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23 pages, 504 KiB

Open AccessArticle

Fractional Reverse Inequalities Involving Generic Interval-Valued Convex Functions and Applications

by Bandar Bin-Mohsin, Muhammad Zakria Javed, Muhammad Uzair Awan, Badreddine Meftah and Artion Kashuri

Fractal Fract. 2024, 8(10), 587; https://doi.org/10.3390/fractalfract8100587 - 3 Oct 2024

Cited by 7 | Viewed by 1390

Abstract

The relation between fractional calculus and convexity significantly impacts the development of the theory of integral inequalities. In this paper, we explore the reverse of Minkowski and Hölder’s inequality, unified Jensen’s inequality, and Hermite–Hadamard (

H

-

H

)-like inequalities using fractional calculus [...] Read more.

The relation between fractional calculus and convexity significantly impacts the development of the theory of integral inequalities. In this paper, we explore the reverse of Minkowski and Hölder’s inequality, unified Jensen’s inequality, and Hermite–Hadamard (

H

-

H

)-like inequalities using fractional calculus and a generic class of interval-valued convexity. We introduce the concept of

I . V

-

(⋏, ℏ)

generic class of convexity, which unifies several existing definitions of convexity. By utilizing Riemann–Liouville (R-L) fractional operators and

I . V

-

(⋏, ℏ)

convexity to derive new improvements of the

H

-

H

- and Fejer and Pachpatte-like inequalities. Our results are quite unified; by substituting the different values of parameters, we obtain a blend of new and existing inequalities. These results are fruitful for establishing bounds for

I . V

R-L integral operators. Furthermore, we discuss various implications of our findings, along with numerical examples and simulations to enhance the reliability of our results. Full article

(This article belongs to the Special Issue Fractional Integral Inequalities and Applications, 3rd Edition)

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19 pages, 345 KiB

Open AccessArticle

Research on New Interval-Valued Fractional Integrals with Exponential Kernel and Their Applications

by Abdulrahman F. Aljohani, Ali Althobaiti and Saad Althobaiti

Axioms 2024, 13(9), 616; https://doi.org/10.3390/axioms13090616 - 11 Sep 2024

Viewed by 787

Abstract

This paper aims to introduce a new fractional extension of the interval Hermite–Hadamard (

H H

),

H H

–Fejér, and Pachpatte-type inequalities for left- and right-interval-valued harmonically convex mappings (

L R I V H

convex mappings) with an exponential function in [...] Read more.

This paper aims to introduce a new fractional extension of the interval Hermite–Hadamard (

H H

),

H H

–Fejér, and Pachpatte-type inequalities for left- and right-interval-valued harmonically convex mappings (

L R I V H

convex mappings) with an exponential function in the kernel. We use fractional operators to develop several generalizations, capturing unique outcomes that are currently under investigation, while also introducing a new operator. Generally, we propose two methods that, in conjunction with more generalized fractional integral operators with an exponential function in the kernel, can address certain novel generalizations of increasing mappings under the assumption of

L R I V

convexity, yielding some noteworthy results. The results produced by applying the suggested scheme show that the computational effects are extremely accurate, flexible, efficient, and simple to implement in order to explore the path of upcoming intricate waveform and circuit theory research. Full article

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

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 975

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)

33 pages, 407 KiB

Open AccessArticle

Some New Estimations of Left and Right Interval Fractional Pachpatte’s Type Integral Inequalities via Rectangle Plane

by Azzh Saad Alshehry, Loredana Ciurdariu, Yaser Saber and Amal F. Soliman

Axioms 2024, 13(7), 417; https://doi.org/10.3390/axioms13070417 - 21 Jun 2024

Viewed by 1019

Abstract

Inequalities involving fractional operators have been an active area of research, which is crucial in establishing bounds, estimates, and stability conditions for solutions to fractional integrals. In this paper, we initially presented a new class that is known as coordinated left and right [...] Read more.

Inequalities involving fractional operators have been an active area of research, which is crucial in establishing bounds, estimates, and stability conditions for solutions to fractional integrals. In this paper, we initially presented a new class that is known as coordinated left and right

ℏ

-pre-invex interval-valued mappings (

C

·

L

·

R

-

ℏ

-pre-invex

Ι

·

V

-

M

), as well classical convex and nonconvex are also obtained. This newly defined class enabled us to derive novel inequalities, such as Hermite–Hadamard and Pachpatte’s type inequalities. Furthermore, the obtained results allowed us to recapture several special cases of known results for different parameter choices, which can be applications of the main results. Finally, we discussed the validity of the main outcomes. Full article

(This article belongs to the Section Mathematical Analysis)

25 pages, 399 KiB

Open AccessArticle

A New Class of Coordinated Non-Convex Fuzzy-Number-Valued Mappings with Related Inequalities and Their Applications

by Aleksandr Rakhmangulov, A. F. Aljohani, Ali Mubaraki and Saad Althobaiti

Axioms 2024, 13(6), 404; https://doi.org/10.3390/axioms13060404 - 16 Jun 2024

Cited by 2 | Viewed by 1071

Abstract

Both theoretical and applied mathematics depend heavily on integral inequalities with generalized convexity. Because of its many applications, the theory of integral inequalities is currently one of the areas of mathematics that is evolving at the fastest pace. In this paper, based on [...] Read more.

Both theoretical and applied mathematics depend heavily on integral inequalities with generalized convexity. Because of its many applications, the theory of integral inequalities is currently one of the areas of mathematics that is evolving at the fastest pace. In this paper, based on fuzzy Aumann’s integral theory, the Hermite–Hadamard’s type inequalities are introduced for a newly defined class of nonconvex functions, which is known as

U

·

D

preinvex fuzzy number-valued mappings (

U

·

D

preinvex

F

·

N

·

V

·

M

s) on coordinates. Some Pachpatte-type inequalities are also established for the product of two

U

·

D

preinvex

F

·

N

·

V

·

M

s, and some Hermite–Hadamard–Fejér-type inequalities are also acquired via fuzzy Aumann’s integrals. Additionally, several new generalized inequalities are also obtained for the special situations of the parameters. Additionally, some of the interesting remarks are provided to acquire the classical and new exceptional cases that can be considered as applications of the main outcomes. Lastly, a few suggested uses for these inequalities in numerical integration are made. Full article

(This article belongs to the Special Issue Analysis of Mathematical Inequalities)

34 pages, 1306 KiB

Open AccessArticle

Hyers–Ulam Stability of 2D-Convex Mappings and Some Related New Hermite–Hadamard, Pachpatte, and Fejér Type Integral Inequalities Using Novel Fractional Integral Operators via Totally Interval-Order Relations with Open Problem

by Waqar Afzal, Daniel Breaz, Mujahid Abbas, Luminiţa-Ioana Cotîrlă, Zareen A. Khan and Eleonora Rapeanu

Mathematics 2024, 12(8), 1238; https://doi.org/10.3390/math12081238 - 19 Apr 2024

Cited by 9 | Viewed by 1689

Abstract

The aim of this paper is to introduce a new type of two-dimensional convexity by using total-order relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves [...] Read more.

The aim of this paper is to introduce a new type of two-dimensional convexity by using total-order relations. In the first part of this paper, we examine the Hyers–Ulam stability of two-dimensional convex mappings by using the sandwich theorem. Our next step involves the development of Hermite–Hadamard inequality, including its weighted and product forms, by using a novel type of fractional operator having non-singular kernels. Moreover, we develop several nontrivial examples and remarks to demonstrate the validity of our main results. Finally, we examine approximate convex mappings and have left an open problem regarding the best optimal constants for two-dimensional approximate convexity. Full article

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

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24 pages, 1954 KiB

Open AccessArticle

New Version of Fractional Pachpatte-Type Integral Inequalities via Coordinated ℏ-Convexity via Left and Right Order Relation

by Tareq Saeed, Eze R. Nwaeze, Muhammad Bilal Khan and Khalil Hadi Hakami

Fractal Fract. 2024, 8(3), 125; https://doi.org/10.3390/fractalfract8030125 - 20 Feb 2024

Cited by 5 | Viewed by 1980

Abstract

In particular, the fractional forms of Hermite–Hadamard inequalities for the newly defined class of convex mappings proposed that are known as coordinated left and right

ℏ

-convexity (

L R

-

ℏ

-convexity) over interval-valued codomain. We exploit the use of double Riemann–Liouville [...] Read more.

In particular, the fractional forms of Hermite–Hadamard inequalities for the newly defined class of convex mappings proposed that are known as coordinated left and right

ℏ

-convexity (

L R

-

ℏ

-convexity) over interval-valued codomain. We exploit the use of double Riemann–Liouville fractional integral to derive the major results of the research. We also examine the key results’ numerical validations that examples are nontrivial. By taking the product of two left and right coordinated

ℏ

-convexity, some new versions of fractional integral inequalities are also obtained. Moreover, some new and classical exceptional cases are also discussed by taking some restrictions on endpoint functions of interval-valued functions that can be seen as applications of these new outcomes. Full article

(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)

18 pages, 545 KiB

Open AccessArticle

Certain Novel Dynamic Inequalities Applicable in the Theory of Retarded Dynamic Equations and Their Applications

by Sujata Bhamre, Nagesh Kale, Subhash Kendre and James Peters

Mathematics 2024, 12(3), 406; https://doi.org/10.3390/math12030406 - 26 Jan 2024

Cited by 1 | Viewed by 1154

Abstract

In this article, we establish certain time-scale-retarded dynamic inequalities that contain nonlinear retarded integral equations on various time scales. These inequalities extend and generalize some significant inequalities existing in the literature to their more general forms. The qualitative and quantitative characteristics of solutions [...] Read more.

In this article, we establish certain time-scale-retarded dynamic inequalities that contain nonlinear retarded integral equations on various time scales. These inequalities extend and generalize some significant inequalities existing in the literature to their more general forms. The qualitative and quantitative characteristics of solutions to various dynamic equations on time scales involving retarded integrals can be studied using these inequalities. The results presented in this manuscript furnish a powerful tool to analyze the boundedness of nonlinear integral equations with retarded integrals on several time scales. In the end, we also include numerical illustrations to signify the applicability of these results to power nonlinear retarded integral equations on real and quantum time scales. Full article

(This article belongs to the Special Issue Solutions of Integrable PDEs: Solving, Properties and Applications)

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27 pages, 930 KiB

Open AccessArticle

Weighted Fractional Hermite–Hadamard Integral Inequalities for up and down Ԓ-Convex Fuzzy Mappings over Coordinates

by Muhammad Bilal Khan, Eze R. Nwaeze, Cheng-Chi Lee, Hatim Ghazi Zaini, Der-Chyuan Lou and Khalil Hadi Hakami

Mathematics 2023, 11(24), 4974; https://doi.org/10.3390/math11244974 - 16 Dec 2023

Cited by 4 | Viewed by 1116

Abstract

Due to its significant influence on numerous areas of mathematics and practical sciences, the theory of integral inequality has attracted a lot of interest. Convexity has undergone several improvements, generalizations, and extensions over time in an effort to produce more accurate variations of [...] Read more.

Due to its significant influence on numerous areas of mathematics and practical sciences, the theory of integral inequality has attracted a lot of interest. Convexity has undergone several improvements, generalizations, and extensions over time in an effort to produce more accurate variations of known findings. This article’s main goal is to introduce a new class of convexity as well as to prove several Hermite–Hadamard type interval-valued integral inequalities in the fractional domain. First, we put forth the new notion of generalized convexity mappings, which is defined as

U D

-

Ԓ

-convexity on coordinates with regard to fuzzy-number-valued mappings and the up and down (

U D

) fuzzy relation. The generic qualities of this class make it novel. By taking into account different values for

Ԓ

, we produce several known classes of convexity. Additionally, we create some new fractional variations of the Hermite–Hadamard (

H H

) and Pachpatte types of inequalities using the concepts of coordinated

U D

-

Ԓ

-convexity and double Riemann–Liouville fractional operators. The results attained here are the most cohesive versions of previous findings. To demonstrate the importance of the key findings, we offer a number of concrete examples. Full article

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

27 pages, 1520 KiB

Open AccessArticle

Some New Fractional Inequalities for Coordinated Convexity over Convex Set Pertaining to Fuzzy-Number-Valued Settings Governed by Fractional Integrals

by Tareq Saeed, Adriana Cătaș, Muhammad Bilal Khan and Ahmed Mohammed Alshehri

Fractal Fract. 2023, 7(12), 856; https://doi.org/10.3390/fractalfract7120856 - 30 Nov 2023

Cited by 6 | Viewed by 1334

Abstract

In this study, we first propose some new concepts of coordinated up and down convex mappings with fuzzy-number values. Then, Hermite–Hadamard-type inequalities via coordinated up and down convex fuzzy-number-valued mapping (coordinated

U D

-convex

F N V M s

) are introduced. By [...] Read more.

In this study, we first propose some new concepts of coordinated up and down convex mappings with fuzzy-number values. Then, Hermite–Hadamard-type inequalities via coordinated up and down convex fuzzy-number-valued mapping (coordinated

U D

-convex

F N V M s

) are introduced. By taking the products of two coordinated

U D

-convex

F N V M s

, Pachpatte-type inequalities are also obtained. Some new conclusions are also derived by making particular decisions with the newly defined inequalities, and it is demonstrated that the recently discovered inequalities are expansions of comparable findings in the literature. It is important to note that the main outcomes are validated using nontrivial examples. Full article

(This article belongs to the Special Issue Mathematical Inequalities in Fractional Calculus and Applications)

21 pages, 457 KiB

Open AccessArticle

On Some New AB-Fractional Inclusion Relations

by Bandar Bin-Mohsin, Muhammad Zakria Javed, Muhammad Uzair Awan and Artion Kashuri

Fractal Fract. 2023, 7(10), 725; https://doi.org/10.3390/fractalfract7100725 - 30 Sep 2023

Cited by 10 | Viewed by 1461

Abstract

The theory of integral inequality has gained considerable attention due to its influential impact on several fields of mathematics and applied sciences. Over the years, numerous refinements, generalizations, and extensions of convexity have been explored to achieve more precise variants of already established [...] Read more.

The theory of integral inequality has gained considerable attention due to its influential impact on several fields of mathematics and applied sciences. Over the years, numerous refinements, generalizations, and extensions of convexity have been explored to achieve more precise variants of already established results. The principal idea of this article is to establish some interval-valued integral inequalities of the Hermite–Hadamard type in the fractional domain. First, we propose the idea of generalized interval-valued convexity with respect to the continuous monotonic functions ⋎, bifunction

ζ

, and based on the containment ordering relation, which is termed as

(⋎, h)

pre-invex functions. This class is innovative due to its generic characteristics. We generate numerous known and new classes of convexity by considering various values for ⋎ and

h

. Moreover, we use the notion of

(⋎, h)

-pre-invexity and Atangana–Baleanu (AB) fractional operators to develop some fresh fractional variants of the Hermite–Hadamard (HH), Pachpatte, and Hermite–Hadamard–Fejer (HHF) types of inequalities. The outcomes obtained here are the most unified forms of existing results. We provide several specific cases, as well as a numerical and graphical study, to show the significance of the major results. Full article

(This article belongs to the Section General Mathematics, Analysis)

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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 2421

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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32 pages, 422 KiB

Open AccessArticle

New Variant of Hermite–Hadamard, Fejér and Pachpatte-Type Inequality and Its Refinements Pertaining to Fractional Integral Operator

by Muhammad Tariq, Sotiris K. Ntouyas and Asif Ali Shaikh

Fractal Fract. 2023, 7(5), 405; https://doi.org/10.3390/fractalfract7050405 - 16 May 2023

Cited by 5 | Viewed by 1843

Abstract

In order to show novel generalizations of mathematical inequality, fractional integral operators are frequently used. Fractional operators are used to simulate a broad range of scientific as well as engineering phenomena such as elasticity, viscous fluid, fracture mechanics, continuous population, equilibrium, visco-elastic deformation, [...] Read more.

In order to show novel generalizations of mathematical inequality, fractional integral operators are frequently used. Fractional operators are used to simulate a broad range of scientific as well as engineering phenomena such as elasticity, viscous fluid, fracture mechanics, continuous population, equilibrium, visco-elastic deformation, heat conduction problems, and others. In this manuscript, we introduce some novel notions of generalized preinvexity, namely the

(m, t g s)

-type s-preinvex function, Godunova–Levin

(s, m)

-preinvex of the 1st and 2nd kind, and a prequasi m-invex. Furthermore, we explore a new variant of the Hermite–Hadamard (H–H), Fejér, and Pachpatte-type inequality via a generalized fractional integral operator, namely, a non-conformable fractional integral operator (NCFIO). In addition, we explore new equalities. With the help of these equalities, we examine and present several extensions of H–H and Fejér-type inequalities involving a newly introduced concept via NCFIO. Finally, we explore some special means as applications in the aspects of NCFIO. The results and the unique situations offered by this research are novel and significant improvements over previously published findings. Full article

(This article belongs to the Special Issue Initial and Boundary Value Problems for Differential Equations)

23 pages, 407 KiB

Open AccessArticle

Some New Fractional Hadamard and Pachpatte-Type Inequalities with Applications via Generalized Preinvexity

by Muhammad Tariq, Asif Ali Shaikh and Sotiris K. Ntouyas

Symmetry 2023, 15(5), 1033; https://doi.org/10.3390/sym15051033 - 7 May 2023

Cited by 1 | Viewed by 1648

Abstract

The term convexity associated with the theory of inequality in the sense of fractional analysis has a broad range of different and remarkable applications in the domain of applied sciences. The prime objective of this article is to investigate some new variants of [...] Read more.

The term convexity associated with the theory of inequality in the sense of fractional analysis has a broad range of different and remarkable applications in the domain of applied sciences. The prime objective of this article is to investigate some new variants of Hermite–Hadamard and Pachpatte-type integral inequalities involving the idea of the preinvex function in the frame of a fractional integral operator, namely the Caputo–Fabrizio fractional operator. By employing our approach, a new fractional integral identity that correlates with preinvex functions for first-order differentiable mappings is presented. Moreover, we derive some refinements of the Hermite–Hadamard-type inequality for mappings, whose first-order derivatives are generalized preinvex functions in the Caputo–Fabrizio fractional sense. From an application viewpoint, to represent the usability of the concerning results, we presented several inequalities by using special means of real numbers. Integral inequalities in association with convexity in the frame of fractional calculus have a strong relationship with symmetry. Our investigation provides a better image of convex analysis in the frame of fractional calculus. Full article

(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry)

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 3117

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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