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15 pages, 292 KB

Open AccessArticle

Weighted Simpson-Type Quantum Integral Inequalities for h-Convex Functions

by Tuncay Köroğlu, Muhammet Yazıcı, Bahadır Özgür Güler and Abdul Wakil Baidar

Mathematics 2026, 14(13), 2436; https://doi.org/10.3390/math14132436 (registering DOI) - 7 Jul 2026

This paper establishes a weighted Simpson-type identity on the parameter domain associated with quantum integral operators. Using this identity together with Hölder’s inequality and the power mean inequality, we derive new estimates for classes of functions whose associated parameter-domain q-derivatives satisfy h [...] Read more.

This paper establishes a weighted Simpson-type identity on the parameter domain associated with quantum integral operators. Using this identity together with Hölder’s inequality and the power mean inequality, we derive new estimates for classes of functions whose associated parameter-domain q-derivatives satisfy h-convexity assumptions. Additional bounds are obtained under boundedness and Lipschitz conditions. Applications to the s-moment of a random variable and to several special means are derived in the classical limit

q \to 1^{-}

. Full article

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

18 pages, 794 KB

Open AccessArticle

Computational Analysis of Newton-Type Inequalities for Differentiable Strongly Convex Functions via RL-Integrals

by Ghulam Abbas, Saima Riaz, Tamador Alihia, Khuram Ali Khan, Saba Yasmin and Ramy M. Hafez

Fractal Fract. 2026, 10(5), 341; https://doi.org/10.3390/fractalfract10050341 - 18 May 2026

Viewed by 491

Abstract

In this paper, new generalizations of Newton-type inequalities for the class of strongly convex functions by utilizing Riemann–Liouville fractional integrals are established. New estimates are obtained for functions that are strongly convex. The established inequalities are further improved. Examples, along with graphs, are [...] Read more.

In this paper, new generalizations of Newton-type inequalities for the class of strongly convex functions by utilizing Riemann–Liouville fractional integrals are established. New estimates are obtained for functions that are strongly convex. The established inequalities are further improved. Examples, along with graphs, are provided to demonstrate the validity of the newly established inequalities and comparisons with existing results. It is expected that the results of this paper will open up new avenues of research and may be generalized to other types of fractional operators and generalized convex functions. Full article

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30 pages, 922 KB

Open AccessArticle

A Comprehensive Analysis of Proportional Caputo-Hybrid Fractional Inequalities and Numerical Verification via Artificial Neural Networks

by Ayed R. A. Alanzi, Mariem Al-Hazmy, Raouf Fakhfakh, Wedad Saleh, Abdellatif Ben Makhlouf and Abdelghani Lakhdari

Fractal Fract. 2026, 10(4), 247; https://doi.org/10.3390/fractalfract10040247 - 8 Apr 2026

Cited by 1 | Viewed by 650

Abstract

Accuracy in fractional numerical integration is often limited by the regularity of the integrand. This work proposes a flexible error estimation framework for proportional Caputo-hybrid integral operators based on

s

-convexity. We introduce a parametric Newton–Cotes formula ( [...] Read more.

Accuracy in fractional numerical integration is often limited by the regularity of the integrand. This work proposes a flexible error estimation framework for proportional Caputo-hybrid integral operators based on

s

-convexity. We introduce a parametric Newton–Cotes formula (

ν \in [0, 1]

) that bridges the gap between classical quadrature rules, recovering the fractional Trapezoidal, Midpoint, and Simpson’s methods as specific instances. In order to confirm the correctness of our results, we provide an illustrative example with graphical representations. Furthermore, we provide some additional results using Hölder’s and power mean inequalities and employ a verification strategy based on an Artificial Neural Networks (ANNs) model. The ANN approach allows for high-dimensional parameter space exploration, demonstrating that the proposed inequalities provide robust and precise error estimates. Full article

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

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34 pages, 1064 KB

Open AccessArticle

On Multiparameter Post-Quantum Fractional Quadrature Inequalities with Simulation

by Sobia Rafeeq, Sabir Hussain, Mariyam Aslam and Youngsoo Seol

Fractal Fract. 2026, 10(4), 242; https://doi.org/10.3390/fractalfract10040242 - 6 Apr 2026

Viewed by 614

Abstract

This paper introduces a comprehensive class of multiparameter post-quantum fractional quadrature inequalities, unifying classical error bounds within the setting of the post-quantum Riemann–Liouville fractional integral. By incorporating multiple parameters, we derive a flexible family of inequalities that generalize well-known quadrature rules such as [...] Read more.

This paper introduces a comprehensive class of multiparameter post-quantum fractional quadrature inequalities, unifying classical error bounds within the setting of the post-quantum Riemann–Liouville fractional integral. By incorporating multiple parameters, we derive a flexible family of inequalities that generalize well-known quadrature rules such as the Boole-type, Bullen–Simpson-type, Maclaurin-type, corrected Euler–Maclaurin-type,

\frac{3}{8}

-Simpson-type, and companion Ostrowski-type estimates. Under assumptions of s-convexity, log-convexity, power mean inequality, and Holder inequality, we establish novel error bounds. Our results provide a unified framework for designing and analyzing post-quantum fractional quadrature inequalities. Applications to special means and numerical and graphic examples are presented to illustrate the applicability and generality of the derived inequalities. This work lays a theoretical foundation for the development of post-quantum fractional quadrature inequalities and offers new tools for error estimation in post-quantum fractional-order models arising in applied sciences and engineering. Full article

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

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15 pages, 561 KB

Open AccessArticle

Parametric Inequalities for s-Convex Stochastic Processes via Caputo Fractional Derivatives

by Ymnah Alruwaily, Rabab Alzahrani, Fatimah Alshahrani, Badreddine Meftah and Raouf Fakhfakh

Axioms 2026, 15(2), 147; https://doi.org/10.3390/axioms15020147 - 17 Feb 2026

Cited by 2 | Viewed by 453

Abstract

This paper establishes a general parametric integral identity involving

(n + 1)

-times differentiable stochastic processes, formulated entirely in terms of stochastic k-Caputo fractional derivatives. This identity serves as a unifying tool for deriving a broad class of parameter-dependent inequalities [...] Read more.

This paper establishes a general parametric integral identity involving

(n + 1)

-times differentiable stochastic processes, formulated entirely in terms of stochastic k-Caputo fractional derivatives. This identity serves as a unifying tool for deriving a broad class of parameter-dependent inequalities for differentiable s-convex stochastic processes. Remarkably, by assigning specific values to the underlying parameter, we have ensured our results specialize to well-known numerical integration inequalities, including those of midpoint, trapezium, Simpson, and Bullen types, in the stochastic fractional context. The findings not only enrich the theory of stochastic fractional calculus but also provide a flexible analytical apparatus for uncertainty quantification in fractional dynamical systems. Full article

(This article belongs to the Special Issue Fractional Differential Equation and Its Applications, 2nd Edition)

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15 pages, 301 KB

Open AccessArticle

On Fractional Simpson-Type Inequalities via Harmonic Convexity

by Li Liao, Abdelghani Lakhdari, Hongyan Xu and Badreddine Meftah

Mathematics 2025, 13(23), 3778; https://doi.org/10.3390/math13233778 - 25 Nov 2025

Viewed by 463

Abstract

In this paper, we establish some Simpson-type inequalities within the framework of Riemann–Liouville fractional calculus, specifically tailored for differentiable harmonically convex functions. By introducing a novel fractional integral identity for differentiable functions with harmonic arguments, we derive several estimates that generalize and refine [...] Read more.

In this paper, we establish some Simpson-type inequalities within the framework of Riemann–Liouville fractional calculus, specifically tailored for differentiable harmonically convex functions. By introducing a novel fractional integral identity for differentiable functions with harmonic arguments, we derive several estimates that generalize and refine existing results in the literature. The theoretical findings are validated through a numerical example supported by graphical illustration, and potential applications in approximation theory and numerical analysis are discussed. Full article

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

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1 pages, 126 KB

Open AccessCorrection

Correction: Javed et al. New Bounds of Hadamard’s and Simpson’s Inequalities Involving Green Functions. Mathematics 2025, 13, 2750

by Muhammad Zakria Javed, Awais Ali, Muhammad Uzair Awan, Lorentz Jäntschi and Omar Mutab Alsalami

Mathematics 2025, 13(18), 3014; https://doi.org/10.3390/math13183014 - 18 Sep 2025

Viewed by 525

Abstract

In the original publication [...] Full article

18 pages, 2166 KB

Open AccessArticle

Error Estimation of Weddle’s Rule for Generalized Convex Functions with Applications to Numerical Integration and Computational Analysis

by Abdul Mateen, Bandar Bin-Mohsin, Ghulam Hussain Tipu and Asia Shehzadi

Mathematics 2025, 13(17), 2874; https://doi.org/10.3390/math13172874 - 5 Sep 2025

Cited by 2 | Viewed by 1357

Abstract

This paper presents new integral inequalities for differentiable generalized convex functions in the second sense, with a focus on improving the accuracy of Weddle’s formula for numerical integration. The study is motivated by the following three key factors: the generalization of convexity through [...] Read more.

This paper presents new integral inequalities for differentiable generalized convex functions in the second sense, with a focus on improving the accuracy of Weddle’s formula for numerical integration. The study is motivated by the following three key factors: the generalization of convexity through s-convex functions, the enhancement of the approximation quality, particularly as

s \to 0^{+}

, and the effectiveness of Weddle’s formula in cases where Simpson’s 1/3 rule fails. An integral identity is derived for differentiable functions, which is then used to establish sharp error bounds for Weddle’s formula under s-convexity. Numerical examples and comparative tables demonstrate that the proposed inequalities yield significantly tighter bounds than those based on classical convexity. Applications to numerical quadrature highlight the practical utility of the results in computational mathematics. Full article

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23 pages, 500 KB

Open AccessArticle

New Bounds of Hadamard’s and Simpson’s Inequalities Involving Green Functions

by Muhammad Zakria Javed, Awais Ali, Muhammad Uzair Awan, Lorentz Jäntschi and Omar Mutab Alsalami

Mathematics 2025, 13(17), 2750; https://doi.org/10.3390/math13172750 - 27 Aug 2025

Cited by 3 | Viewed by 992 | Correction

Abstract

This manuscript aims to assess some new refinements of right Hadamard’s and Simpson’s-like inequalities by bridging the concepts of Green function theory and convexity framework. It is a known fact that Green functions are convex and symmetric. By considering the identities based on [...] Read more.

This manuscript aims to assess some new refinements of right Hadamard’s and Simpson’s-like inequalities by bridging the concepts of Green function theory and convexity framework. It is a known fact that Green functions are convex and symmetric. By considering the identities based on Green functions for second-order differentiable functions and elementary results of inequalities, convexity and bounded variation of functions, we present various new upper estimates of trapezoidal and Simpson’s inequalities. Also, the accuracy of the results is determined by illustrative numerical examples and simulations. Lastly, we furnish some novel applications to linear combinations of means and composite error estimates. Full article

(This article belongs to the Special Issue Current Topics in Optimization, Inequalities and Convex Function Theory)

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48 pages, 1213 KB

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

Cited by 3 | Viewed by 1080

Abstract

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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21 pages, 5704 KB

Open AccessArticle

A Novel Framework for Assessing Urban Green Space Equity Integrating Accessibility and Diversity: A Shenzhen Case Study

by Fei Chang, Zhengdong Huang, Wen Liu and Jiacheng Huang

Remote Sens. 2025, 17(15), 2551; https://doi.org/10.3390/rs17152551 - 23 Jul 2025

Cited by 8 | Viewed by 4403

Abstract

Urban green spaces (UGS) are essential for residents’ well-being, environmental quality, and social cohesion. However, previous studies have typically employed undifferentiated analytical frameworks, overlooking UGS types and failing to adequately measure the structural disparities of different UGS types within residents’ walking distance. To [...] Read more.

Urban green spaces (UGS) are essential for residents’ well-being, environmental quality, and social cohesion. However, previous studies have typically employed undifferentiated analytical frameworks, overlooking UGS types and failing to adequately measure the structural disparities of different UGS types within residents’ walking distance. To address this, this study integrates Gaussian Two-Step Floating Catchment Area models, Simpson’s index, and the Gini coefficient to construct an accessibility–diversity–equality assessment framework for UGS. This study conducted an analysis of accessibility, diversity, and equity for various types of UGSs under pedestrian conditions, using the high-density city of Shenzhen, China as a case study. Results reveal high inequality in accessibility to most UGS types within 15 min to 30 min walking range, except residential green spaces, which show moderate-high inequality (Gini coefficient: 0.4–0.6). Encouragingly, UGS diversity performs well, with over 80% of residents able to access three or more UGS types within walking distance. These findings highlight the heterogeneous UGS supply and provide actionable insights for optimizing green space allocation to support healthy urban development. Full article

(This article belongs to the Special Issue Estimating the Ecological Services of Urban Green Infrastructures Using Remote Sensing)

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20 pages, 325 KB

Open AccessArticle

Development of Fractional Newton-Type Inequalities Through Extended Integral Operators

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

Fractal Fract. 2025, 9(7), 443; https://doi.org/10.3390/fractalfract9070443 - 4 Jul 2025

Cited by 2 | Viewed by 1035

Abstract

This paper introduces a new class of Newton-type inequalities (NTIs) within the framework of extended fractional integral operators. This study begins by establishing a fundamental identity for generalized fractional Riemann–Liouville (FR-L) operators, which forms the basis for deriving various inequalities under different assumptions [...] Read more.

This paper introduces a new class of Newton-type inequalities (NTIs) within the framework of extended fractional integral operators. This study begins by establishing a fundamental identity for generalized fractional Riemann–Liouville (FR-L) operators, which forms the basis for deriving various inequalities under different assumptions on the integrand. In particular, fractional counterparts of the classical

1 / 3

and

3 / 8

Simpson rules are obtained when the modulus of the first derivative is convex. The analysis is further extended to include functions that satisfy a Lipschitz condition or have bounded first derivatives. Moreover, an additional NTI is presented for functions of bounded variation, expressed in terms of their total variation. In all scenarios, the proposed results reduce to classical inequalities when the fractional parameters are specified accordingly, thus offering a unified perspective on numerical integration through fractional operators. Full article

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

13 pages, 441 KB

Open AccessFeature PaperArticle

Some New and Sharp Inequalities of Composite Simpson’s Formula for Differentiable Functions with Applications

by Wei Liu, Yu Wang, Ifra Bashir Sial and Loredana Ciurdariu

Mathematics 2025, 13(11), 1814; https://doi.org/10.3390/math13111814 - 29 May 2025

Cited by 1 | Viewed by 1390

Abstract

Composite integral formulas offer greater accuracy by dividing the interval into smaller subintervals, which better capture the local behavior of function. In the finite volume method for solving differential equations, composite formulas are mostly used on control volumes to achieve high-accuracy solutions. In [...] Read more.

Composite integral formulas offer greater accuracy by dividing the interval into smaller subintervals, which better capture the local behavior of function. In the finite volume method for solving differential equations, composite formulas are mostly used on control volumes to achieve high-accuracy solutions. In this work, error estimates of the composite Simpson’s formula for differentiable convex functions are established. These error estimates can be applied to general subdivisions of the integration interval, provided the integrand satisfies a first-order differentiability condition. To this end, a novel and general integral identity for differentiable functions is established by considering general subdivisions of the integration interval. The new integral identity is proved in a manner that allows it to be transformed into different identities for different subdivisions of the integration interval. Then, under the convexity assumption on the integrand, sharp error bounds for the composite Simpson’s formula are proved. Moreover, the well-known Hölder’s inequality is applied to obtain sharper error bounds for differentiable convex functions, which represents a significant finding of this study. Finally, to support the theoretical part of this work, some numerical examples are tested and demonstrate the efficiency of the new bounds for different partitions of the integration interval. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

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28 pages, 397 KB

Open AccessArticle

Hybrid Integral Inequalities on Fractal Set

by Badreddine Meftah, Wedad Saleh, Muhammad Uzair Awan, Loredana Ciurdariu and Abdelghani Lakhdari

Axioms 2025, 14(5), 358; https://doi.org/10.3390/axioms14050358 - 9 May 2025

Cited by 1 | Viewed by 859

Abstract

In this study, we introduce a new hybrid identity that effectively combines Newton–Cotes and Gauss quadrature, allowing us to recover well-known formulas such as Simpson’s second rule and the left- and right-Radau two-point rules, among others. Building upon this flexible framework, we establish [...] Read more.

In this study, we introduce a new hybrid identity that effectively combines Newton–Cotes and Gauss quadrature, allowing us to recover well-known formulas such as Simpson’s second rule and the left- and right-Radau two-point rules, among others. Building upon this flexible framework, we establish several new biparametrized fractal integral inequalities for functions whose local fractional derivatives are of a generalized convex type. In addition to employing tools from local fractional calculus, our approach utilizes the Hölder inequality, the power mean inequality, and a refined version of the latter. Further results are also derived using the concept of generalized concavity. To support our theoretical findings, we provide a graphical example that illustrates the validity of the obtained results, along with some practical applications that demonstrate their effectiveness. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities, 2nd Edition)

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25 pages, 382 KB

Open AccessArticle

New Numerical Quadrature Functional Inequalities on Hilbert Spaces in the Framework of Different Forms of Generalized Convex Mappings

by Waqar Afzal and Luminita-Ioana Cotîrlă

Symmetry 2025, 17(1), 146; https://doi.org/10.3390/sym17010146 - 20 Jan 2025

Cited by 3 | Viewed by 1714

Abstract

The purpose of this article is to investigate some tensorial norm inequalities for continuous functions of self-adjoint operators in Hilbert spaces. Our first approach is to develop a gradient descent inequality and some relational properties for continuous functions involving Huber convex functions, as [...] Read more.

The purpose of this article is to investigate some tensorial norm inequalities for continuous functions of self-adjoint operators in Hilbert spaces. Our first approach is to develop a gradient descent inequality and some relational properties for continuous functions involving Huber convex functions, as well as several new bounds for Simpson type inequality that is twice differentiable using different types of generalized convex mappings. It is believed that this study will provide a valuable contribution towards developing a new perspective on functional inequalities by utilizing some other types of generalized mappings. Full article

(This article belongs to the Special Issue Advance in Functional Equations, Second Edition)

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