Quantum Inequalities of Hermite–Hadamard Type for <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2724395pt" id="M1" height="8.14084pt" version="1.1" viewBox="-0.0657574 -7.8684 7.40017 8.14084" width="7.40017pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-115" d="M393 379C402 394 400 411 393 422C384 437 365 448 348 448C301 448 237 372 186 285H182L193 335C210 408 205 448 178 448C150 448 80 402 29 344L45 321C80 355 114 373 122 373C128 373 130 365 124 330C106 228 76 98 50 -5L57 -12C82 -5 112 3 132 6L172 203C196 256 234 304 254 329C275 355 293 367 306 367C318 367 330 360 342 348C347 343 355 343 365 350S386 367 393 379Z"/></g></svg>-</span>Convex Functions

You, Xuexiao; Kara, Hasan; Budak, Hüseyin; Kalsoom, Humaira

doi:https://doi.org/10.1155/2021/6634614

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Volume 2021 | Article ID 6634614 | https://doi.org/10.1155/2021/6634614

Quantum Inequalities of Hermite–Hadamard Type for -Convex Functions

Xuexiao You,¹Hasan Kara,²Hüseyin Budak,²and Humaira Kalsoom³

Academic Editor: Ahmet Ocak Akdemir

Received29 Dec 2020

Revised29 Jan 2021

Accepted04 Feb 2021

Published23 Feb 2021

Abstract

In this present study, we first establish Hermite–Hadamard type inequalities for -convex functions via -definite integrals. Then, we prove some quantum inequalities of Hermite–Hadamard type for product of two -convex functions. Finally, by using these established inequalities and the results given by (Brahim et al. 2015), we prove several quantum Hermite–Hadamard type inequalities for coordinated -convex functions and for the product of two coordinated -convex functions.

1. Introduction

Quantum calculus research is an unlimited analysis of calculus and is known as -calculus. We get the initial mathematical formulas in -calculus as reaches . The commencement of the analysis of -calculus was initiated by Euler (1707–1783). The aforementioned results lead to an intensive investigation on -calculus in the twentieth century. The concept of -calculus is used in many areas in mathematics and physics such as theory, orthogonal polynomials, integration, basic hypergeometric functions, mechanical theory, and quantum and relativity theory. For more information about -calculus, one can refer to [1–10].

Mathematically, convexity is very simple and natural which plays a very important role in various fields of pure and applied science, such as in the field of practicality, engineering science, and management science. In the recent past, the classical concept of convexity has been extended and generalized in different directions. Another factor that makes the theory of the most popular convex works is its relationship to the concept of inequality. Many inequalities can be achieved using the definition of convex functions. One of the widely studied inequalities involving convex works is the Hermite–Hadamard inequality, which is the first basic result of convex design with natural geometric descriptions and multiple uses and has attracted great interest in elementary mathematics. Many mathematicians have devoted their efforts to generalization, refinement, modelling, and multiplication of various fields of work such as the use of convex mappings (see, e.g., [11], p.137, and [12]).

The classical Hermite–Hadamard inequality states that if is a convex function on the interval of real numbers and with , then

The inequality holds in the reversed direction if is concave. We see that the Hermite–Hadamard inequality can be regarded as a refinement of the concept of integration and is easily followed by Jensen’s inequality. The Hermite–Hadamard inequality of convex works has received renewed attention in recent years and has been studied in significant and practical variations.

In [13], Pachpatte proved the following inequalities for products of convex functions.

Theorem 1. Let and be real-valued, nonnegative, and convex functions on . Then, we havewhere and .

A positive function is called -convex on , if for all and ,

It is obvious if , then the inequality classical convex functions. It should be noted that if is -convex function, then is convex function. We have that 0-convex functions are simply -convex functions and 1-convex functions are ordinary convex functions [14].

In [15], the definition of -convex functions on coordinates is given, such that

Definition 1. A function will be called -convex on for all and , if the following inequality holds:

It is simply to see that if we choose , we have coordinated -convex functions and if we choose , we have coordinated convex functions. In [15], Ekinci et al. also prove several Hermite–Hadamard type inequalities for coordinated -convex functions. In literature, many studies have been done on -convex functions. For some of them, one can see [16–23].

2. Preliminaries of -Calculus and Some Inequalities

In this section, we present some required definitions and related inequalities about -calculus. For more information about -calculus, one can refer to [1–10, 24, 25]. Also, here and further, we use the following notation (see [5]):

In [4], Jackson gave the -Jackson integral from 0 to for as follows:provided the sum converges absolutely.

Jackson in [4] gave the -Jackson integral in a generic interval as

Definition 2 (see [9]). For a continuous function , then -derivative of at for is characterized by the expressionSince is a continuous function, thus we have. The function is said to be -differentiable on if exists for all . If in (9), then , where is familiar -derivative of at defined by the expression (see [5])

Definition 3 (see [9]). Let be a continuous function. Then, the -definite integral on and are defined as

In [26], Alp et al. proved the following -Hermite–Hadamard inequality for convex functions in the setting of quantum calculus.

Theorem 2. If is a convex differentiable function on and . Then, -Hermite–Hadamard inequalities are as follows:

On the other hand, Bermudo et al. gave the following new definition and related Hermite–Hadamard type inequalities.

Definition 4 (see [27]). Let be a continuous function. Then, the -definite integral on for is defined as

Theorem 3 (see [27]). If is a convex differentiable function on and . Then, -Hermite–Hadamard inequalities are as follows:

From Theorem 2 and Theorem 3, one can get the following inequalities.

Corollary 1 (see [27]). For any convex function and , we have

Brahim et al. prove the following lemma and theorem for -convex functions.

Lemma 1 (see [28]). For and , the following inequality is valid:

Theorem 4 (see [28]). Let be -convex on . Then, the following inequality holds for and :

Theorem 5 (see [28]). Let be -convex and -convex functions, respectively, on . Then, the following inequality holds for and :

Theorem 6 (see [28]). Let be -convex and -convex functions, respectively, on and . Then, the following inequality holds if and :

In [29], Latif defined the -integral and related properties for two variable functions as follows.

Definition 5. Suppose that is continuous function and . Then, the definite -integral on is defined byfor .

In [29], Latif et al. also proved a -Hermite–Hadamard inequality for coordinated convex functions.

By Definitions 4 and 5, Budak et al. defined the following , and integrals.

Definition 6 (see [30]). Suppose that is a continuous function and . Then, the following , , and integrals on are defined byrespectively, for .

Budak et al. also proved some quantum Hermite–Hadamard type inequalities for coordinated convex functions. For other similar quantum inequalities, please see [31,32].

In this paper, we first prove the new variant of results of Brahim et al. for -integrals. We also obtain quantum versions of the inequalities in [15].

3. Quantum Hermite–Hadamard Type Inequalities for -Convex Functions

In this section, we obtain some quantum inequalities of Hermite–Hadamard type for -convex functions and for product of two -convex functions.

Theorem 7. Let be a -convex function on . Then, the following inequality holds for :where .

Proof. According to definition -convex, for all , we haveBy integrating the inequality on , we obtainFrom Definition 4, we getUsing Minkowski’s inequality for right side of inequality (26), By Lemma 1, we haveThus, by substituting (29) and (30) in (28), we obtainThe proof is completed.

Remark 1. If we take the limit in Theorem 7, then Theorem 7 reduces to Theorem 2.1 in [33].

Remark 2. If we choose in Theorem 7, then inequality (24) reduces to the second inequality in (14).

Theorem 8. Let be -convex and -convex functions, respectively, on . Then, the following inequality holds for :where .

Proof. By the assumptions that is an -convex function and is an -convex function, we can writefor all and .
Then,Integrating both sides with respect to on and from Definition 4, we obtainUsing Cauchy’s inequality for right side of inequality (36), we obtainBy using Minkowski’s inequality, we haveSimilarly, we haveThus, from the inequalities (36)–(39), we obtain the desired result.

Remark 3. If we take the limit in Theorem 8, then Theorem 8 reduces to Theorem 2.3 in [33].

Corollary 2. If we choose in Theorem 8, then we have the inequality

Particularly, if for all , then we get

Theorem 9. Let be -convex and -convex functions, respectively, on . Then, we get the following inequality:where and with .

Proof. From (36), we haveUsing Hölder inequality for quantum integrals, we haveThis completes the proof.

Remark 4. If we take the limit in Theorem 9, then Theorem 9 reduces to Theorem 2.6 in [33].

Corollary 3. If we choose in Theorem 9, then we have the inequality

Particularly, if for all , then we get

4. Quantum Hermite–Hadamard Type Inequalities for Coordinated -Convex Functions

In this section, we present several Hermite–Hadamard type inequalities for coordinated -convex functions via , , , and integrals. We also prove some quantum inequalities of Hermite–Hadamard type for the product of two coordinated -convex functions.where and .

Theorem 10. Suppose that is a positive coordinated -convex function on . Then, one has the inequality

Proof. Since is a coordinated -convex function, then the partial mappings,are -convex. By inequality (24), we can writei.e.,Dividing both sides of the inequality and -integrating with respect to on , we getBy a similar argument for the mappingwe haveBy adding inequalities (52) and (54), we can obtain inequality (47).

Remark 5. If we take the limit and in Theorem 10, then Theorem 10 reduces to Theorem 5 in [15].

Remark 6. If we choose in Theorem 10, then inequality (47) reduces to the third inequality of Theorem 3.6 in [30].

Theorem 11. Suppose that is a positive coordinated -convex function on . Then, one has the inequalitywhere and .

Proof. The proof is similar to the proof of Theorem 10 by using Theorem 4.

Theorem 12. Suppose that is a positive coordinated convex function on . Then, one has the inequalitywhere and .

Proof. The proof is similar to the proof of Theorem 10 by using Theorems 4 and 7.

Theorem 13. Suppose that is a positive coordinated -convex function on . Then, one has the inequalitywhere and .

Proof. The proof is similar to the proof of Theorem 10 by using Theorems 4 and 7.

Theorem 14. Suppose that is a coordinated -convex function and coordinated -convex function, respectively, on . Then, we have the inequalitywhere , , and .

Proof. Since is a coordinated -convex on , then the partial mappings,are -convex on . On the other hand, if is a coordinated -convex function, then the partial mappings,are -convex on . From inequality (32), we geti.e.,Dividing both sides of the inequality and -integrating with respect to on , we haveBy a similar argument, we obtainBy adding inequalities (63) and (64), we obtain the required result.

Remark 7. If we take the limit and in Theorem 14, then Theorem 14 reduces to Theorem 6 in [15].

Corollary 4. If we choose in Theorem 14, then we have the inequalityParticularly, if for all , then we get

Theorem 15. Suppose that is a coordinated -convex function and coordinated -convex function, respectively, on . Then, we have the inequalitywhere , , and .

Proof. The proof is similar to the proof of Theorem 14 by using Theorems 5 and 8.

Theorem 16. Suppose that is a coordinated -convex function and coordinated -convex function, respectively, on . Then, we have the inequalitywhere , , and .

Proof. The proof is similar to the proof of Theorem 14 by using Theorem 5.

Theorem 17. Suppose that is a coordinated -convex function and coordinated -convex function, respectively, on . Then, we have the inequalitywhere , , and .

Proof. The proof is similar to the proof of Theorem 14 by using Theorems 5 and 8.

Theorem 18. Suppose that is a coordinated -convex function and coordinated -convex function, respectively, on . Then, we have the inequalitywhere , and with .

Proof. By applying inequality (42) for the partial mapping and , we can writeBy using -integral, we obtainSimilarly, by applying inequality (42) for the partial mapping and , we can writeBy adding inequalities (72) and (73), we obtain the desired result (70).

Remark 8. If we take the limit and in Theorem 18, then Theorem 18 reduces to Theorem 7 in [15].

Corollary 5. If we choose in Theorem 18, then we have the inequality

Theorem 19. Suppose that is a coordinated -convex function and coordinated -convex function, respectively, on . Then, we have the inequalitywhere , and with .

Proof. The proof is similar to the proof of Theorem 18 by using Theorems 6 and 9.

Theorem 20. Suppose that is a coordinated -convex function and coordinated -convex function, respectively, on . Then, we have the inequalitywhere , and with .

Proof. The proof is similar to the proof of Theorem 18 by using Theorems 6 and 9.

Theorem 21. Suppose that is a coordinated -convex function and coordinated -convex function, respectively, on . Then, we have the inequalitywhere , and with .

Proof. The proof is similar to the proof of Theorem 18 by using Theorem 6.

5. Conclusions

In this study, we present several quantum Hermite–Hadamard type inequalities for -convex functions and coordinated -convex functions. We also give some quantum inequalities for the product of two -convex functions and for the product of two coordinated -convex functions. In the future work, we can establish the similar quantum inequalities by using generalized -convex functions.

Data Availability

No datasets were generated or analysed 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. All authors read and approved the final manuscript.

Acknowledgments

This research was supported by Philosophy and Social Sciences of Educational Commission of Hubei Province of China (20Y109), Key Projects of Educational Commission of Hubei Province of China (D20192501), and Special Soft Science Project of Technological Innovation in Hubei Province (2019ADC146).

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