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48 pages, 1213 KiB

Open AccessArticle

Parameterized Fractal–Fractional Analysis of Ostrowski- and Simpson-Type Inequalities with Applications

by Saad Ihsan Butt, Muhammad Mehtab and Youngsoo Seol

Fractal Fract. 2025, 9(8), 494; https://doi.org/10.3390/fractalfract9080494 - 28 Jul 2025

Viewed by 197

In this paper, we first introduce a parametric identity for generalized differentiable functions using a generalized fractal–fractional integral operators. Based on this identity, we establish several variants of parameterized inequalities for functions whose local fractional derivatives in absolute value satisfy generalized convexity conditions. [...] Read more.

In this paper, we first introduce a parametric identity for generalized differentiable functions using a generalized fractal–fractional integral operators. Based on this identity, we establish several variants of parameterized inequalities for functions whose local fractional derivatives in absolute value satisfy generalized convexity conditions. Furthermore, we demonstrate that our main results reduce to well-known Ostrowski- and Simpson-type inequalities by selecting suitable parameters. These inequalities contribute to finding tight bounds for various integrals over fractal spaces. By comparing the classical Hölder and Power mean inequalities with their new generalized versions, we show that the improved forms yield sharper and more refined upper bounds. In particular, we illustrate that the generalizations of Hölder and Power mean inequalities provide better results when applied to fractal integrals, with their tighter bounds supported by graphical representations. Finally, a series of applications are discussed, including generalized special means, generalized probability density functions and generalized quadrature formulas, which highlight the practical significance of the proposed results in fractal analysis. Full article

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

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27 pages, 957 KiB

Open AccessArticle

Complex-Valued Multivariate Neural Network (MNN) Approximation by Parameterized Half-Hyperbolic Tangent Function

by Seda Karateke

Mathematics 2025, 13(3), 453; https://doi.org/10.3390/math13030453 - 29 Jan 2025

Viewed by 773

Abstract

This paper deals with a family of normalized multivariate neural network (MNN) operators of complex-valued continuous functions for a multivariate context on a box of

R^{\bar{N}}

,

\bar{N} \in N

. Moreover, we consider the case of approximation employing iterated [...] Read more.

This paper deals with a family of normalized multivariate neural network (MNN) operators of complex-valued continuous functions for a multivariate context on a box of

R^{\bar{N}}

,

\bar{N} \in N

. Moreover, we consider the case of approximation employing iterated MNN operators. In addition, pointwise and uniform convergence results are obtained in Banach spaces thanks to the multivariate versions of trigonometric and hyperbolic-type Taylor formulae on the corresponding feed-forward neural networks (FNNs) based on one or more hidden layers. Full article

(This article belongs to the Special Issue Approximation Theory and Applications)

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27 pages, 326 KiB

Open AccessArticle

Parametrized Half-Hyperbolic Tangent Function-Activated Complex-Valued Neural Network Approximation

by George A. Anastassiou and Seda Karateke

Symmetry 2024, 16(12), 1568; https://doi.org/10.3390/sym16121568 - 23 Nov 2024

Cited by 1 | Viewed by 815

Abstract

In this paper, we create a family of neural network (NN) operators employing a parametrized and deformed half-hyperbolic tangent function as an activation function and a density function produced by the same activation function. Moreover, we consider the univariate quantitative approximations by complex-valued [...] Read more.

In this paper, we create a family of neural network (NN) operators employing a parametrized and deformed half-hyperbolic tangent function as an activation function and a density function produced by the same activation function. Moreover, we consider the univariate quantitative approximations by complex-valued neural network (NN) operators of complex-valued functions on a compact domain. Pointwise and uniform convergence results on Banach spaces are acquired through trigonometric, hyperbolic, and hybrid-type hyperbolic–trigonometric approaches. Full article

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

29 pages, 608 KiB

Open AccessArticle

Novel Fuzzy Ostrowski Integral Inequalities for Convex Fuzzy-Valued Mappings over a Harmonic Convex Set: Extending Real-Valued Intervals Without the Sugeno Integrals

by Mesfer H. Alqahtani, Der-Chyuan Lou, Fahad Sikander, Yaser Saber and Cheng-Chi Lee

Mathematics 2024, 12(22), 3495; https://doi.org/10.3390/math12223495 - 8 Nov 2024

Viewed by 914

Abstract

This study presents new fuzzy adaptations of Ostrowski’s integral inequalities through a novel class of convex fuzzy-valued mappings defined over a harmonic convex set, avoiding the use of the Sugeno integral. These innovative inequalities generalize the recently developed interval forms of real-valued Ostrowski [...] Read more.

This study presents new fuzzy adaptations of Ostrowski’s integral inequalities through a novel class of convex fuzzy-valued mappings defined over a harmonic convex set, avoiding the use of the Sugeno integral. These innovative inequalities generalize the recently developed interval forms of real-valued Ostrowski inequalities. Their formulations incorporate integrability concepts for fuzzy-valued mappings (FVMs), applying the Kaleva integral and a Kulisch–Miranker fuzzy order relation. The fuzzy order relation is constructed via a level-wise approach based on the Kulisch–Miranker order within the fuzzy number space. Additionally, numerical examples illustrate the effectiveness and significance of the proposed theoretical model. Various applications are explored using different means, and some complex cases are derived. Full article

(This article belongs to the Special Issue Novel Approaches in Fuzzy Sets and Metric Spaces)

22 pages, 343 KiB

Open AccessArticle

Novel Ostrowski–Type Inequalities for Generalized Fractional Integrals and Diverse Function Classes

by Areej A. Almoneef, Abd-Allah Hyder, Mohamed A. Barakat and Hüseyin Budak

Fractal Fract. 2024, 8(9), 534; https://doi.org/10.3390/fractalfract8090534 - 13 Sep 2024

Cited by 1 | Viewed by 849

Abstract

In this work, novel Ostrowski-type inequalities for dissimilar function classes and generalized fractional integrals (FITs) are presented. We provide a useful identity for differentiable functions under FITs, which results in special expressions for functions whose derivatives have convex absolute values. A new condition [...] Read more.

In this work, novel Ostrowski-type inequalities for dissimilar function classes and generalized fractional integrals (FITs) are presented. We provide a useful identity for differentiable functions under FITs, which results in special expressions for functions whose derivatives have convex absolute values. A new condition for bounded variation functions is examined, as well as expansions to bounded and Lipschitzian derivatives. Our comprehension is improved by comparison with current findings, and recommendations for future study areas are given. Full article

(This article belongs to the Special Issue Fractional Systems, Integrals and Derivatives: Theory and Application)

22 pages, 286 KiB

Open AccessArticle

Ostrowski-Type Inequalities for Functions of Two Variables in Banach Spaces

by Muhammad Amer Latif and Ohud Bulayhan Almutairi

Mathematics 2024, 12(17), 2748; https://doi.org/10.3390/math12172748 - 4 Sep 2024

Cited by 1 | Viewed by 713

Abstract

In this paper, we offer Ostrowski-type inequalities that extend the findings that have been proven for functions of one variable with values in Banach spaces, conducted in a remarkable study by Dragomir, to functions of two variables containing values in the product Banach [...] Read more.

In this paper, we offer Ostrowski-type inequalities that extend the findings that have been proven for functions of one variable with values in Banach spaces, conducted in a remarkable study by Dragomir, to functions of two variables containing values in the product Banach spaces. Our findings are also an extension of several previous findings that have been established for functions of two variable functions. Prior studies on Ostrowski-type inequalities incriminated functions that have values in Banach spaces or Hilbert spaces. This study is unique and significant in the field of mathematical inequalities, and specifically in the study of Ostrowski-type inequalities, because they have been established for functions having values in a product of two Banach spaces. Full article

(This article belongs to the Special Issue Mathematical Analysis and Functional Analysis and Their Applications)

31 pages, 1687 KiB

Open AccessArticle

Some Classical Inequalities Associated with Generic Identity and Applications

by Muhammad Zakria Javed, Muhammad Uzair Awan, Bandar Bin-Mohsin, Hüseyin Budak and Silvestru Sever Dragomir

Axioms 2024, 13(8), 533; https://doi.org/10.3390/axioms13080533 - 6 Aug 2024

Cited by 2 | Viewed by 1090

Abstract

In this paper, we derive a new generic equality for the first-order differentiable functions. Through the utilization of the general identity and convex functions, we produce a family of upper bounds for numerous integral inequalities like Ostrowski’s inequality, trapezoidal inequality, midpoint inequality, Simpson’s [...] Read more.

In this paper, we derive a new generic equality for the first-order differentiable functions. Through the utilization of the general identity and convex functions, we produce a family of upper bounds for numerous integral inequalities like Ostrowski’s inequality, trapezoidal inequality, midpoint inequality, Simpson’s inequality, Newton-type inequalities, and several two-point open trapezoidal inequalities. Also, we provide the numerical and visual explanation of our principal findings. Later, we provide some novel applications to the theory of means, special functions, error bounds of composite quadrature schemes, and parametric iterative schemes to find the roots of linear functions. Also, we attain several already known and new bounds for different values of

γ

and parameter

ξ

. Full article

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

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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 971

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)

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 825

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)

17 pages, 1033 KiB

Open AccessArticle

Some New Estimations of Ostrowski-Type Inequalities for Harmonic Fuzzy Number Convexity via Gamma, Beta and Hypergeometric Functions

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

Axioms 2024, 13(7), 455; https://doi.org/10.3390/axioms13070455 - 4 Jul 2024

Cited by 1 | Viewed by 820

Abstract

This paper demonstrates several of Ostrowski-type inequalities for fuzzy number functions and investigates their connections with other inequalities. Specifically, employing the Aumann integral and the Kulisch–Miranker order, as well as the inclusion order on the space of real and compact intervals, we establish [...] Read more.

This paper demonstrates several of Ostrowski-type inequalities for fuzzy number functions and investigates their connections with other inequalities. Specifically, employing the Aumann integral and the Kulisch–Miranker order, as well as the inclusion order on the space of real and compact intervals, we establish various Ostrowski-type inequalities for fuzzy-valued mappings (

F

·

V

·

M

s). Furthermore, by employing diverse orders, we establish connections with the classical versions of Ostrowski-type inequalities. Additionally, we explore new ideas and results rooted in submodular measures, accompanied by examples and applications to illustrate our findings. Moreover, by using special functions, we have provided some applications of Ostrowski-type inequalities. Full article

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

23 pages, 384 KiB

Open AccessArticle

Some Simpson- and Ostrowski-Type Integral Inequalities for Generalized Convex Functions in Multiplicative Calculus with Their Computational Analysis

by Xinlin Zhan, Abdul Mateen, Muhammad Toseef and Muhammad Aamir Ali

Mathematics 2024, 12(11), 1721; https://doi.org/10.3390/math12111721 - 31 May 2024

Cited by 10 | Viewed by 1169

Abstract

Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas [...] Read more.

Integral inequalities are very useful in finding the error bounds for numerical integration formulas. In this paper, we prove some multiplicative integral inequalities for first-time differentiable s-convex functions. These new inequalities help in finding the error bounds for different numerical integration formulas in multiplicative calculus. The use of s-convex function extends the results for convex functions and covers a large class of functions, which is the main motivation for using s-convexity. To prove the inequalities, we derive two different integral identities for multiplicative differentiable functions in the setting of multiplicative calculus. Then, with the help of these integral identities, we prove some integral inequalities of the Simpson and Ostrowski types for multiplicative generalized convex functions. Moreover, we provide some numerical examples and computational analysis of these newly established inequalities, to show the validity of the results for multiplicative s-convex functions. We also give some applications to quadrature formula and special means of real numbers within the framework of multiplicative calculus. Full article

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

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15 pages, 272 KiB

Open AccessArticle

Inequalities of Ostrowski Type for Functions Whose Derivative Module Is Relatively Convex on Time Scales

by Haytham M. Rezk, Ahmed I. Saied, Maha Ali, Ghada AlNemer and Mohammed Zakarya

Axioms 2024, 13(4), 235; https://doi.org/10.3390/axioms13040235 - 2 Apr 2024

Cited by 2 | Viewed by 1189

Abstract

In this article, we discuss several novel generalized Ostrowski-type inequalities for functions whose derivative module is relatively convex in time scales calculus. Our core findings are proved by using the integration by parts technique, Hölder’s inequality, and the chain rule on time scales. [...] Read more.

In this article, we discuss several novel generalized Ostrowski-type inequalities for functions whose derivative module is relatively convex in time scales calculus. Our core findings are proved by using the integration by parts technique, Hölder’s inequality, and the chain rule on time scales. These derived inequalities expand the existing literature, enriching specific integral inequalities within this domain. Full article

(This article belongs to the Special Issue Current Research on Mathematical Inequalities II)

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 2763

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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18 pages, 309 KiB

Open AccessArticle

More General Ostrowski-Type Inequalities in the Fuzzy Context

by Muhammad Amer Latif

Mathematics 2024, 12(3), 500; https://doi.org/10.3390/math12030500 - 5 Feb 2024

Cited by 1 | Viewed by 1138

Abstract

In this study, Ostrowski-type inequalities in fuzzy settings were investigated. A detailed theory of fuzzy analysis is provided and utilized to establish the Ostrowski-type inequality in the fuzzy number-valued space. The results obtained in this research not only provide a generalization of the [...] Read more.

In this study, Ostrowski-type inequalities in fuzzy settings were investigated. A detailed theory of fuzzy analysis is provided and utilized to establish the Ostrowski-type inequality in the fuzzy number-valued space. The results obtained in this research not only provide a generalization of the results of Dragomir but also give an extended version of the Ostrowski-type inequalities obtained by Anastassiou. Full article

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

64 pages, 648 KiB

Open AccessReview

Ostrowski-Type Fractional Integral Inequalities: A Survey

by Muhammad Tariq, Sotiris K. Ntouyas and Bashir Ahmad

Foundations 2023, 3(4), 660-723; https://doi.org/10.3390/foundations3040040 - 13 Nov 2023

Viewed by 1963

Abstract

This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions,

(ζ, m)

-convex [...] Read more.

This paper presents an extensive review of some recent results on fractional Ostrowski-type inequalities associated with a variety of convexities and different kinds of fractional integrals. We have taken into account the classical convex functions, quasi-convex functions,

(ζ, m)

-convex functions, s-convex functions,

(s, r)

-convex functions, strongly convex functions, harmonically convex functions, h-convex functions, Godunova-Levin-convex functions,

M T

-convex functions, P-convex functions, m-convex functions,

(s, m)

-convex functions, exponentially s-convex functions,

(β, m)

-convex functions, exponential-convex functions,

\bar{ζ}, β, γ, δ

-convex functions, quasi-geometrically convex functions,

s - e

-convex functions and n-polynomial exponentially s-convex functions. Riemann–Liouville fractional integral, Katugampola fractional integral, k-Riemann–Liouville, Riemann–Liouville fractional integrals with respect to another function, Hadamard fractional integral, fractional integrals with exponential kernel and Atagana-Baleanu fractional integrals are included. Results for Ostrowski-Mercer-type inequalities, Ostrowski-type inequalities for preinvex functions, Ostrowski-type inequalities for Quantum-Calculus and Ostrowski-type inequalities of tensorial type are also presented. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions II)

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