculus, which is a generalization of classical integral inequalities. In 2013, Tariboon et al. presented -calculus principles over finite intervals, explored their characteristics, and applied impulsive difference equations in [5]. In 2015, Sudsutad et al. [6] proved quantum integral inequalities for convex functions. Shortly afterward, certain -Hermite–Hadamard form inequalities are acquired by Alp in [7]. Recently, Lou et al. [8] presented basic properties of -calculus and derived -Hermite–Hadamard inequalities for convex interval-valued functions. For more details, see [9–13].

Postquantum calculus theory, prefixed by the -calculus, is a native -calculus generalization. We deal with -number with one base in a recent development in the study of quantum calculus, but postquantum calculus includes and numbers with two independent and variables. Chakarabarti and Jagannathan [14] was the first to consider this. Inspired by the current research on Tunc and Gov [15], the definitions of -derivatives and -integrals have been adopted on finite intervals; interested readers are referred to [16–18]. A good deal of the book by Moore [19] is a narrative of the methods used by Moore to find an unknown variable and substitute it with an interval of real numbers and an arithmetic interval used in error analysis, which has a significant effect on the outcome of the calculation and automatic error analysis. It has been used extensively in several countries in recent days to address a variety of uncertain topics. In particular, Costa et al. [20] developed convex function understandings in the field of inequality and provided Jensen inequality in 2017 for the interval-valued functions. Therefore, some scientists have combined classical inequalities with interval values to achieve several extensive inequalities, see [21, 22].

The paper is summarized as follows. We review some basic properties of interval analysis in Section 2. In Section 3, we put forward the concepts of -derivative and give some properties. Similarly, the concepts of -integral and some properties are presented in Section 4. In Sections 5 and 6, we give some new -Hermite–Hadamard-type inequalities and some results related to upper and lower bounds of -Hermite–Hadamard. Briefly, conclusion has been discussed in Section 7.

2. Preliminaries

Throughout this paper, we suppose that closed interval . You can describe the length of interval as . In addition, we conclude that seems to be positive if , and we present that all positive intervals belong to .

For some kind of and ; then, we have the following properties:

Definition 1. (see [23]). For some kind of , we denote the -difference of and as the set , and we haveIt seems beyond controversy thatSuppose that if we take a consent , thenThe relation between and can be described by the relation of “”:The distance of Hausdorff–Pompeiu between and is denoted as . The later result is that is a complete metric space, as proven in [24].

Definition 2. Suppose that a continuous function at ifWe denote and to show the collection of all continuous interval- and real-valued functions on , respectively.
For much more simple notations with interval analysis, see [23, 25, 26].
In this paper, the symbols and are used to refer to functions with interval values. If a function and , then is -increasing (or -decreasing) on if is increasing (or decreasing) on . If is monotone on , then is -monotone on .

3. -Derivative for Interval-Valued Functions

In this portion, we introduce the -derivative concepts and give some properties. Firstly, let us study the -derivative concept. Let any constant be .

Definition 3. (see [15]). Let and ; the -derivative of function at is defined byIf, for all and exists, then we called as a -differentiable on . If in (9), then ; then,For more details, see [15].
Now, we are adding the -derivative for the interval-valued functions and some related properties.

Definition 4. Suppose that and , and the -derivative of at is denoted aswhere is said -derivative of denoted as

Theorem 1. Suppose that . is -differentiable at if and only if and are -differentiable at , and

Proof. Suppose is -differentiable at ; then, there exist such that . According to Definition 4,Exist; then, equation (11) is proved by using the above derivatives.
Conversely, suppose and are -differentiable at .
If , thenSo, is -differentiable at . Similarly, if , then .
Show the above result in the next example.

Example 1. Let , taking . It shows that is -differentiable. By Definition 4, if , we haveand taking , thenIn the meantime, we realize that and are -differentiable at 0. In the same way, taking , we haveand taking , thenIt show that if , then . And, if , then .
We include the following findings to more clearly explain the existence of the derivatives.

Theorem 2. Let . If is -differentiable on , then we have that(i), for all , if is -increasing(ii), for all , if is -decreasing

Proof. First, suppose is -increasing and -differentiable is on . For any , we have . Since is increasing, then we haveTherefore,The other condition can be proved, similarly.

Remark 1. Let be a given point. If is -increasing on and -decreasing on , then on and on .

Example 2. Suppose that a function ; then, we take . We know that , and it shows that is -decreasing on and -increasing on . We know that and are -differentiable on ; then, we haveTherefore,

Theorem 3. Suppose that is a -differentiable on with and ; then, functions and are -differentiable on , such that and .

Proof. For any ,

Theorem 4. Suppose that is a -differentiable on . Let ; if has a constant sign on , then functions is -differentiable on and .

Proof. For any ,

Theorem 5. Suppose that . Let be -differentiable on ; then, is -differentiable on ; then, we have the following properties:(i)If are equally -monotonic on , for all , then(ii)If and are differently -monotonic on , for all , then Therefore, we have

Proof. (i)Suppose that two functions are -differentiable and -increasing on ; taking , , , and are -differentiable,(ii)It shows that and are -differentiable functions on , and also, is -differentiable on and(iii)Correspondingly, both and can be shown to be -decreasing.(iv)Suppose that is -increasing and is -decreasing. Then,(v)Moreover,(vi)We get (25) by comparing (30) with (31). Additionally,(v)We obtain if it is -increasing or -decreasing; we can obtain(vi)The opposite case, similarly, can be proved.

Theorem 6. Let . If are -differentiable and has a constant sign on , then is -differentiable on , and one of the following cases holds:(i)If are equally -monotonic on , for all , then(ii)If and are differently -monotonic on , for all , then

Proof. We now assume that on , and .(i)Suppose are -increasing on . Since are -differentiable, we have that , , , and are -differentiable and(ii)Then, and are -differentiable functions on . So, is -differentiable on and(iii)The case of and are both -decreasing can be proved, similarly.(iv)Suppose is -increasing and is -decreasing. From (i), we have that(v)For , on the one hand,(vi)On the other hand,(vii)Comparing (39) with (40), we get (35). The opposite case, similarly, can be proved.

Example 3. Let , given by and . Since and , then , for all . We have that is -increasing on and -decreasing on . is -decreasing on and -increasing on .
Furthermore, we have that and . Since and , then , are -decreasing on and -increasing on . For all , we get thatThen, from (31) and (40),Furthermore, for all , similarly, we obtain thatObviously, we can see that and .

4. -Integral for Interval-Valued Functions

In this section, we present the concepts of -integral and give some properties. Firstly, let us review the definition of -integral.

Definition 5. (see [15]). Let and ; then, the expression -integral is defined byfor all .
Additionally, if , then the definite -integral on is defined byNote that if , then (44) reduces to the classical -Jackson integral of a function , defined by for . For more details, see [15].
Next, we give the concept of the -integral and discuss some basic properties.

Definition 6. Let and ; then, the expression -integral is defined byfor all .

Theorem 7. Let and . If , then we have that

Proof.

Theorem 8. Let . If , then is -integral if and only if and are -integral over . Moreover,

Proof. The proof can be obtained by combining Definitions 5 and 6 and, hence, is omitted.

Example 4. Let , given by . For , we have

Theorem 9. Let and let . If , for , then we have that(i)(ii)

Proof. From Definition 6, we have that

Theorem 10. Let . If , thenMoreover, if has a constant sign on , then

Proof. First, we haveIt implies thatMoreover, if , or if . We now assume that on , and . So, we have . It implies that

Theorem 11. Let . If is -differentiable on , then is -integral. Moreover, if is -monotone on , then

Proof. If is -differentiable on , then, from Theorem 1, it follows that and are -differentiable. Hence, and exist on . Meanwhile, and are -integral. Therefore, Theorem 8 implies that is -integral. If is -increasing on , then , for all . Then, we have thatIt follows thatSince is -increasing on , by (2), thenIf is -decreasing on , then , for all . Then, we get that

Remark 2. We remark that if is -increasing on , then (57) is equivalent withand if is -decreasing on , then (57) is equivalent withfor all . Also, we remark that relation (57) can be false if is not -monotone on . Indeed, let , given by. For and , we have that (see Example 3)Then, we get thatTherefore, (57) is not true for all .

Example 5. Let , given by . Since is -differentiable and -increasing on , then is -integral and . Let ; then,

Theorem 12. (see [27]). Let be a convex differentiable function on . Then, the following inequalities holds for -integrals:

Theorem 13. (see [27]). Let be a convex differentiable function on . Then, the following inequalities holds for -integrals:

Theorem 14. (see [27]). Let be a differentiable function on and is continuous and integrable on . If is convex function over , then we have the following -midpoint inequality:whereand .

Theorem 15. (see [28]). Let be a differentiable function on and be continuous and integrable on . If is convex function over , then we have the following new -trapezoidal inequality:where

5. -Hermite–Hadamard Inequalities for Interval-Valued Functions

Now, we review the content of the convex interval-valued functions.

Definition 7. (see [21]). Suppose that . Take is convex if, for all and , we haveWe use to represent the set of all convex interval-valued functions.

Theorem 16. (see [21]). Let . Then, is said to be convex if and only if is convex and is concave on .

Theorem 17. Let be a differentiable interval-valued convex function; then, the following inequalities hold for the -integral:

Proof. Since is an interval-valued convex function, therefore is a convex function and is a concave function. So, from and inequality (67), we haveand from concavity of and (67), we haveFrom (75) and (76), we obtainand hence, we haveAlso, from (75) and (76), we obtainand hence, we haveBy combining (78) and (80), we obtain the required inequality which accomplishes the proof.

Theorem 18. Let be a differentiable interval-valued convex function on ; then, the following inequalities hold for the -integral:

Proof. Since is an interval-valued convex function, therefore, is a convex function and is a concave function. Because of convexity of and from inequalities (68), we obtain thatNow, using the fact that is concave function and from inequality (68), we obtain thatThe rest part of the proof can be done by applying the same lines of previous theorem and considering inequalities (82) and (83). Thus, the proof is completed.

Theorem 19. Let be a differentiable interval-valued convex function on ; then, the following inequalities hold for the -integral:where

Proof. From inequalities (74) and (75), we have the required inequalities (84). Thus, the proof is finished.

6. Midpoint- and Trapezoidal-Type Inequalities for -Integral

In this section, some new inequalities of midpoint and trapezoidal type for interval-valued functions are obtained.

Theorem 20. Let be a -differentiable function. If and are convex functions on , then the following midpoint inequality holds for interval-valued functions:where - are defined in Theorem 14 and is Pompeiu–Hausdorff distance between the intervals.

Proof. Using the definition of distance between intervals, one can easily obtain thatNow, using the fact that is a convex function and from inequality (69), we haveSimilarly, considering that is convex on and using inequality (69), we haveSo, from inequalities (88) and (89), we havesinceTherefore, the proof is completed.

Corollary 1. If we set in Theorem 20, then we have the following new -midpoint inequality for interval-valued functions:where and both are convex functions.

Corollary 2. If we set and in Theorem 20, then we have following midpoint inequality for interval-valued functions:where and both are convex functions.

Theorem 21. Let be a -differentiable function. If and are convex functions on , then the following trapezoidal inequality holds for interval-valued functions:where and are defined in Theorem 15 and is Pompeiu–Hausdorff distance between the intervals.

Proof. From definition of distance between the intervals and inequality (71) and using the strategies that were followed in Theorem 21, one can easily obtain inequality (94).

Corollary 3. If we set in Theorem 21, then we have following new -trapezoidal inequality for interval-valued functions:where and both are convex functions.

Corollary 4. If we set and in Theorem 21, then we have the following new trapezoidal inequality for interval-valued functions:where and both are convex functions.

7. Conclusions

In this work, the concept of -derivative and -integral are introduced and some fundamental properties are discussed. Furthermore, some new -Hermite–Hadamard type inequalities are established and we proved some results for midpoint- and trapezoidal-type inequalities by using the concept of Pompeiu–Hausdorff distance between the intervals. We intend to study the integral inequalities of fuzzy-interval-valued functions and some applications in interval optimizations by using -integral.

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed to each part of this study equally and have read and approved the final manuscript.

Acknowledgments

The work was supported by Zhejiang Normal University.

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Copyright © 2021 Humaira Kalsoom 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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