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37 pages, 776 KiB

Open AccessArticle

Fractional Inclusion Analysis of Superquadratic Stochastic Processes via Center-Radius Total Order Relation with Applications in Information Theory

by Mohsen Ayyash, Dawood Khan, Saad Ihsan Butt and Youngsoo Seol

Fractal Fract. 2025, 9(6), 375; https://doi.org/10.3390/fractalfract9060375 - 12 Jun 2025

Viewed by 319

Abstract

This study presents, for the first time, a new class of interval-valued superquadratic stochastic processes and examines their core properties through the lens of the center-radius total order relation on intervals. These processes serve as a powerful tool for modeling uncertainty in stochastic [...] Read more.

This study presents, for the first time, a new class of interval-valued superquadratic stochastic processes and examines their core properties through the lens of the center-radius total order relation on intervals. These processes serve as a powerful tool for modeling uncertainty in stochastic systems involving interval-valued data. By utilizing their intrinsic structure, we derive sharpened versions of Jensen-type and Hermite–Hadamard-type inequalities, along with their fractional extensions, within the framework of mean-square stochastic Riemann–Liouville fractional integrals. The theoretical findings are validated through extensive graphical representations and numerical simulations. Moreover, the applicability of the proposed processes is demonstrated in the domain of information theory by constructing novel stochastic divergence measures and Shannon’s entropy grounded in interval calculus. The outcomes of this work lay a solid foundation for further exploration in stochastic analysis, particularly in advancing generalized integral inequalities and formulating new stochastic models under uncertainty. Full article

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

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17 pages, 335 KiB

Open AccessArticle

Significant Study of Fuzzy Fractional Inequalities with Generalized Operators and Applications

by Rana Safdar Ali, Humira Sif, Gauhar Rehman, Ahmad Aloqaily and Nabil Mlaiki

Fractal Fract. 2024, 8(12), 690; https://doi.org/10.3390/fractalfract8120690 - 24 Nov 2024

Cited by 1 | Viewed by 654

Abstract

There are many techniques for the extension and generalization of fractional theories, one of which improves fractional operators by means of their kernels. This paper is devoted to the most general concept of interval-valued functions, studying fractional integral operators for interval-valued functions, along [...] Read more.

There are many techniques for the extension and generalization of fractional theories, one of which improves fractional operators by means of their kernels. This paper is devoted to the most general concept of interval-valued functions, studying fractional integral operators for interval-valued functions, along with the multi-variate extension of the Bessel–Maitland function, which acts as kernel. We discuss the behavior of Hermite–Hadamard Fejér (HHF)-type inequalities by using the convex fuzzy interval-valued function (C-FIVF) with generalized fuzzy fractional operators. Also, we obtain some refinements of Hermite–Hadamard(H-H)-type inequalities via convex fuzzy interval-valued functions (C-FIVFs). Our results extend and generalize existing findings from the literature. Full article

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

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 6 | Viewed by 1385

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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25 pages, 728 KiB

Open AccessArticle

On Extended Class of Totally Ordered Interval-Valued Convex Stochastic Processes and Applications

by Muhammad Zakria Javed, Muhammad Uzair Awan, Loredana Ciurdariu, Silvestru Sever Dragomir and Yahya Almalki

Fractal Fract. 2024, 8(10), 577; https://doi.org/10.3390/fractalfract8100577 - 30 Sep 2024

Cited by 7 | Viewed by 1065

Abstract

The intent of the current study is to explore convex stochastic processes within a broader context. We introduce the concept of unified stochastic processes to analyze both convex and non-convex stochastic processes simultaneously. We employ weighted quasi-mean, non-negative mapping

γ

, and center-radius [...] Read more.

The intent of the current study is to explore convex stochastic processes within a broader context. We introduce the concept of unified stochastic processes to analyze both convex and non-convex stochastic processes simultaneously. We employ weighted quasi-mean, non-negative mapping

γ

, and center-radius ordering relations to establish a class of extended

c r

-interval-valued convex stochastic processes. This class yields a combination of innovative convex and non-convex stochastic processes. We characterize our class by illustrating its relationships with other classes as well as certain key attributes and sufficient conditions for this class of processes. Additionally, leveraging Riemann–Liouville stochastic fractional operators and our proposed class, we prove parametric fractional variants of Jensen’s inequality, Hermite–Hadamard’s inequality, Fejer’s inequality, and product Hermite–Hadamard’s like inequality. We establish an interesting relation between means by means of Hermite–Hadamard’s inequality. We utilize the numerical and graphical approaches to showcase the significance and effectiveness of primary findings. Also, the proposed results are powerful tools to evaluate the bounds for stochastic Riemann–Liouville fractional operators in different scenarios for a larger space of processes. Full article

(This article belongs to the Special Issue New Trends on Generalized Fractional Calculus, 2nd Edition)

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25 pages, 4236 KiB

Open AccessArticle

A New Contribution in Fractional Integral Calculus and Inequalities over the Coordinated Fuzzy Codomain

by Zizhao Zhou, Ahmad Aziz Al Ahmadi, Alina Alb Lupas and Khalil Hadi Hakami

Axioms 2024, 13(10), 666; https://doi.org/10.3390/axioms13100666 - 26 Sep 2024

Cited by 1 | Viewed by 1068

Abstract

The correct derivation of integral inequalities on fuzzy-number-valued mappings depends on applying fractional calculus to fuzzy number analysis. The purpose of this article is to introduce a new class of convex mappings and generalize various previously published results on the fuzzy number and [...] Read more.

The correct derivation of integral inequalities on fuzzy-number-valued mappings depends on applying fractional calculus to fuzzy number analysis. The purpose of this article is to introduce a new class of convex mappings and generalize various previously published results on the fuzzy number and interval-valued mappings via fuzzy-order relations using fuzzy coordinated ỽ-convexity mappings so that the new version of the well-known Hermite–Hadamard (H-H) inequality can be presented in various variants via the fractional integral operators (Riemann–Liouville). Some new product forms of these inequalities for coordinated ỽ-convex fuzzy-number-valued mappings (coordinated ỽ-convex FNVMs) are also discussed. Additionally, we provide several fascinating non-trivial examples and exceptional cases to show that these results are accurate. Full article

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

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 776

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)

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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12 pages, 250 KiB

Open AccessArticle

Right Quantum Calculus on Finite Intervals with Respect to Another Function and Quantum Hermite–Hadamard Inequalities

by Asawathep Cuntavepanit, Sotiris K. Ntouyas and Jessada Tariboon

Axioms 2024, 13(7), 466; https://doi.org/10.3390/axioms13070466 - 10 Jul 2024

Cited by 1 | Viewed by 867

Abstract

In this paper, we study right quantum calculus on finite intervals with respect to another function. We present new definitions on the right quantum derivative and right quantum integral of a function with respect to another function and study their basic properties. The [...] Read more.

In this paper, we study right quantum calculus on finite intervals with respect to another function. We present new definitions on the right quantum derivative and right quantum integral of a function with respect to another function and study their basic properties. The new definitions generalize the previous existing results in the literature. We provide applications of the newly defined quantum calculus by obtaining new Hermite–Hadamard-type inequalities for convex, h-convex, and modified h-convex functions. Full article

(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)

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 1017

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)

20 pages, 2547 KiB

Open AccessArticle

The Estimation of Different Kinds of Integral Inequalities for a Generalized Class of Convex Mapping and a Harmonic Set via Fuzzy Inclusion Relations and Their Applications in Quadrature Theory

by Ali Althobaiti, Saad Althobaiti and Miguel Vivas Cortez

Axioms 2024, 13(6), 344; https://doi.org/10.3390/axioms13060344 - 22 May 2024

Cited by 2 | Viewed by 1117

Abstract

The relationship between convexity and symmetry is widely recognized. In fuzzy theory, both concepts exhibit similar behavior. It is crucial to remember that real and interval-valued mappings are special instances of fuzzy-number-valued mappings (

F - N - V - M s

), [...] Read more.

The relationship between convexity and symmetry is widely recognized. In fuzzy theory, both concepts exhibit similar behavior. It is crucial to remember that real and interval-valued mappings are special instances of fuzzy-number-valued mappings (

F - N - V - M s

), as fuzzy theory relies on the unit interval, which is crucial to resolving problems with interval analysis and fuzzy number theory. In this paper, a new harmonic convexities class of fuzzy numbers has been introduced via up and down relation. We show several Hermite–Hadamard (

H \cdot H

) and Fejér-type inequalities by the implementation of fuzzy Aumann integrals using the newly defined class of convexities. Some nontrivial examples are also presented to validate the main outcomes. Full article

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

29 pages, 514 KiB

Open AccessArticle

Symmetric Quantum Inequalities on Finite Rectangular Plane

by Saad Ihsan Butt, Muhammad Nasim Aftab and Youngsoo Seol

Mathematics 2024, 12(10), 1517; https://doi.org/10.3390/math12101517 - 13 May 2024

Cited by 5 | Viewed by 1226

Abstract

Finding the range of coordinated convex functions is yet another application for the symmetric Hermite–Hadamard inequality. For any two-dimensional interval

[a_{0}, a_{1}] \times [c_{0}, c_{1}] \subseteq ℜ^{2}

, we introduce the notion [...] Read more.

Finding the range of coordinated convex functions is yet another application for the symmetric Hermite–Hadamard inequality. For any two-dimensional interval

[a_{0}, a_{1}] \times [c_{0}, c_{1}] \subseteq ℜ^{2}

, we introduce the notion of partial

q_{θ}

-,

q_{ϕ}

-, and

q_{θ} q_{ϕ}

-symmetric derivatives and a

q_{θ} q_{ϕ}

-symmetric integral. Moreover, we will construct the

q_{θ} q_{ϕ}

-symmetric Hölder’s inequality, the symmetric quantum Hermite–Hadamard inequality for the function of two variables in a rectangular plane, and address some of its related applications. Full article

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

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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 1681

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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12 pages, 268 KiB

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 8 | Viewed by 1797

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)

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 1976

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)

21 pages, 674 KiB

Open AccessArticle

Properties and Applications of Symmetric Quantum Calculus

by Miguel Vivas-Cortez, Muhammad Zakria Javed, Muhammad Uzair Awan, Silvestru Sever Dragomir and Ahmed M. Zidan

Fractal Fract. 2024, 8(2), 107; https://doi.org/10.3390/fractalfract8020107 - 12 Feb 2024

Cited by 7 | Viewed by 2759

Abstract

Symmetric derivatives and integrals are extensively studied to overcome the limitations of classical derivatives and integral operators. In the current investigation, we explore the quantum symmetric derivatives on finite intervals. We introduced the idea of right quantum symmetric derivatives and integral operators and [...] Read more.

Symmetric derivatives and integrals are extensively studied to overcome the limitations of classical derivatives and integral operators. In the current investigation, we explore the quantum symmetric derivatives on finite intervals. We introduced the idea of right quantum symmetric derivatives and integral operators and studied various properties of both operators as well. Using these concepts, we deliver new variants of Young’s inequality, Hölder’s inequality, Minkowski’s inequality, Hermite–Hadamard’s inequality, Ostrowski’s inequality, and Gruss–Chebysev inequality. We report the Hermite–Hadamard’s inequalities by taking into account the differentiability of convex mappings. These fundamental results are pivotal to studying the various other problems in the field of inequalities. The validation of results is also supported with some visuals. Full article

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

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