Some Further Results Using Green’s Function 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 6.59063 8.14084" width="6.59063pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-116" d="M352 391C352 416 319 448 267 448C236 448 173 423 147 400C107 364 96 332 96 304C96 248 143 210 193 181C241 153 258 124 258 100C258 72 232 38 184 38C151 38 107 66 81 108C77 114 64 116 55 111C34 99 23 84 23 65C23 29 81 -12 134 -12C220 -12 325 61 325 141C325 184 297 215 234 256C194 282 161 309 161 346C161 380 188 401 217 401C255 401 279 380 301 353C308 344 313 341 325 347C341 355 352 371 352 391Z"/></g></svg>-</span>Convexity

Yildiz, Çetin; Akgün, Levent; Khan, Muhammad Adil; Rehman, Nasir

doi:https://doi.org/10.1155/2023/3848846

Journal of Mathematics

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Abstract Introduction Preliminaries Results and Discussion Conclusion Data Availability Conflicts of Interest Authors’ Contributions References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 3848846 | https://doi.org/10.1155/2023/3848846

Some Further Results Using Green’s Function for -Convexity

Çetin Yildiz,¹Levent Akgün,¹Muhammad Adil Khan,²and Nasir Rehman³

Academic Editor: Firdous A. Shah

Received20 May 2023

Revised10 Jul 2023

Accepted12 Jul 2023

Published25 Jul 2023

Abstract

For s-convex functions, the Hermite–Hadamard inequality is already well-known in convex analysis. In this regard, this work presents new inequalities associated with the left-hand side of the Hermite–Hadamard inequality for s-convexity by utilizing a novel technique based on Green’s function. Also, Hölder, Young, and power-mean inequalities are used to obtain these new inequalities. Finally, some applications to special means of real numbers are provided. In conclusion, we think that the methodology used in this work will encourage more research in this field.

1. Introduction

Convexity is a fundamental idea in both applied and pure mathematics, acting as a powerful tool for evaluating functions and sets, proving inequalities, and modeling and resolving practical issues. In many areas of mathematics and beyond, this idea is essential for estimating integrals and setting boundaries. Relevant researchers can access some articles on convex functions from references [1–9].

Thus, we recall the elementary notation in convex analysis.

Definition 1. A set is said to be convex function iffor each and .

Definition 2. The mapping is said to be a convex function if the following inequality holds:for all and . If is convex, then is said to be concave. In terms of geometry, this indicates that if , , and are three separate locations on the graph of , with between and , then is on or below the chord .
In many different fields, convex functions are crucial. For instance, the concavity of is described in terms of declining returns for a production function = in economics. If is convex, then increasing returns are shown. In addition, a convex function applied to the expected value of a random variable is always limited beyond the convex function’s expected value, according to the theory of probability. It is possible to derive other inequalities, including the geometric-arithmetic mean inequality and Hölder’s inequality, using this conclusion, known as Jensen’s inequality. The idea of convexity has developed into a rich source of inspiration and a fascinating topic for scholars because of its widespread viewpoints, resilience, and plenty of applications. Mathematicians have developed incredible tools and numerical methods using the notion of convexity to deal with and resolve an enormous number of issues that emerge in the pure and applied sciences. This theory has a long and important history, and for more than a century, mathematics has focused on and concentrated on it. On the other hand, there are a lot of new issues in applied mathematics where the idea of convexity is insufficient to adequately characterize them in order to have beneficial consequences. Because of this, the idea of convexity has been expanded upon and developed in various research studies; see [10–16].
Utilizing various forms of convexity, some important inequalities have been observed. -convexity is one of the several varieties of convexity. Hudzik and Maligranda in reference [17] took into account, among other things, the class of functions that are -convex in the second sense. The following is the definition of this class: a function is -convex in the second sense ifholds for all and for some fixed . The class of -convexity is frequently denoted by the symbol . It is obvious that converts -convexity into the typical convexity of functions defined on .
The authors of the same paper, namely, [17], demonstrated that all functions from , , are nonnegative if implies .

Example 3. (see [17]). Let and . We define function as follows:It can be simply confirmed that(i)If and , then (ii)If and , then

Remark 4. Throughout the article, the Hermite–Hadamard inequality will be denoted by inequality.
Recently, there are many studies on -convexity in the literature. A few new general type inequalities for -convex mappings were developed in [18] by Yildiz et al. The Hölder inequality, the power-mean integral inequality, and certain extensions connected to these inequalities were utilized to establish these inequalities. In addition, they compared some inequalities. In [19], a new definition for -convex functions is given and some properties of this definition are investigated. In addition, extended versions of the previously well-known conclusions for harmonically convex functions, such as , various refinements, and Ostrowski-type inequalities, are developed. In [20], the expression “extended -convex functions” was introduced by the authors, who also developed some inequalities of the type for extended -convex functions. The authors then used these newly discovered integral inequalities to deduce certain specific mean inequalities. In reference [21], the authors established an equation for a function whose third derivative is integrable, developed some novel integral inequalities of the type for extended -convex mappings using the Hölder inequality, and then used these integral inequalities to produce inequalities for various kinds of special means. In reference [22], the authors established some new inequalities of the type for extended -convex mappings and obtained new inequalities with respect to and using Lemma 2.1. Finally, utilizing the -convexity for the Raina function, different inequalities are obtained with fractional integral operators in reference [23].
Convex mappings and sets have been improved and expanded in many disciplines of mathematics due to their robustness (as was described above); in particular, the convexity theory has been used to prove a number of inequalities that are prevalent in the literature. In the practical literature on mathematical inequalities, the type integral inequality is, to the best of our knowledge, a well-known, important, and incredibly helpful inequality. There are several classical inequalities that are closely associated with the classical type integral inequality, such as Simpson, Opial, Hardy, Hölder, Ostrowski, Minkowski, arithmetic-geometric, Young, and Gagliardo–Nirenberg inequalities. These inequalities are of pivotal significance. Following is a statement of this double inequality: assume that is a convex mapping on , where . Therefore,The reader who is interested can refer to references [24–27] for a number of recent findings pertaining to inequality.
In [28], the researchers proved a different form of inequality which holds for -convex mappings in the second sense.

Theorem 5. Suppose that is an -convex function in the second sense, where , and let , . If , then the following inequalities hold:

In the second inequality in [29], the constant is the best possibility.

The famous Young inequality is defined as follows:where and are nonnegative numbers, , and , in [30].

The reversed version of inequality (7) reads

The definition of the Hölder inequality is as follows.

Let and . If and are real functions defined on such that and are integrable functions on , then

The well-known Hölder inequality, one of the most significant inequalities in analysis, was demonstrated in this way using inequality (7). It makes a significant contribution to many fields of applied and pure mathematics and is essential in helping to solve several issues in the social, cultural, and natural sciences.

The most popular form of Young’s inequality, which is frequently used to demonstrate the well-known inequality for functions, is as follows:where and .

2. Preliminaries

This paper uses a relatively new approach based on Green’s function to illustrate the inequalities for -convex functions. First, we will start by giving the definition of Green’s function.

Let and the following four new Green’s functions defined on are defined by Mehmood et al. in [31] as follows:

In [31], the authors developed the following Lemma, which we will utilize to demonstrate our main conclusions.

Lemma 6. Let and be a twice differentiable function and be the new Green’s functions defined by (12). Then,

Proof. By using the procedures of integration by parts in , the above equation may be easily computed. The specifics of proof are thus left to readers who are interested. Likewise, a similar method can be used in other equations.

Remark 7. Throughout this study, will be used.
There are many studies on Green’s function in the literature. In these studies, different inequalities were obtained by using different methods as well as Green’s function. For example, in [32], by utilizing Green’s function, Jensen’s inequality, convexity, and monotone functions, the authors developed the left Riemann–Liouville fractional type inequalities as well as the extended type inequalities. In [33], the inequalities have again been established using Green’s function and convexity. In [34], the authors revisited the inequalities for the Riemann–Liouville fractional operators with the help of Green’s function, and finally, in [35], Li et al. established inequalities for the left generalized fractional integral via Green’s function.
The aim of this study is to obtain new integral inequalities with the use of Green’s function for functions whose -th power is -convex and -concave. In other words, it is to develop a new method using Green’s function. In addition to the definition of Green’s function, Young, Hölder, and power-mean inequalities were used to obtain these new identities. As a consequence, these inequalities are associated with the left-hand side of inequality. Finally, new propositions are given for special means.

3. Main Results

Theorem 8. Let be a twice differentiable and be a function in the second sense on that satisfy the relation given in (13). If , then the following inequality holds:for some fixed .

Proof. Substituting in (12), we haveIntegrating both sides of identity (13) with respect to and multiplying by , we obtainSubtracting (15) from (14), we haveNow, from the property of integral and definition Green’s function, we obtainUsing the properties of absolute value and triangle inequality for integrals in (17), then we haveLet and in inequality (18), then we clearly see thatSince is -convex functions in the second sense on , we can write inequality (19) as follows:which is the desired inequality.

Remark 9. If we choose in inequality (20), we obtain the following result presented in [36] (Corollary 11, inequality (2.9)).

Theorem 10. Let be a twice differentiable and be a function in the second sense on . If , then the inequalityholds for some fixed and such that .

Proof. Utilizing the definition of Green’s function, we have identity (17). From the properties of absolute value and triangle inequality for integrals in (17), we obtain inequality (18). By the change of variable in (18), we haveUsing the Young inequality in (23), we getBy using the -convexity of in (24), we haveThis completes the proof.

Corollary 11. Let all the assumptions of Theorem 10 be satisfied and since , for , we obtain

Corollary 12. If we choose in inequality (22), we get

For -convex maps, the following theorem provides a new upper bound for the left-hand side of inequality.

Theorem 13. Let be a twice differentiable and be a function in the second sense on . If , then we have the following inequality:for and .

Proof. By using the same procedure in Theorem 8, we have inequality (19) as follows:Utilizing the Hölder inequality for and in (29), we getBecause is -convex, we haveHence, the proof is done.

Corollary 14. Assume that Theorem 13’s assumptions are all true and since , for , we obtain

Corollary 15. Assume that Theorem 13’s assumptions are all true. Then,

Proof. Using the fact, we getFor and in (32), we obtain inequality (33).

Corollary 16. Let us choose in ineqıality (28). Then, we obtain

Theorem 17. Let be a twice differentiable and be a function in the second sense on . If and , then we obtain the following inequality:

Proof. Similarly, from the property of the Green function and inequality (19), we getUsing the -convexity and power-mean inequality in (37), we obtainwhich is our required inequality.

Corollary 18. If we choose in ineqıality (36), we have

Theorem 19. Let be a twice differentiable and be a function in the second sense on . If , then we have the following inequality:for and .

Proof. By using the same procedure in Theorem 13 and the Hölder inequality for and in (29), we obtainBecause is -concave, using inequality (6), we haveandFrom (41)–(43), we getwhich completes the proof.

Corollary 20. If we take in ineqıality (40), we have

Theorem 21. Let be a twice differentiable and be a function in the second sense on . If , then the inequalityholds for some fixed and such that .

Proof. Let . Similarly, using inequality (19) and Young inequality, we obtainSince is -concave functions, from inequality (6), we can write inequalities (42) and (43). Therefore, we havewhich concludes all of the proof.

Corollary 22. In Theorem 21, if we take , we obtainAnd since , for , then

4. Applications for Special Means

Now, let us look at the means for the random real numbers and .(1):(2):(3):

Now, we provide some applications to special means of real numbers based on the Section 3 results.

Proposition 23. Let and . Then, we have

Proof. The assumption follows from (13) applied to the -convex function , .

Proposition 24. Let and . Then, we have

Proof. The assumption follows from (28) applied to the -convex function , .

5. Results and Discussion

If we use Green’s functions , , and for various convexities, we may obtain the same conclusions as in this article. The inequality for -convex functions, as well as the inequality for fractional operators, preinvex, co-ordinate convex functions, and so on, may all be studied using Green’s function or any other new Green’s function. Readers who are interested in an exercise can utilize the other three Green’s functions to produce the results that correspond to them.

6. Conclusion

Producing novel and special integral inequalities is the primary motivation for inequality theory, one of the most crucial areas of mathematical analysis. Researchers sometimes utilize novel function classes, sometimes new integral operators, and sometimes try to obtain modifications of a few well-known inequalities in different spaces for this aim. Especially in the last decade, many methods have been developed with different results, and many remarkable refinements, extensions, and generalizations have been obtained. Using these different methods, many different types of inequalities, lemmas, and different identities can be found in articles, and relevant researchers can access these studies in the literature. One of these important methods is the well-known Green’s function.

In this study, a new method was developed using Green’s function to prove the new results obtained. In addition, conclusions pertaining to the left-hand side of were obtained for derivatives of the -th power of -convex functions. Using the identity (2.11) (in [37]) and different types of convexity, researchers can derive both type and different type well-known inequalities. In this sense, we hope that this study will inspire researchers to obtain further results.

Data Availability

No data were used to support the findings of this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors contributed equally and significantly in writing this paper. All authors have read and approved the final manuscript.

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Copyright © 2023 Çetin Yildiz 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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