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25 pages, 486 KiB

Open AccessArticle

I.V-

CR

-γ-Convex Functions and Their Application in Fractional Hermite–Hadamard Inequalities

by Miguel Vivas-Cortez, Sofia Ramzan, Muhammad Uzair Awan, Muhammad Zakria Javed, Awais Gul Khan and Muhammad Aslam Noor

Symmetry 2023, 15(7), 1405; https://doi.org/10.3390/sym15071405 - 12 Jul 2023

Cited by 15 | Viewed by 2418

Abstract

In recent years, the theory of convexity has influenced every field of mathematics due to its unique characteristics. Numerous generalizations, extensions, and refinements of convexity have been introduced, and one of them is set-valued convexity. Interval-valued convex mappings are a special type of set-valued maps. These have a close relationship with symmetry analysis. One of the important aspects of the relationship between convex and symmetric analysis is the ability to work on one field and apply its principles to another. In this paper, we introduce a novel class of interval-valued (I.V.) functions called

CR

γ

-convex functions based on a non-negative mapping

γ

and center-radius ordering relation. Due to its generic property, a set of new and known forms of convexity can be obtained. First, we derive new generalized discrete and integral forms of Jensen’s inequalities using

CR

γ

-convex I.V. functions. We employ this definition and Riemann-Liouville fractional operators to develop new fractional versions of Hermite-Hadamard’s, Hermite-Hadamard-Fejer, and Pachpatte’s type integral inequalities. We examine various key properties of this class of functions by considering them as special cases. Finally, we support our findings with interesting examples and graphical representations. Full article

(This article belongs to the Special Issue Functional Equations and Inequalities in 2022)

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17 pages, 366 KiB

Open AccessArticle

Some Novel Estimates of Hermite–Hadamard and Jensen Type Inequalities for (h₁,h₂)-Convex Functions Pertaining to Total Order Relation

by Tareq Saeed, Waqar Afzal, Khurram Shabbir, Savin Treanţă and Manuel De la Sen

Mathematics 2022, 10(24), 4777; https://doi.org/10.3390/math10244777 - 15 Dec 2022

Cited by 18 | Viewed by 2312

Abstract

There are different types of order relations that are associated with interval analysis for determining integral inequalities. The purpose of this paper is to connect the inequalities terms to total order relations, often called (CR)-order. In contrast to classical interval-order relations, total order [...] Read more.

ω = ⟨ ω_{c}, ω_{r} ⟩ = ⟨ \frac{\bar{ω} + \underset{̲}{ω}}{2}, \frac{\bar{ω} - \underset{̲}{ω}}{2} ⟩

. A major benefit of total order relations is that they produce more efficient results than other order relations. This study introduces the notion of CR-

(h_{1}, h_{2})

-convex function using total order relations. Center and Radius order relations are a powerful tool for studying inequalities based on their properties and widespread application. Using this novel notion, we first developed some variants of Hermite–Hadamard inequality and then constructed Jensen inequality. Based on the results, this new concept is extremely useful in connection with a variety of inequalities. There are many new and well-known convex functions unified by this type of convexity. These results will stimulate further research on inequalities for fractional interval-valued functions and fuzzy interval-valued functions, as well as the optimization problems associated with them. For the purpose of verifying our main findings, we provide some nontrivial examples. Full article

(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)

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16 pages, 362 KiB

Open AccessArticle

Some New Generalizations of Integral Inequalities for Harmonical cr-(h₁,h₂)-Godunova–Levin Functions and Applications

by Tareq Saeed, Waqar Afzal, Mujahid Abbas, Savin Treanţă and Manuel De la Sen

Mathematics 2022, 10(23), 4540; https://doi.org/10.3390/math10234540 - 1 Dec 2022

Cited by 22 | Viewed by 2091

Abstract

The interval analysis is famous for its ability to deal with uncertain data. This method is useful for addressing models with data that contain inaccuracies. Different concepts are used to handle data uncertainty in an interval analysis, including a pseudo-order relation, inclusion relation, and center–radius (cr)-order relation. This study aims to establish a connection between inequalities and a cr-order relation. In this article, we developed the Hermite–Hadamard (

H . H

) and Jensen-type inequalities using the notion of harmonical

(h_{1}, h_{2})

-Godunova–Levin (GL) functions via a cr-order relation which is very novel in the literature. These new definitions have allowed us to identify many classical and novel special cases that illustrate our main findings. It is possible to unify a large number of well-known convex functions using the principle of this type of convexity. Furthermore, for the sake of checking the validity of our main findings, some nontrivial examples are given. Full article

(This article belongs to the Special Issue Variational Problems and Applications, 2nd Edition)

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14 pages, 316 KiB

Open AccessArticle

Some H-Godunova–Levin Function Inequalities Using Center Radius (Cr) Order Relation

by Waqar Afzal, Mujahid Abbas, Jorge E. Macías-Díaz and Savin Treanţă

Fractal Fract. 2022, 6(9), 518; https://doi.org/10.3390/fractalfract6090518 - 14 Sep 2022

Cited by 38 | Viewed by 2217

Abstract

Interval analysis distinguishes between different types of order relations. As a result of these order relations, convexity and nonconvexity contribute to different kinds of inequalities. Despite this, convex theory is commonly known to rely on Godunova–Levin functions because their properties make it more efficient for determining inequality terms than convex ones. The purpose of this study is to introduce the notion of cr-h-Godunova–Levin functions by using total order relation between two intervals. Considering their properties and widespread use, center-radius order relation appears to be ideally suited for the study of inequalities. In this paper, various types of inequalities are introduced using center-radius order (cr) relation. The cr-order relation enables us firstly to derive some Hermite–Hadamard (

H . H

) inequalities, and then to present Jensen-type inequality for h-Godunova–Levin interval-valued functions (GL-

IVFS

) using a Riemann integral operator. This kind of convexity unifies several new and well-known convex functions. Additionally, the study includes useful examples to support its findings. These results confirm that this new concept is useful for addressing a wide range of inequalities. We hope that our results will encourage future research into fractional versions of these inequalities and optimization problems associated with them. Full article

(This article belongs to the Topic Advances in Optimization and Nonlinear Analysis Volume II)
(This article belongs to the Section Engineering)

15 pages, 309 KiB

Open AccessArticle

The Properties of Harmonically cr-h-Convex Function and Its Applications

by Wei Liu, Fangfang Shi, Guoju Ye and Dafang Zhao

Mathematics 2022, 10(12), 2089; https://doi.org/10.3390/math10122089 - 16 Jun 2022

Cited by 28 | Viewed by 2219

Abstract

In this paper, the definition of the harmonically

c r

-h-convex function is given, and its important properties are discussed. Jensen type inequality, Hermite–Hadamard type inequalities and Fejér type inequalities for harmonically

c r

-h-convex functions are also established. [...] Read more.

In this paper, the definition of the harmonically

c r

-h-convex function is given, and its important properties are discussed. Jensen type inequality, Hermite–Hadamard type inequalities and Fejér type inequalities for harmonically

c r

-h-convex functions are also established. In addition, some numerical examples are given to verify the accuracy of the results. Full article

(This article belongs to the Section D2: Operations Research and Fuzzy Decision Making)

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