Axioms

Research

25 pages, 2510 KB

Open AccessArticle

ANN-Assisted Sharp Bounds for Higher-Order Euler–Maclaurin Inequalities

by Muhammad Zakria Javed, Muhammad Uzair Awan, Loredana Ciurdariu, Eugenia Grecu and Hala Mostafa

Axioms 2026, 15(5), 358; https://doi.org/10.3390/axioms15050358 - 11 May 2026

Viewed by 197

This study presents some novel sharp estimates of the Euler–Maclaurin inequality using a new higher-order derivative Maclaurin identity. By utilizing the properties of convexity and classical inequalities, we exploit various novel tight boundaries of the Euler–Maclaurin inequality. They offer alternatives to measuring the [...] Read more.

This study presents some novel sharp estimates of the Euler–Maclaurin inequality using a new higher-order derivative Maclaurin identity. By utilizing the properties of convexity and classical inequalities, we exploit various novel tight boundaries of the Euler–Maclaurin inequality. They offer alternatives to measuring the sharp bounds of the mean integral of the higher-order differentiable mappings. In order to prove the importance and precision of the key findings, we apply graphical and numerical techniques. Another important section evaluates the behavior and validity of inequalities using a neural network model. The method is not only utilized to authenticate the results but also brings out the practical advancements of the study within a computational framework. The method and results of the article provide an insight and develop a solid connection between inequalities, higher-order derivative convex mappings, numerical analysis, approximation theory, and artificial neural networking. Full article

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

► Show Figures

Figure 1

23 pages, 362 KB

Open AccessArticle

On Proportional Caputo-Hybrid Fractional Milne-Type Inequalities: Theory, Numerical Simulations, and Applications

by Mariem Al-Hazmy, Yazeed Alkhrijah, Wedad Saleh, Borhen Louhichi and Badreddine Meftah

Axioms 2026, 15(4), 280; https://doi.org/10.3390/axioms15040280 - 12 Apr 2026

Cited by 1 | Viewed by 392

Abstract

The goal of this study is to establish a new type of Milne-type inequality in the scope of fractional calculus with the aid of proportional Caputo-hybrid operators. We will focus on two different scopes of regularity, which contain functions whose first and second [...] Read more.

The goal of this study is to establish a new type of Milne-type inequality in the scope of fractional calculus with the aid of proportional Caputo-hybrid operators. We will focus on two different scopes of regularity, which contain functions whose first and second derivatives are convex, and functions whose first and second derivatives are Lipschitz continuous. We will base these estimates on a new integral identity of proportional Caputo-hybrid integrals. We will show that the smoothness of the derivative influences the shape of the bounds. Convexity will cause symmetry. Lipschitz continuity will contain bounds on the modulus of continuity. To show that our results are accurate and easy to obtain, we included a full numerical example with graphics and applications to quadrature error estimation. Full article

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

► Show Figures

Figure 1

16 pages, 307 KB

Open AccessArticle

Integral Inequalities for Vector (Multi)functions

by Cristina Stamate and Anca Croitoru

Axioms 2025, 14(12), 915; https://doi.org/10.3390/axioms14120915 - 12 Dec 2025

Cited by 1 | Viewed by 612

Abstract

We present some integral inequalities such as Minkowski-type and optimal bound-type for vector functions and vector multifunctions for different kinds of integrals:

G

-integral, Choquet-type integral, and Sugeno-type integral. Full article

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

15 pages, 337 KB

Open AccessArticle

Fractional Error Bounds for Lobatto Quadrature: A Convexity Approach via Riemann–Liouville Integrals

by Li Liao, Abdelghani Lakhdari, Muhammad Uzair Awan, Hongyan Xu and Badreddine Meftah

Axioms 2025, 14(11), 823; https://doi.org/10.3390/axioms14110823 - 7 Nov 2025

Viewed by 603

Abstract

In this paper, we establish a new fractional integral identity linked to the 4-point Lobatto quadrature rule within the Riemann–Liouville fractional calculus framework. Building on this identity, we derive several Lobatto-type inequalities under convexity assumptions, yielding error bounds that involve only first-order derivatives, [...] Read more.

In this paper, we establish a new fractional integral identity linked to the 4-point Lobatto quadrature rule within the Riemann–Liouville fractional calculus framework. Building on this identity, we derive several Lobatto-type inequalities under convexity assumptions, yielding error bounds that involve only first-order derivatives, thereby improving practical applicability. A numerical example with graphical illustration confirms the theoretical findings and demonstrates their accuracy. We also present applications to special means, highlighting the utility of the obtained inequalities. The integration of fractional analysis, quadrature theory, and numerical validation provides a robust methodology for refining and analyzing high-order integration rules. Full article

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

► Show Figures

Figure 1

22 pages, 463 KB

Open AccessArticle

Improved Bounds for Integral Jensen’s Inequality Through Fifth-Order Differentiable Convex Functions and Applications

by Sidra Nisar, Fiza Zafar and Hind Alamri

Axioms 2025, 14(8), 602; https://doi.org/10.3390/axioms14080602 - 2 Aug 2025

Cited by 1 | Viewed by 1397

Abstract

The main objective of this research is to obtain interesting estimates for Jensen’s gap in the integral sense, along with their applications. The convexity of a fifth-order absolute function is used to established proposed estimates of Jensen’s gap. We performed numerical computations to [...] Read more.

The main objective of this research is to obtain interesting estimates for Jensen’s gap in the integral sense, along with their applications. The convexity of a fifth-order absolute function is used to established proposed estimates of Jensen’s gap. We performed numerical computations to compare our estimates with previous findings. With the use of the primary findings, we are able to obtain improvements of the Hölder inequality and Hermite–Hadamard inequality. Furthermore, the primary results lead to some inequalities for power means and quasi-arithmetic means. We conclude by outlining the information theory applications of our primary inequalities. Full article

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

► Show Figures

Figure 1

26 pages, 355 KB

Open AccessArticle

Extension of an Inequality on Three Intervals and Applications to Csiszár ϕ-Divergence and Landau–Kolmogorov Inequality

by Ðilda Pečarić, Josip Pečarić and Jinyan Miao

Axioms 2025, 14(8), 563; https://doi.org/10.3390/axioms14080563 - 24 Jul 2025

Cited by 1 | Viewed by 929

Abstract

In this paper, we generalize an inequality for a convex function in one dimension

R^{1}

on three intervals to a function with nondecreasing increments in k dimensions

R^{k}

on

(2 n + 1)

intervals. We prove all the situations [...] Read more.

In this paper, we generalize an inequality for a convex function in one dimension

R^{1}

on three intervals to a function with nondecreasing increments in k dimensions

R^{k}

on

(2 n + 1)

intervals. We prove all the situations when

n = 1, 2

and prove a very special case for a general n as well as the discrete version. The proofs are based on a general conclusion for convex functions, and analogues of this conclusion are established. We apply the discrete case of the inequality to Csiszár

ϕ

-divergence

I_{ϕ} (p, q)

in information theory, and the continuous case

I_{ϕ} (p_{1}, q_{1}) \leq I_{ϕ} (p_{2}, q_{2})

on a measurable set is also established. The same inequality for an

ϵ

-approximately convex function on a discrete set is also established and can be used to prove a similar Landau–Kolmogorov-type inequality. Full article

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

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 805

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)

► Show Figures

Figure 1

13 pages, 257 KB

Open AccessArticle

Investigating the Hyers–Ulam Stability of the Generalized Drygas Functional Equation: New Results and Methods

by Gang Lyu, Yang Liu, Yuanfeng Jin and Yingxiu Jiang

Axioms 2025, 14(4), 315; https://doi.org/10.3390/axioms14040315 - 21 Apr 2025

Viewed by 827

Abstract

In this paper, we explore the Hyers–Ulam stability of a generalized Drygas functional equation, which extends the classical Drygas equation by incorporating additional parameters and conditions. Our investigation focuses on mappings from a real vector space into a Banach space and employs the [...] Read more.

In this paper, we explore the Hyers–Ulam stability of a generalized Drygas functional equation, which extends the classical Drygas equation by incorporating additional parameters and conditions. Our investigation focuses on mappings from a real vector space into a Banach space and employs the fixed-point method to establish stability criteria. Our findings provide new insights into the conditions under which the generalized Drygas equation maintains stability, contributing to the broader understanding of functional equations in mathematical analysis. The results have implications for the study of functional equations and their applications in various mathematical contexts. Full article

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

12 pages, 236 KB

Open AccessArticle

Spatial Decay Estimates for the Moore–Gibson–Thompson Heat Equation Based on an Integral Differential Inequality

by Naiqiao Qing, Jincheng Shi and Yunfeng Wen

Axioms 2025, 14(4), 265; https://doi.org/10.3390/axioms14040265 - 1 Apr 2025

Cited by 2 | Viewed by 720

Abstract

The present work investigates the spatial evolution characteristics of solutions to the Moore–Gibson–Thompson heat equation within a three-dimensional cylindrical geometry. By constructing an integral-differential inequality framework, we establish rigorous estimates demonstrating the exponential spatial decay of the solution as the axial distance from [...] Read more.

The present work investigates the spatial evolution characteristics of solutions to the Moore–Gibson–Thompson heat equation within a three-dimensional cylindrical geometry. By constructing an integral-differential inequality framework, we establish rigorous estimates demonstrating the exponential spatial decay of the solution as the axial distance from the inlet boundary increases without bound. This finding aligns with a generalized interpretation of the Saint-Venant principle, demonstrating its applicability under the present asymptotic conditions. The integral-differential inequality method proposed in this paper can also be used for the study of the Saint-Venant principle for other equations. Full article

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

Journal Menu

Journal Browser

Theory and Application of Integral Inequalities, 2nd Edition

Share This Special Issue

Special Issue Editor

Special Issue Information

Keywords

Benefits of Publishing in a Special Issue

Published Papers (9 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI