A Generalization of Hermite–Hadamard–Fejer Type Inequalities for the <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.52687pt" id="M1" height="13.4804pt" version="1.1" viewBox="-0.0657574 -8.95353 10.3455 13.4804" width="10.3455pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-113" d="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"/></g></svg>-</span>Convex Function via <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="M2" height="8.14084pt" version="1.1" viewBox="-0.0657574 -7.8684 9.91493 8.14084" width="9.91493pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-223" d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"/></g></svg>-</span>Generator

Ünlüyol, Erdal; Erdaş, Yeter

doi:https://doi.org/10.1155/2023/1185960

Journal of Mathematics

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 1185960 | https://doi.org/10.1155/2023/1185960

A Generalization of Hermite–Hadamard–Fejer Type Inequalities for the -Convex Function via -Generator

Erdal Ünlüyol¹and Yeter Erdaş¹

Academic Editor: Chiranjibe Jana

Received13 Jan 2023

Revised19 Feb 2023

Accepted01 Mar 2023

Published15 Apr 2023

Abstract

In the century, I. Newton and G. Leibniz found independently each other the basic operations of calculus, i.e., differentiation and integration. And this development broke new ground in mathematics. From 1967 to 1970, Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral, converting the roles of subtraction and addition into division and multiplication, respectively. And then they generalised this operation. Later, they named this analysis non-Newtonian calculus. This calculus is basically generated by generators. So, in this article, first, we give the definition of the -convex function due to -generator. Second, we obtain some new theorems for this function with respect to the -generator. Third, we get some new theorems using Hermite–Hadamard–Fejer inequality for the -convex function. Finally, we show that our obtained results are reduced to the classical case in the special conditions.

1. Introduction

Inequalities play an important role in almost all areas of mathematics. The first basic work on inequalities was the book “Inequalities” written by Hardy et al. [1]. This is the first reference book on pure inequalities and includes many new inequalities and their applications. The second book “Inequalities” was written by Beckenbach and Bellman [2], which contains some interesting results on inequalities in the period 1934–1960. Mitrinovic’s book [3], “Analytic Inequalities,” published in 1970, contains new topics that were not included in the two books mentioned above. Besides these three main sources, Mitrinovic et al. [4], “Classical and New Inequalities in Analysis” by Pachpatte, and “Mathematical Inequalities” by Pachpatte [5], in the recent years, many books and articles on inequalities have been published by researchers such as S. S. Dragomir, V. Lakshmikantham, R. P. Agarwal, M. E. Özdemir, E. Set, İşcan, M. Z. Sarkaya, and A. O. Akdemir.

Although the history of convex functions is very old, its beginning can be shown as the end of the 19^th century. The basis of such functions is mentioned in Hadamard’s work in 1893 [6], although it is not explicitly stated. After this date, although results implying convex functions were encountered in the literature, convex functions were first systematically used in 1905 and 1906 by J. L. W. V. Jensen. It is accepted that the theory of convex functions has developed rapidly since Jensen’s pioneering work. Many researchers such as Beckenbach and Bellman [2] and Mitrinovic [3] have discussed the issue of inequalities for convex functions in their books. Also, Roberts and Varberg [7], Pecaric et al. [8], and Niculescu and Persson [9] have done many studies on inequalities on convex functions. Integral inequalities constitute some of these studies.

In the literature, well-known inequalities related to the integral mean of a convex function are the Hermite–Hadamard inequalities [6] or its weighted versions, namely, Hermite–Hadamard–Fejer inequality [10]. Also, there are many kinds of convex functions. -convex functions were first defined by Kunt and İşcan [11] for Hermite–Hadamard–Fejer inequality.

Sir I. Newton and G. Leibniz developed mathematics by establishing differentiation and integration in the century. These developments were very useful for mathematics and related sciences. From 1967 to 1970, Grossman and Katz gave new definitions for derivative and integral, converting the roles of subtraction and addition into division and multiplication, respectively. Then, they established geometric, bigeometric, harmonic, biharmonic, quadratic, and biquadratic calculus, and they named these calculi non-Newtonian calculi. Grossman and Katz published the first book concerning with non-Newtonian calculus in 1972 [12] and then wrote nine books related to the non-Newtonian calculi.

In geometric calculus and bigeometric calculus from within these calculi, the derivative and integral are both multiplicative. The geometric derivative and the bigeometric derivative are closely related to the well-known logarithmic derivative and elasticity, respectively. Also, the linear functions of classical calculus are the functions which have a constant derivative, and besides the exponential functions in the geometric calculus which have a constant derivative, the power functions in the bigeometric calculus are the functions which have a constant derivative. Among the non-Newtonian calculi, geometric and bigeometric calculi have been often used.

Since these calculi emerged, it has become a serious alternative to the classical analysis developed by Newton and Leibniz. Just like the classical analysis, non-Newtonian calculi have many varieties as a derivative, an integral, a natural average, a special class of functions having a constant derivative, and two fundamental theorems which reveal that the derivative and integral are inversely related. However, the results obtained by non-Newtonian calculus are also significantly different from the classical analysis. For example, infinitely, many non-Newtonian calculi have a nonlinear derivative or integral.

The non-Newtonian calculi are useful mathematical tools in science, engineering, and mathematics and provide a wide variety of possibilities, as a different perspective. Specific fields of application include fractal theory, image analysis (e.g., in biomedicine), growth/decay processes (e.g., in economic growth, bacterial growth, and radioactive decay), finance (e.g., rates of return), the theory of elasticity in economics, marketing, the economics of climate change, atmospheric temperature, wave theory in physics, quantum physics and the Gauge theory, signal processing, information technology, pathogen counts in treated water, actuarial science, tumor therapy in medicine, materials science/engineering, demographics, and differential equations [13–29].

2. Preliminaries

Hermite–Hadamard inequality is one of the well-known inequalities concern with the integral mean for convex functions. And the other one is Hermite–Hadamard–Fejer inequality [10]. It is the weighted version of Hermite–Hadamard inequality. If is a convex function, function is integrable on , nonnegative, and symmetric to ; then, the following inequality holds for all

-convex function was firstly defined on by Zhang and Wan [30]. And İşcan also defined the -convex function on in a different way as follows.

Definition 1 (see [31]). Let be a real interval and . A function is said to be a -convex function iffor all and . If inequality (2) is reversed, then is said to be -concave.

Theorem 1 (see [31]). Let be a -convex function, , with . If , then we have

Now, we give about a short brief of non-Newtonian calculus [12].

2.1. Systems of Arithmetic

Arithmetic is any system that satisfies the whole of the ordered field axiom whose domain is a subset of . There are many types arithmetic, all of which are isomorphic, that is, structurally equivalent.

A generator is a one-to-one function whose domain is and whose range is a subset of where . Each generator generates exactly one arithmetic, and conversely, each arithmetic is generated by exactly one generator.

Throughout this article, the identity function is denoted by because we do not want to mix an interval. When we take identity and exponential functions as a generator, then and , and generates the classical and geometric arithmetic, respectively.

2.1.1. -Arithmetics

We suppose that be a generator and . Then, by -arithmetic, we mean the arithmetic whose domain is and whose operations are defined as follows:

As a generator, we choose the function acting from into the set as follows:

It is obvious that arithmetic reduces to the geometric arithmetic as follows:

Definition 2 (see [12]). Let for all . Set of -integers is defined and denoted by as can be seen in the figure below:Namely, .

Remark 1 (see [28]). Let be the generator. Then, is said to be an -interval on if for all ;(1)(2)(3)(4)(5)(6)(7)(8)We can also show the -intervals as follows:

Remark 2 (see [28]). Afterwards, and are said to be -closed interval and -open interval, respectively.

Definition 3 (see [32]). Let be an -closed interval and be a function. If the following inequality holds for all and ,we say that is an -convex function.

Definition 4 (see [33]). Let be an -interval. A function is said to be -harmonically convex if the following inequality holds for all and

Definition 5 (see [33]). Let be a function. If the function holds the following equalitythen we say that the function is an -symmetric according to .

Definition 6 (see [33]). A function is said to be -harmonically symmetric with respect to ifholds for all .

Theorem 2 (see [28]). Let be an -closed interval in , and also be any -convex function. Then, the following double inequality holds for all :

3. Main Results

Definition 7. Let be an -interval and . A function is said to be -convex ifinequality holds for all and .

Definition 8. Let . A function is said to be -symmetric with respect to if the relation holds for all .

Theorem 3. Let be an -convex function and is a nonnegative, integrable, and non-Newtonian-symmetric to . In this case, the following inequality holds:

Proof. According to the claim of the theorem, for all , we can write the below inequality:Multiplying both sides of (16) by and integrating to over , we have the following inequality:Then, function is -symmetric according to , and inequality (12) becomes as follows for So, the proof of the left hand of (15) is completed. Now, we prove the right hand. Since is an -convex function, we can write the below inequality:Multiplying both sides of (19) by and integrating to over , we get the following inequalities:Similarly, function is -symmetric according to , and inequality (21) becomes as follows for Then, we haveConsequently, the proof is completed.

Corollary 1. If we take in Theorem 3, then we obtain the Hermite–Hadamard–Fejer inequality in [10], i.e.,

Corollary 2. If we take in Theorem 3, then we obtain the following inequality:

Corollary 3. If we take in Theorem 3, then we obtain the inequality in [27]:

Theorem 4. Let be an -harmonically convex function. Then, for all we have

Proof. Since is an -harmonically convex function, for all and we can writeIf we choose in (28), then we get the below inequality:In other words, if we choosein (29), then we have the following inequality:If we integrate to over of (31), then we haveWe obtain the following inequality for in (32):And then we can writeThis completes the proof.

Corollary 4. If we take in Theorem 4, then we obtain the inequality in [34]:

Corollary 5. If we take in Theorem 3, then we obtain the following inequality:

Theorem 5. Let be an -harmonically convex function. And and nonnegative, integrable, and -symmetric to . Then, we have

Proof. Because is an -harmonically convex function, we can write the following inequality:In other words, if we choosein (38), then we have the following inequality:Multiplying both sides of (40) byand integrating to over , we obtainIn this case, we can write the following inequality for in (43):Then, we obtainThis completes the proof.

Corollary 6. If we take in Theorem 5, then we obtain the inequality in [35]:

Corollary 7. If we take in Theorem 5, then we obtain the following inequality:

Theorem 6. Let be a -convex function, , with . If and is nonnegative, integrable, and -symmetric with respect to , then the following inequality holds:

Proof. Since is a -convex function, we can write the following inequality for all , and for in (9):If we choosein (49), then we haveMultiplying both sides of (51) by and integrating with respect to over we get below the inequality:If we take in (53), then we can write below the inequality:So, the proof of the left hand of (31) is completed. Now, we prove the right hand. We know that the following inequality holds:Multiplying both sides of (55) byintegrating respect to over andchanging variables, we have the following inequality:So, the proof is completed.

Corollary 8. If we take in Theorem 6, then we obtain the inequality in [36]:

Corollary 9. If we take in Theorem 6, then we obtain the following inequality:

Corollary 10. If we take and in Theorem 6, then we obtain the following inequality:

Remark 3. In inequality (48), one can see the following inequalities:(1)If one takes and , one has (8).(2)If one takes , one has (10).(3)If one takes and , one has (27).(4)If one takes , one has (24).(5)If one takes and , one has (26).(6)If one takes , and , one has (35).(7)If one takes and , one has (46).(8)If one takes and , one has (3).

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2023 Erdal Ünlüyol and Yeter Erdaş. 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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