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27 pages, 460 KiB

Open AccessArticle

A New Inclusion on Inequalities of the Hermite–Hadamard–Mercer Type for Three-Times Differentiable Functions

by Talib Hussain, Loredana Ciurdariu and Eugenia Grecu

Mathematics 2024, 12(23), 3711; https://doi.org/10.3390/math12233711 - 26 Nov 2024

Cited by 1 | Viewed by 573

The goal of this study is to develop numerous Hermite–Hadamard–Mercer (H–H–M)-type inequalities involving various fractional integral operators, including classical, Riemann–Liouville (R.L), k-Riemann–Liouville (k-R.L), and their generalized fractional integral operators. In addition, we establish a number of corresponding fractional integral inequalities for three-times differentiable [...] Read more.

The goal of this study is to develop numerous Hermite–Hadamard–Mercer (H–H–M)-type inequalities involving various fractional integral operators, including classical, Riemann–Liouville (R.L), k-Riemann–Liouville (k-R.L), and their generalized fractional integral operators. In addition, we establish a number of corresponding fractional integral inequalities for three-times differentiable convex functions that are connected to the right side of the H–H–M-type inequality. For these results, further remarks and observations are provided. Following that, a couple of graphical representations are shown to highlight the key findings of our study. Finally, some applications on special means are shown to demonstrate the effectiveness of our inequalities. Full article

(This article belongs to the Special Issue Fractional Calculus and Mathematical Applications, 2nd Edition)

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27 pages, 1378 KiB

Open AccessArticle

Generalized Fuzzy-Valued Convexity with Ostrowski’s, and Hermite-Hadamard Type Inequalities over Inclusion Relations and Their Applications

by Miguel Vivas Cortez, Ali Althobaiti, Abdulrahman F. Aljohani and Saad Althobaiti

Axioms 2024, 13(7), 471; https://doi.org/10.3390/axioms13070471 - 12 Jul 2024

Viewed by 965

Abstract

Convex inequalities and fuzzy-valued calculus converge to form a comprehensive mathematical framework that can be employed to understand and analyze a broad spectrum of issues. This paper utilizes fuzzy Aumman’s integrals to establish integral inequalities of Hermite-Hahadard, Fejér, and Pachpatte types within up [...] Read more.

Convex inequalities and fuzzy-valued calculus converge to form a comprehensive mathematical framework that can be employed to understand and analyze a broad spectrum of issues. This paper utilizes fuzzy Aumman’s integrals to establish integral inequalities of Hermite-Hahadard, Fejér, and Pachpatte types within up and down (

U

·

D

) relations and over newly defined class

U

·

D

-ħ-Godunova–Levin convex fuzzy-number mappings. To demonstrate the unique properties of

U

·

D

-relations, recent findings have been developed using fuzzy Aumman’s, as well as various other fuzzy partial order relations that have notable deficiencies outlined in the literature. Several compelling examples were constructed to validate the derived results, and multiple notes were provided to illustrate, depending on the configuration, that this type of integral operator generalizes several previously documented conclusions. This endeavor can potentially advance mathematical theory, computational techniques, and applications across various fields. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

26 pages, 565 KiB

Open AccessArticle

Fractional Hermite–Hadamard–Mercer-Type Inequalities for Interval-Valued Convex Stochastic Processes with Center-Radius Order and Their Related Applications in Entropy and Information Theory

by Ahsan Fareed Shah, Serap Özcan, Miguel Vivas-Cortez, Muhammad Shoaib Saleem and Artion Kashuri

Fractal Fract. 2024, 8(7), 408; https://doi.org/10.3390/fractalfract8070408 - 11 Jul 2024

Cited by 3 | Viewed by 1658

Abstract

We propose a new definition of the

γ

-convex stochastic processes

(C SP)

using center and radius

(CR)

order with the notion of interval valued functions

(_{C . R}^{I . V})

. By utilizing this definition [...] Read more.

We propose a new definition of the

γ

-convex stochastic processes

(C SP)

using center and radius

(CR)

order with the notion of interval valued functions

(_{C . R}^{I . V})

. By utilizing this definition and Mean-Square Fractional Integrals, we generalize fractional Hermite–Hadamard–Mercer-type inclusions for generalized

_{C . R}^{I . V}

versions of convex, tgs-convex, P-convex, exponential-type convex, Godunova–Levin convex, s-convex, Godunova–Levin s-convex, h-convex, n-polynomial convex, and fractional n-polynomial

(C SP)

. Also, our work uses interesting examples of

_{C . R}^{I . V} (C SP)

with Python-programmed graphs to validate our findings using an extension of Mercer’s inclusions with applications related to entropy and information theory. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 2nd Edition)

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13 pages, 297 KiB

Open AccessArticle

Some Fractional Integral Inequalities by Way of Raina Fractional Integrals

by Miguel Vivas-Cortez, Asia Latif and Rashida Hussain

Symmetry 2023, 15(10), 1935; https://doi.org/10.3390/sym15101935 - 19 Oct 2023

Cited by 2 | Viewed by 1235

Abstract

In this research, some novel Hermite–Hadamard–Fejér-type inequalities using Raina fractional integrals for the class of

ϑ

-convex functions are obtained. These inequalities are more comprehensive and inclusive than the corresponding ones present in the literature. Full article

(This article belongs to the Special Issue Symmetry Applied in Fractional Dynamics, Fractional Calculus and Inequalities)

33 pages, 514 KiB

Open AccessArticle

Some New Properties of Exponential Trigonometric Convex Functions Using up and down Relations over Fuzzy Numbers and Related Inequalities through Fuzzy Fractional Integral Operators Having Exponential Kernels

by Muhammad Bilal Khan, Jorge E. Macías-Díaz, Ali Althobaiti and Saad Althobaiti

Fractal Fract. 2023, 7(7), 567; https://doi.org/10.3390/fractalfract7070567 - 24 Jul 2023

Cited by 6 | Viewed by 1372

Abstract

The concept of convexity is fundamental in order to produce various types of inequalities. Thus, convexity and integral inequality are closely related. The objectives of this paper are to present a new class of up and down convex fuzzy number valued functions known [...] Read more.

The concept of convexity is fundamental in order to produce various types of inequalities. Thus, convexity and integral inequality are closely related. The objectives of this paper are to present a new class of up and down convex fuzzy number valued functions known as up and down exponential trigonometric convex fuzzy number valued mappings (

U D E T -

convex FNVMs) and, with the help of this newly defined class, Hermite–Hadamard-type inequalities (H–H-type inequalities) via fuzzy inclusion relation and fuzzy fractional integral operators having exponential kernels. This fuzzy inclusion relation is level-wise defined by the interval-based inclusion relation. Furthermore, we have shown that our findings apply to a significant class of both novel and well-known inequalities for

U D E T - convex F N V M

s. The application of the theory developed in this study is illustrated with useful instances. Some very interesting examples are provided to discuss the validation of our main results. These results and other approaches may open up new avenues for modeling, interval-valued functions, and fuzzy optimization problems. Full article

(This article belongs to the Special Issue Advances in Variable-Order Fractional Calculus and Its Applications)

23 pages, 874 KiB

Open AccessArticle

Some New Properties of Convex Fuzzy-Number-Valued Mappings on Coordinates Using Up and Down Fuzzy Relations and Related Inequalities

by Muhammad Bilal Khan, Ali Althobaiti, Cheng-Chi Lee, Mohamed S. Soliman and Chun-Ta Li

Mathematics 2023, 11(13), 2851; https://doi.org/10.3390/math11132851 - 25 Jun 2023

Cited by 5 | Viewed by 1158

Abstract

The symmetric function class interacts heavily with other types of functions. One of these is the convex function class, which is strongly related to symmetry theory. In this study, we define a novel class of convex mappings on planes using a fuzzy inclusion [...] Read more.

The symmetric function class interacts heavily with other types of functions. One of these is the convex function class, which is strongly related to symmetry theory. In this study, we define a novel class of convex mappings on planes using a fuzzy inclusion relation, known as coordinated up and down convex fuzzy-number-valued mapping. Several new definitions are introduced by placing some moderate restrictions on the notion of coordinated up and down convex fuzzy-number-valued mapping. Other uncommon examples are also described using these definitions, which can be viewed as applications of the new outcomes. Moreover, Hermite–Hadamard–Fejér inequalities are acquired via fuzzy double Aumann integrals, and the validation of these outcomes is discussed with the help of nontrivial examples and suitable choices of coordinated up and down convex fuzzy-number-valued mappings. Full article

(This article belongs to the Special Issue Fuzzy Modeling and Fuzzy Control Systems)

21 pages, 463 KiB

Open AccessArticle

Generalized AB-Fractional Operator Inclusions of Hermite–Hadamard’s Type via Fractional Integration

by Bandar Bin-Mohsin, Muhammad Uzair Awan, Muhammad Zakria Javed, Awais Gul Khan, Hüseyin Budak, Marcela V. Mihai and Muhammad Aslam Noor

Symmetry 2023, 15(5), 1012; https://doi.org/10.3390/sym15051012 - 1 May 2023

Cited by 9 | Viewed by 3115

Abstract

The aim of this research is to explore fractional integral inequalities that involve interval-valued preinvex functions. Initially, a new set of fractional operators is introduced that uses the extended generalized Mittag-Leffler function [...] Read more.

The aim of this research is to explore fractional integral inequalities that involve interval-valued preinvex functions. Initially, a new set of fractional operators is introduced that uses the extended generalized Mittag-Leffler function

E_{μ, α, l}^{γ, δ, k, c} (τ; p)

as a kernel in the interval domain. Additionally, a new form of Atangana–Baleanu operator is defined using the same kernel, which unifies multiple existing integral operators. By varying the parameters in

E_{μ, α, l}^{γ, δ, k, c} (τ; p)

, several new fractional operators are obtained. This study then utilizes the generalized AB integral operators and the preinvex interval-valued property of functions to establish new Hermite–Hadamard, Pachapatte, and Hermite–Hadamard–Fejer inequalities. The results are supported by numerical examples, graphical illustrations, and special cases. Full article

(This article belongs to the Special Issue Symmetry in Functional Equations and Analytic Inequalities III)

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19 pages, 928 KiB

Open AccessArticle

Properties of Convex Fuzzy-Number-Valued Functions on Harmonic Convex Set in the Second Sense and Related Inequalities via Up and Down Fuzzy Relation

by Muhammad Bilal Khan, Željko Stević, Abdulwadoud A. Maash, Muhammad Aslam Noor and Mohamed S. Soliman

Axioms 2023, 12(4), 399; https://doi.org/10.3390/axioms12040399 - 20 Apr 2023

Cited by 3 | Viewed by 1552

Abstract

In this paper, we provide different variants of the Hermite–Hadamard (

H \cdot H

) inequality using the concept of a new class of convex mappings, which is referred to as up and down harmonically

s

-convex fuzzy-number-valued functions (

U D H

[...] Read more.

In this paper, we provide different variants of the Hermite–Hadamard (

H \cdot H

) inequality using the concept of a new class of convex mappings, which is referred to as up and down harmonically

s

-convex fuzzy-number-valued functions (

U D H

s

-convex

F N V M

) in the second sense based on the up and down fuzzy inclusion relation. The findings are confirmed with certain numerical calculations that take a few appropriate examples into account. The results deal with various integrals of the

\frac{2 ρ σ}{ρ + σ}

type and are innovative in the setting of up and down harmonically

s

-convex fuzzy-number-valued functions. Moreover, we acquire classical and new exceptional cases that can be seen as applications of our main outcomes. In our opinion, this will make a significant contribution to encouraging more research. Full article

(This article belongs to the Special Issue Editorial Board Members’ Collection Series: Fuzzy Modeling, Optimization and Computational Intelligence)

18 pages, 393 KiB

Open AccessArticle

Some New Estimates of Fuzzy Integral Inequalities for Harmonically Convex Fuzzy-Number-Valued Mappings via up and down Fuzzy Relation

by Muhammad Bilal Khan, Aziz Ur Rahman, Abdulwadoud A. Maash, Savin Treanțǎ and Mohamed S. Soliman

Axioms 2023, 12(4), 365; https://doi.org/10.3390/axioms12040365 - 10 Apr 2023

Cited by 8 | Viewed by 1447

Abstract

In this article, the up and down harmonically convex fuzzy-number-valued mapping which is a novel kind of harmonically convex fuzzy-number-valued mapping is introduced. In addition, it is highlighted that the new idea of up and down harmonically convex fuzzy-number-valued mapping ( [...] Read more.

In this article, the up and down harmonically convex fuzzy-number-valued mapping which is a novel kind of harmonically convex fuzzy-number-valued mapping is introduced. In addition, it is highlighted that the new idea of up and down harmonically convex fuzzy-number-valued mapping (

U - O - H

convex

F - N - V - M

), which is a generalization of the previous class, describes a variety of new and classical classes as special cases by employing some mild restrictions. With the help of fuzzy inclusion relation, the new versions of the Hermite–Hadamard-type (HH-type) inequalities for up and down harmonically convex fuzzy-number-valued mappings are established. Then, we introduce a new version of Hermite–Hadamard Fejér-type inequality via fuzzy inclusion relation by using up and down harmonically convex fuzzy-number-valued mapping. Additionally, several instances are given to illustrate our main findings. Full article

17 pages, 345 KiB

Open AccessArticle

Some New Estimates of Hermite–Hadamard, Ostrowski and Jensen-Type Inclusions for h-Convex Stochastic Process via Interval-Valued Functions

by Waqar Afzal, Evgeniy Yu. Prosviryakov, Sheza M. El-Deeb and Yahya Almalki

Symmetry 2023, 15(4), 831; https://doi.org/10.3390/sym15040831 - 30 Mar 2023

Cited by 19 | Viewed by 2002

Abstract

Mathematical programming and optimization problems related to fluid dynamics are heavily influenced by stochastic processes associated with integral and variational inequalities. Furthermore, symmetry and convexity are intrinsically related. Over the last few years, both have become increasingly interconnected so that we can learn [...] Read more.

Mathematical programming and optimization problems related to fluid dynamics are heavily influenced by stochastic processes associated with integral and variational inequalities. Furthermore, symmetry and convexity are intrinsically related. Over the last few years, both have become increasingly interconnected so that we can learn from one and apply it to the other. The objective of this note is to convert ordinary stochastic processes into interval stochastic processes due to the wide range of applications in various disciplines. We have developed Hermite–Hadamard (

H . H

), Ostrowski-, and Jensen-type inequalities using interval h-convex stochastic processes. Our main results can be applied to a variety of new and well-known outcomes as specific situations. The results of this study are expected to stimulate future research on inequalities using fractional and fuzzy integral operators. Furthermore, we validate our main findings by providing some non-trivial examples. To demonstrate their general properties, we illustrate the connections between the examined results and those that have already been published. The results discussed in this article can be seen as improvements and refinements to results that have already been published. This is a fascinating subject that can be investigated in the future to identify equivalent inequalities for various convexity types. Full article

(This article belongs to the Special Issue Symmetry in CFD: Convection, Diffusion and Dynamics)

23 pages, 1113 KiB

Open AccessArticle

Some Certain Fuzzy Aumann Integral Inequalities for Generalized Convexity via Fuzzy Number Valued Mappings

by Muhammad Bilal Khan, Hakeem A. Othman, Michael Gr. Voskoglou, Lazim Abdullah and Alia M. Alzubaidi

Mathematics 2023, 11(3), 550; https://doi.org/10.3390/math11030550 - 19 Jan 2023

Cited by 9 | Viewed by 3619

Abstract

The topic of convex and nonconvex mapping has many applications in engineering and applied mathematics. The Aumann and fuzzy Aumann integrals are the most significant interval and fuzzy operators that allow the classical theory of integrals to be generalized. This paper considers the [...] Read more.

The topic of convex and nonconvex mapping has many applications in engineering and applied mathematics. The Aumann and fuzzy Aumann integrals are the most significant interval and fuzzy operators that allow the classical theory of integrals to be generalized. This paper considers the well-known fuzzy Hermite–Hadamard (HH) type and associated inequalities. With the help of fuzzy Aumann integrals and the newly introduced fuzzy number valued up and down convexity (

U D

-convexity), we increase this mileage even further. Additionally, with the help of definitions of lower

U D

-concave (lower

U D

-concave) and upper

U D

-convex (concave) fuzzy number valued mappings (

F N V M

s), we have gathered a sizable collection of both well-known and new extraordinary cases that act as applications of the main conclusions. We also offer a few examples of fuzzy number valued

U D

-convexity to further demonstrate the validity of the fuzzy inclusion relations presented in this study. Full article

(This article belongs to the Special Issue Fuzzy Sets, Fuzzy Logic and Their Applications 2021)

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 2089

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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24 pages, 744 KiB

Open AccessArticle

New Class Up and Down

λ

-Convex Fuzzy-Number Valued Mappings and Related Fuzzy Fractional Inequalities

by Muhammad Bilal Khan, Hatim Ghazi Zaini, Gustavo Santos-García, Muhammad Aslam Noor and Mohamed S. Soliman

Fractal Fract. 2022, 6(11), 679; https://doi.org/10.3390/fractalfract6110679 - 16 Nov 2022

Cited by 12 | Viewed by 2091

Abstract

The fuzzy-number valued up and down

λ

-convex mapping is originally proposed as an intriguing generalization of the convex mappings. The newly suggested mappings are then used to create certain Hermite–Hadamard- and Pachpatte-type integral fuzzy inclusion relations in fuzzy fractional calculus. It is [...] Read more.

The fuzzy-number valued up and down

λ

-convex mapping is originally proposed as an intriguing generalization of the convex mappings. The newly suggested mappings are then used to create certain Hermite–Hadamard- and Pachpatte-type integral fuzzy inclusion relations in fuzzy fractional calculus. It is also suggested to revise the Hermite–Hadamard integral fuzzy inclusions with regard to the up and down

λ

-convex fuzzy-number valued mappings (U∙D

λ

-convex F-N∙V∙Ms). Moreover, Hermite–Hadamard–Fejér has been proven, and some examples are given to demonstrate the validation of our main results. The new and exceptional cases are presented in terms of the change of the parameters

“ i ”

and

“ α ”

in order to assess the accuracy of the obtained fuzzy inclusion relations in this study. Full article

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

23 pages, 344 KiB

Open AccessArticle

Some Hermite–Hadamard and Hermite–Hadamard–Fejér Type Fractional Inclusions Pertaining to Different Kinds of Generalized Preinvexities

by Muhammad Tariq, Soubhagya Kumar Sahoo, Sotiris K. Ntouyas, Omar Mutab Alsalami, Asif Ali Shaikh and Kamsing Nonlaopon

Symmetry 2022, 14(10), 1957; https://doi.org/10.3390/sym14101957 - 20 Sep 2022

Cited by 5 | Viewed by 1850

Abstract

Fractional derivative and integral operators are often employed to present new generalizations of mathematical inequalities. The introduction of new fractional operators has prompted another direction in different branches of mathematics and applied sciences. First, we investigate and prove new fractional equality. Considering this [...] Read more.

Fractional derivative and integral operators are often employed to present new generalizations of mathematical inequalities. The introduction of new fractional operators has prompted another direction in different branches of mathematics and applied sciences. First, we investigate and prove new fractional equality. Considering this equality as the auxiliary result, we attain some estimations of a Hermite–Hadamard type inequality involving s-preinvex, s-Godunova–Levin preinvex, and prequasi invex functions. In addition, we investigate a fractional order Hadamard–Fejér inequality and some of its refinements pertaining to h-preinvexity via a non-conformable fractional integral operator. Finally, we present a Pachpatte type inequality for the product of two preinvex functions. The findings as well as the special cases presented in this research are new and applications of our main results. Full article

(This article belongs to the Special Issue Symmetry in Functional Equations and Analytic Inequalities III)

25 pages, 790 KiB

Open AccessArticle

Hermite–Hadamard and Pachpatte Type Inequalities for Coordinated Preinvex Fuzzy-Interval-Valued Functions Pertaining to a Fuzzy-Interval Double Integral Operator

by Gustavo Santos-García, Muhammad Bilal Khan, Hleil Alrweili, Ahmad Aziz Alahmadi and Sherif S. M. Ghoneim

Mathematics 2022, 10(15), 2756; https://doi.org/10.3390/math10152756 - 3 Aug 2022

Cited by 12 | Viewed by 1727

Abstract

Many authors have recently examined the relationship between symmetry and generalized convexity. Generalized convexity and symmetry have become a new area of study in the field of inequalities as a result of this close relationship. In this article, we introduce the idea of [...] Read more.

Many authors have recently examined the relationship between symmetry and generalized convexity. Generalized convexity and symmetry have become a new area of study in the field of inequalities as a result of this close relationship. In this article, we introduce the idea of preinvex fuzzy-interval-valued functions (preinvex F∙I-V∙F) on coordinates in a rectangle drawn on a plane and show that these functions have Hermite–Hadamard-type inclusions. We also develop Hermite–Hadamard-type inclusions for the combination of two coordinated preinvex functions with interval values. The weighted Hermite–Hadamard-type inclusions for products of coordinated convex interval-valued functions discussed in a recent publication by Khan et al. in 2022 served as the inspiration for our conclusions. Our proven results expand and generalize several previous findings made in the body of literature. Additionally, we offer appropriate examples to corroborate our theoretical main findings. Full article

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

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