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18 pages, 288 KiB

Open AccessFeature PaperArticle

Majorization-Type Integral Inequalities Related to a Result of Bennett with Applications

by László Horváth

Mathematics 2025, 13(10), 1563; https://doi.org/10.3390/math13101563 - 9 May 2025

Viewed by 281

In this paper, starting from abstract versions of a result of Bennett given by Niculescu, we derive new majorization-type integral inequalities for convex functions using finite signed measures. The proof of the main result is based on a generalization of a recently discovered [...] Read more.

In this paper, starting from abstract versions of a result of Bennett given by Niculescu, we derive new majorization-type integral inequalities for convex functions using finite signed measures. The proof of the main result is based on a generalization of a recently discovered majorization-type integral inequality. As applications of the results, we give simple proofs of the integral Jensen and Lah–Ribarič inequalities for finite signed measures, generalize and extend known results, and obtain an interesting new refinement of the Hermite–Hadamard–Fejér inequality. Full article

19 pages, 370 KiB

Open AccessArticle

On Quantum Hermite-Hadamard-Fejer Type Integral Inequalities via Uniformly Convex Functions

by Hasan Barsam, Somayeh Mirzadeh, Yamin Sayyari and Loredana Ciurdariu

Fractal Fract. 2025, 9(2), 108; https://doi.org/10.3390/fractalfract9020108 - 12 Feb 2025

Cited by 2 | Viewed by 812

Abstract

The main goal of this study is to provide new q-Fejer and q-Hermite-Hadamard type integral inequalities for uniformly convex functions and functions whose second quantum derivatives in absolute values are uniformly convex. Two basic inequalities as power mean inequality and Holder’s [...] Read more.

The main goal of this study is to provide new q-Fejer and q-Hermite-Hadamard type integral inequalities for uniformly convex functions and functions whose second quantum derivatives in absolute values are uniformly convex. Two basic inequalities as power mean inequality and Holder’s inequality are used in demonstrations. Some particular functions are chosen to illustrate the investigated results by two examples analyzed and the result obtained have been graphically visualized. Full article

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33 pages, 641 KiB

Open AccessArticle

Computational Representation of Fractional Inequalities Through 2D and 3D Graphs with Applications

by Muhammad Younis, Ahsan Mehmood, Muhammad Samraiz, Gauhar Rahman, Salma Haque, Ahmad Aloqaily and Nabil Mlaiki

Computation 2025, 13(2), 46; https://doi.org/10.3390/computation13020046 - 7 Feb 2025

Cited by 1 | Viewed by 650

Abstract

The aim of this research article is to use the extended fractional operators involving the multivariate Mittag–Leffler (M-M-L) function, we provide the generalization of the Hermite–Hadamard–Fejer (H-H-F) inequalities. We relate these inequalities to previously published disparities in the literature by making appropriate substitutions. [...] Read more.

The aim of this research article is to use the extended fractional operators involving the multivariate Mittag–Leffler (M-M-L) function, we provide the generalization of the Hermite–Hadamard–Fejer (H-H-F) inequalities. We relate these inequalities to previously published disparities in the literature by making appropriate substitutions. In the last section, we analyze several inequalities related to the H-H-F inequalities, focusing on generalized h-convexity associated with extended fractional operators involving the M-M-L function. To achieve this, we derive two identities for locally differentiable functions, which allows us to provide specific estimates for the differences between the left, middle, and right terms in the H-H-F inequalities. Also, we have constructed specific inequalities and visualized them through graphical representations to facilitate their applications in analysis. The research bridges theoretical advancements with practical applications, providing high-accuracy bounds for complex systems involving fractional calculus. Full article

(This article belongs to the Topic Fractional Calculus: Theory 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 660

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 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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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 1074

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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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 784

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)

31 pages, 431 KiB

Open AccessArticle

Fuzzy Milne, Ostrowski, and Hermite–Hadamard-Type Inequalities for ħ-Godunova–Levin Convexity and Their Applications

by Juan Wang, Valer-Daniel Breaz, Yasser Salah Hamed, Luminita-Ioana Cotirla and Xuewu Zuo

Axioms 2024, 13(7), 465; https://doi.org/10.3390/axioms13070465 - 10 Jul 2024

Viewed by 828

Abstract

In this paper, we establish several Milne-type inequalities for fuzzy number mappings and investigate their relationships with other inequalities. Specifically, we utilize Aumann’s integral and the fuzzy Kulisch–Miranker order, as well as the newly defined class,

ħ

-Godunova–Levin convex fuzzy number mappings, to [...] Read more.

In this paper, we establish several Milne-type inequalities for fuzzy number mappings and investigate their relationships with other inequalities. Specifically, we utilize Aumann’s integral and the fuzzy Kulisch–Miranker order, as well as the newly defined class,

ħ

-Godunova–Levin convex fuzzy number mappings, to derive Ostrowski’s and Hermite–Hadamard-type inequalities for fuzzy number mappings. Using the fuzzy Kulisch–Miranker order, we also establish connections with Hermite–Hadamard-type inequalities. Furthermore, we explore novel ideas and results based on Hermite–Hadamard–Fejér and provide examples and applications to illustrate our findings. Some very interesting examples are also provided to discuss the validation of the main results. Additionally, some new exceptional and classical outcomes have been obtained, which can be considered as applications of our main results. Full article

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

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 1068

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)

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 1121

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)

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 1686

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 1805

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)

28 pages, 431 KiB

Open AccessArticle

Weighted Fejér, Hermite–Hadamard, and Trapezium-Type Inequalities for

(h_{1}, h_{2})

–Godunova–Levin Preinvex Function with Applications and Two Open Problems

by Abdullah Ali H. Ahmadini, Waqar Afzal, Mujahid Abbas and Elkhateeb S. Aly

Mathematics 2024, 12(3), 382; https://doi.org/10.3390/math12030382 - 24 Jan 2024

Cited by 20 | Viewed by 1401

Abstract

This note introduces a new class of preinvexity called

(h_{1}, h_{2})

-Godunova-Levin preinvex functions that generalize earlier findings. Based on these notions, we developed Hermite-Hadamard, weighted Fejér, and trapezium type inequalities. Furthermore, we constructed some non-trivial examples in [...] Read more.

This note introduces a new class of preinvexity called

(h_{1}, h_{2})

-Godunova-Levin preinvex functions that generalize earlier findings. Based on these notions, we developed Hermite-Hadamard, weighted Fejér, and trapezium type inequalities. Furthermore, we constructed some non-trivial examples in order to verify all the developed results. In addition, we discussed some applications related to the trapezoidal formula, probability density functions, special functions and special means. Lastly, we discussed the importance of order relations and left two open problems for future research. As an additional benefit, we believe that the present work can provide a strong catalyst for enhancing similar existing literature. Full article

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

25 pages, 1038 KiB

Open AccessArticle

New Versions of Fuzzy-Valued Integral Inclusion over p-Convex Fuzzy Number-Valued Mappings and Related Fuzzy Aumman’s Integral Inequalities

by Nasser Aedh Alreshidi, Muhammad Bilal Khan, Daniel Breaz and Luminita-Ioana Cotirla

Symmetry 2023, 15(12), 2123; https://doi.org/10.3390/sym15122123 - 28 Nov 2023

Cited by 3 | Viewed by 1115

Abstract

It is well known that both concepts of symmetry and convexity are directly connected. Similarly, in fuzzy theory, both ideas behave alike. It is important to note that real and interval-valued mappings are exceptional cases of fuzzy number-valued mappings ( [...] Read more.

It is well known that both concepts of symmetry and convexity are directly connected. Similarly, in fuzzy theory, both ideas behave alike. It is important to note that real and interval-valued mappings are exceptional cases of fuzzy number-valued mappings (

F N V M

s) because fuzzy theory depends upon the unit interval that make a significant contribution to overcoming the issues that arise in the theory of interval analysis and fuzzy number theory. In this paper, the new class of

p

-convexity over up and down (

U D

) fuzzy relation has been introduced which is known as

U D

-

p

-convex fuzzy number-valued mappings (

U D

-

p

-convex

F N V M

s). We offer a thorough analysis of Hermite–Hadamard-type inequalities for

F N V M

s that are

U D

-

p

-convex using the fuzzy Aumann integral. Some previous results from the literature are expanded upon and broadly applied in our study. Additionally, we offer precise justifications for the key theorems that Kunt and İşcan first deduced in their article titled “Hermite–Hadamard–Fejer type inequalities for p-convex functions”. Some new and classical exceptional cases are also discussed. Finally, we illustrate our findings with well-defined examples. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

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 1239

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)

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