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12 pages, 268 KB

Open AccessArticle

Some New Fractional Inequalities Defined Using cr-Log-h-Convex Functions and Applications

by Sikander Mehmood, Pshtiwan Othman Mohammed, Artion Kashuri, Nejmeddine Chorfi, Sarkhel Akbar Mahmood and Majeed A. Yousif

Symmetry 2024, 16(4), 407; https://doi.org/10.3390/sym16040407 - 1 Apr 2024

Cited by 9 | Viewed by 1999

Abstract

There is a strong correlation between the concept of convexity and symmetry. One of these is the class of interval-valued cr-log-h-convex functions, which is closely related to the theory of symmetry. In this paper, we obtain Hermite–Hadamard and its weighted version inequalities that [...] Read more.

There is a strong correlation between the concept of convexity and symmetry. One of these is the class of interval-valued cr-log-h-convex functions, which is closely related to the theory of symmetry. In this paper, we obtain Hermite–Hadamard and its weighted version inequalities that are related to interval-valued cr-log-h-convex functions, and some known results are recaptured. To support our main results, we offer three examples and two applications related to modified Bessel functions and special means as well. Full article

(This article belongs to the Special Issue Symmetries of Difference Equations, Special Functions and Orthogonal Polynomials II)

21 pages, 420 KB

Open AccessArticle

Some New Estimates of Hermite–Hadamard Inequalities for Harmonical cr-h-Convex Functions via Generalized Fractional Integral Operator on Set-Valued Mappings

by Yahya Almalki and Waqar Afzal

Mathematics 2023, 11(19), 4041; https://doi.org/10.3390/math11194041 - 23 Sep 2023

Cited by 15 | Viewed by 1486

Abstract

The application of fractional calculus to interval analysis is vital for the precise derivation of integral inequalities on set-valued mappings. The objective of this article is to reformulated the well-known Hermite–Hadamard inequality into various new variants via fractional integral operator (Riemann–Liouville) and generalize [...] Read more.

The application of fractional calculus to interval analysis is vital for the precise derivation of integral inequalities on set-valued mappings. The objective of this article is to reformulated the well-known Hermite–Hadamard inequality into various new variants via fractional integral operator (Riemann–Liouville) and generalize the various previously published results on set-valued mappings via center and radius order relations using harmonical h-convex functions. First, using these notions, we developed the Hermite–Hadamard (

H - H

) inequality, and then constructed some product form of these inequalities for harmonically convex functions. Moreover, to demonstrate the correctness of these results, we constructed some interesting non-trivial examples. Full article

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

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25 pages, 486 KB

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 18 | Viewed by 2576

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 [...] Read more.

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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13 pages, 418 KB

Open AccessArticle

New Fractional Integral Inequalities Pertaining to Center-Radius (cr)-Ordered Convex Functions

by Soubhagya Kumar Sahoo, Hleil Alrweili, Savin Treanţă and Zareen A. Khan

Fractal Fract. 2023, 7(1), 81; https://doi.org/10.3390/fractalfract7010081 - 11 Jan 2023

Cited by 1 | Viewed by 2202

Abstract

In this work, we use the idea of interval-valued convex functions of Center-Radius (

cr

)-order to give fractional versions of Hermite–Hadamard inequality. The results are supported by some numerical estimations and graphical representations considering some suitable examples. The results are novel in [...] Read more.

In this work, we use the idea of interval-valued convex functions of Center-Radius (

cr

)-order to give fractional versions of Hermite–Hadamard inequality. The results are supported by some numerical estimations and graphical representations considering some suitable examples. The results are novel in the context of

cr

-convex interval-valued functions and deal with differintegrals of the

(\frac{p + s}{2})

type. We believe this will be an important contribution to spurring additional research. Full article

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

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17 pages, 366 KB

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 2501

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.

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 relations are quite different and novel in the literature and are calculated as

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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18 pages, 404 KB

Open AccessArticle

Modified Inequalities on Center-Radius Order Interval-Valued Functions Pertaining to Riemann–Liouville Fractional Integrals

by Soubhagya Kumar Sahoo, Eman Al-Sarairah, Pshtiwan Othman Mohammed, Muhammad Tariq and Kamsing Nonlaopon

Axioms 2022, 11(12), 732; https://doi.org/10.3390/axioms11120732 - 15 Dec 2022

Cited by 7 | Viewed by 2074

Abstract

In this paper, we shall discuss a newly introduced concept of center-radius total-ordered relations between two intervals. Here, we address the Hermite–Hadamard-, Fejér- and Pachpatte-type inequalities by considering interval-valued Riemann–Liouville fractional integrals. Interval-valued fractional inequalities for a new class of preinvexity, i.e.,

cr

[...] Read more.

In this paper, we shall discuss a newly introduced concept of center-radius total-ordered relations between two intervals. Here, we address the Hermite–Hadamard-, Fejér- and Pachpatte-type inequalities by considering interval-valued Riemann–Liouville fractional integrals. Interval-valued fractional inequalities for a new class of preinvexity, i.e.,

cr

-h-preinvexity, are estimated. The fractional operator is used for the first time to prove such inequalities involving center–radius-ordered functions. Some numerical examples are also provided to validate the presented inequalities. Full article

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16 pages, 362 KB

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 23 | Viewed by 2217

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, [...] Read more.

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 KB

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 42 | Viewed by 2572

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 [...] Read more.

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)

24 pages, 1007 KB

Open AccessArticle

Hermite–Hadamard, Fejér and Pachpatte-Type Integral Inequalities for Center-Radius Order Interval-Valued Preinvex Functions

by Soubhagya Kumar Sahoo, Muhammad Amer Latif, Omar Mutab Alsalami, Savin Treanţă, Weerawat Sudsutad and Jutarat Kongson

Fractal Fract. 2022, 6(9), 506; https://doi.org/10.3390/fractalfract6090506 - 10 Sep 2022

Cited by 18 | Viewed by 2323

Abstract

The objective of this manuscript is to establish a link between the concept of inequalities and Center-Radius order functions, which are intriguing due to their properties and widespread use. We introduce the notion of the

CR

(Center-Radius)-order interval-valued preinvex function with the help [...] Read more.

The objective of this manuscript is to establish a link between the concept of inequalities and Center-Radius order functions, which are intriguing due to their properties and widespread use. We introduce the notion of the

CR

(Center-Radius)-order interval-valued preinvex function with the help of a total order relation between two intervals. Furthermore, we discuss some properties of this new class of preinvexity and show that the new concept unifies several known concepts in the literature and also gives rise to some new definitions. By applying these new definitions, we have amassed many classical and novel special cases that serve as applications of the key findings of the manuscript. The computations of

cr

-order intervals depend upon the following concept

B = ⟨ B_{c}, B_{r} ⟩ = ⟨ \frac{\bar{B} + \underset{̲}{B}}{2}, \frac{\bar{B} - \underset{̲}{B}}{2} ⟩ .

Then, for the first time, inequalities such as Hermite–Hadamard, Pachpatte, and Fejér type are established for

CR

-order in association with the concept of interval-valued preinvexity. Some numerical examples are given to validate the main results. The results confirm that this new concept is very useful in connection with various inequalities. A fractional version of the Hermite–Hadamard inequality is also established to show how the presented results can be connected to fractional calculus in future developments. Our presented results will motivate further research on inequalities for fractional interval-valued functions, fuzzy interval-valued functions, and their associated optimization problems. Full article

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

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15 pages, 309 KB

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 2432

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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