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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

Viewed by 254

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)

19 pages, 289 KB

Open AccessArticle

Some New Sobolev-Type Theorems for the Rough Riesz Potential Operator on Grand Variable Herz Spaces

by Ghada AlNemer, Ghada Ali Basendwah, Babar Sultan and Ioan-Lucian Popa

Mathematics 2025, 13(11), 1873; https://doi.org/10.3390/math13111873 - 3 Jun 2025

Cited by 2 | Viewed by 835

Abstract

In this paper, our first objective is to define the idea of grand variable Herz spaces. Then, our main goal is to prove boundedness results for operators, including the rough Riesz potential operator of variable order and the fractional Hardy operators, on grand [...] Read more.

In this paper, our first objective is to define the idea of grand variable Herz spaces. Then, our main goal is to prove boundedness results for operators, including the rough Riesz potential operator of variable order and the fractional Hardy operators, on grand variable Herz spaces under some proper assumptions. To prove the boundedness results, we use Holder-type and Minkowski inequalities. In the proof of the main result, we use different techniques. We divide the summation into different terms and estimate each term under different conditions. Then, by combining the estimates, we prove that the rough Riesz potential operator of variable order and the fractional Hardy operators are bounded on grand variable Herz spaces. It is easy to show that the rough Riesz potential operator of variable order generalizes the Riesz potential operator and that the fractional Hardy operators are generalized versions of simple Hardy operators. So, our results extend some previous results to the more generalized setting of grand variable Herz spaces. Full article

(This article belongs to the Special Issue Advances on Complex Analysis, 2nd Edition)

17 pages, 430 KB

Open AccessArticle

Young and Inverse Young Inequalities on Euclidean Jordan Algebra

by Chien-Hao Huang

Axioms 2025, 14(4), 312; https://doi.org/10.3390/axioms14040312 - 18 Apr 2025

Viewed by 778

Abstract

This paper mainly focuses on in-depth research on inequalities on symmetric cones. We will further analyze and discuss the inequalities we developed on the second-order cone and develop more inequalities. According to our past research in dealing with second-order cone inequalities, we derive [...] Read more.

This paper mainly focuses on in-depth research on inequalities on symmetric cones. We will further analyze and discuss the inequalities we developed on the second-order cone and develop more inequalities. According to our past research in dealing with second-order cone inequalities, we derive more inequalities concerning the eigenvalue version of Young’s inequality and trace a version of an inverse Young inequality and its applications. These conclusions align with the results established for the positive semidefinite cone, which is also a symmetric cone. It is of considerable help to the establishment of inequalities on symmetric cones and the analysis of their derivative algorithms. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization, 2nd Edition)

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11 pages, 284 KB

Open AccessFeature PaperArticle

Lightweight Implicit Approximation of the Minkowski Sum of an N-Dimensional Ellipsoid and Hyperrectangle

by Martijn Courteaux, Bert Ramlot, Peter Lambert and Glenn Van Wallendael

Mathematics 2025, 13(8), 1326; https://doi.org/10.3390/math13081326 - 18 Apr 2025

Viewed by 1810

Abstract

This work considers the Minkowski sum of an N-dimensional ellipsoid and hyperrectangle, a combination that is extremely relevant due to the usage of ellipsoid-adjacent primitives in computer graphics for work such as 3D Gaussian splatting. While parametric representations of this Minkowski sum are [...] Read more.

This work considers the Minkowski sum of an N-dimensional ellipsoid and hyperrectangle, a combination that is extremely relevant due to the usage of ellipsoid-adjacent primitives in computer graphics for work such as 3D Gaussian splatting. While parametric representations of this Minkowski sum are available, they are often difficult or too computationally intensive to work with when, for example, performing an inclusion test. For performance-critical applications, a lightweight approximation of this Minkowski sum is preferred over its exact form. To this end, we propose a fast, computationally lightweight, non-iterative algorithm that approximates the Minkowski sum through the intersection of two carefully constructed bounding boxes. Our approximation is a super-set that completely envelops the exact Minkowski sum. This approach yields an implicit representation that is defined by a logical conjunction of linear inequalities. For applications where a tight super-set of the Minkowski sum is acceptable, the proposed algorithm can substantially improve the performance of common operations such as intersection testing. Full article

(This article belongs to the Section B: Geometry and Topology)

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10 pages, 253 KB

Open AccessArticle

The Dual Hamilton–Jacobi Equation and the Poincaré Inequality

by Rigao He, Wei Wang, Jianglin Fang and Yuanlin Li

Mathematics 2024, 12(24), 3927; https://doi.org/10.3390/math12243927 - 13 Dec 2024

Viewed by 1364

Abstract

Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity shown by L. Gross, and applying the ideas and methods of the work by Bobkov, Gentil and Ledoux, we would like to establish a new connection between the logarithmic Sobolev inequalities and the hypercontractivity [...] Read more.

Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity shown by L. Gross, and applying the ideas and methods of the work by Bobkov, Gentil and Ledoux, we would like to establish a new connection between the logarithmic Sobolev inequalities and the hypercontractivity of solutions of dual Hamilton–Jacobi equations. In addition, Poincaré inequality is also recovered by the dual Hamilton–Jacobi equations. Full article

7 pages, 220 KB

Open AccessArticle

An Information-Theoretic Proof of a Hypercontractive Inequality

by Ehud Friedgut

Entropy 2024, 26(11), 966; https://doi.org/10.3390/e26110966 - 11 Nov 2024

Viewed by 1391

Abstract

The famous hypercontractive estimate discovered independently by Gross, Bonami and Beckner has had a great impact on combinatorics and theoretical computer science since it was first used in this setting in a seminal paper by Kahn, Kalai and Linial. The usual proofs of [...] Read more.

The famous hypercontractive estimate discovered independently by Gross, Bonami and Beckner has had a great impact on combinatorics and theoretical computer science since it was first used in this setting in a seminal paper by Kahn, Kalai and Linial. The usual proofs of this inequality begin with two-point space, where some elementary calculus is used and then generalised immediately by introducing another dimension using submultiplicativity (Minkowski’s integral inequality). In this paper, we prove this inequality using information theory. We compare the entropy of a pair of correlated vectors in

{0, 1}^{n}

to their separate entropies, analysing them bit by bit (not as a figure of speech, but as the bits are revealed) using the chain rule of entropy. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

23 pages, 504 KB

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 9 | Viewed by 1749

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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13 pages, 286 KB

Open AccessArticle

Stability of the Borell–Brascamp–Lieb Inequality for Multiple Power Concave Functions

by Meng Qin, Zhuohua Zhang, Rui Luo, Mengjie Ren and Denghui Wu

Axioms 2024, 13(5), 320; https://doi.org/10.3390/axioms13050320 - 11 May 2024

Cited by 1 | Viewed by 1509

Abstract

In this paper, we prove the stability of the Brunn–Minkowski inequality for multiple convex bodies in terms of the concept of relative asymmetry. Using these stability results and the relationship of the compact support of functions, we establish the stability of the Borell–Brascamp–Lieb [...] Read more.

In this paper, we prove the stability of the Brunn–Minkowski inequality for multiple convex bodies in terms of the concept of relative asymmetry. Using these stability results and the relationship of the compact support of functions, we establish the stability of the Borell–Brascamp–Lieb inequality for multiple power concave functions via relative asymmetry. Full article

(This article belongs to the Special Issue Advances in Convex Geometry and Analysis)

15 pages, 355 KB

Open AccessArticle

New Inequalities Using Multiple Erdélyi–Kober Fractional Integral Operators

by Asifa Tassaddiq, Rekha Srivastava, Rabab Alharbi, Ruhaila Md Kasmani and Sania Qureshi

Fractal Fract. 2024, 8(4), 180; https://doi.org/10.3390/fractalfract8040180 - 22 Mar 2024

Cited by 10 | Viewed by 2043

Abstract

The role of fractional integral inequalities is vital in fractional calculus to develop new models and techniques in the most trending sciences. Taking motivation from this fact, we use multiple Erdélyi–Kober (M-E-K) fractional integral operators to establish Minkowski fractional inequalities. Several other new [...] Read more.

The role of fractional integral inequalities is vital in fractional calculus to develop new models and techniques in the most trending sciences. Taking motivation from this fact, we use multiple Erdélyi–Kober (M-E-K) fractional integral operators to establish Minkowski fractional inequalities. Several other new and novel fractional integral inequalities are also established. Compared to the existing results, these fractional integral inequalities are more general and substantial enough to create new and novel results. M-E-K fractional integral operators have been previously applied for other purposes but have never been applied to the subject of this paper. These operators generalize a popular class of fractional integrals; therefore, this approach will open an avenue for new research. The smart properties of these operators urge us to investigate more results using them. Full article

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

20 pages, 308 KB

Open AccessArticle

Generalized Dynamic Inequalities of Copson Type on Time Scales

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

Symmetry 2024, 16(3), 288; https://doi.org/10.3390/sym16030288 - 1 Mar 2024

Cited by 2 | Viewed by 1229

Abstract

This paper introduces novel generalizations of dynamic inequalities of Copson type within the framework of time scales delta calculus. The proposed generalizations leverage mathematical tools such as Hölder’s inequality, Minkowski’s inequality, the chain rule on time scales, and the properties of power rules [...] Read more.

This paper introduces novel generalizations of dynamic inequalities of Copson type within the framework of time scales delta calculus. The proposed generalizations leverage mathematical tools such as Hölder’s inequality, Minkowski’s inequality, the chain rule on time scales, and the properties of power rules on time scales. As special cases of our results, particularly when the time scale

T

equals the real line (

T = R

), we derive some classical continuous analogs of previous inequalities. Furthermore, when

T

corresponds to the set of natural numbers including zero (

T = N_{0}

), the obtained results, to the best of the authors’ knowledge, represent innovative contributions to the field. Full article

(This article belongs to the Special Issue Fractional Dynamic Inequalities with Numerical Techniques and Its Application to Arbitrary Time Scales)

21 pages, 674 KB

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 9 | Viewed by 3202

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, 337 KB

Open AccessFeature PaperArticle

Numerical Approximation for a Stochastic Fractional Differential Equation Driven by Integrated Multiplicative Noise

by James Hoult and Yubin Yan

Mathematics 2024, 12(3), 365; https://doi.org/10.3390/math12030365 - 23 Jan 2024

Cited by 2 | Viewed by 1761

Abstract

We consider a numerical approximation for stochastic fractional differential equations driven by integrated multiplicative noise. The fractional derivative is in the Caputo sense with the fractional order

α \in (0, 1)

, and the non-linear terms satisfy the global Lipschitz [...] Read more.

We consider a numerical approximation for stochastic fractional differential equations driven by integrated multiplicative noise. The fractional derivative is in the Caputo sense with the fractional order

α \in (0, 1)

, and the non-linear terms satisfy the global Lipschitz conditions. We first approximate the noise with the piecewise constant function to obtain the regularized stochastic fractional differential equation. By applying Minkowski’s inequality for double integrals, we establish that the error between the exact solution and the solution of the regularized problem has an order of

O (Δ t^{α})

in the mean square norm, where

Δ t

denotes the step size. To validate our theoretical conclusions, numerical examples are presented, demonstrating the consistency of the numerical results with the established theory. Full article

12 pages, 275 KB

Open AccessArticle

Inequalities in Riemann–Lebesgue Integrability

by Anca Croitoru, Alina Gavriluţ, Alina Iosif and Anna Rita Sambucini

Mathematics 2024, 12(1), 49; https://doi.org/10.3390/math12010049 - 22 Dec 2023

Cited by 2 | Viewed by 1654

Abstract

In this paper, we prove some inequalities for Riemann–Lebesgue integrable functions when the considered integration is obtained via a non-additive measure, including the reverse Hölder inequality and the reverse Minkowski inequality. Then, we generalize these inequalities to the framework of a multivalued case, [...] Read more.

In this paper, we prove some inequalities for Riemann–Lebesgue integrable functions when the considered integration is obtained via a non-additive measure, including the reverse Hölder inequality and the reverse Minkowski inequality. Then, we generalize these inequalities to the framework of a multivalued case, in particular for Riemann–Lebesgue integrable interval-valued multifunctions, and obtain some inequalities, such as a Minkowski-type inequality, a Beckenbach-type inequality and some generalizations of Hölder inequalities. Full article

27 pages, 474 KB

Open AccessArticle

Certain New Reverse Hölder- and Minkowski-Type Inequalities for Modified Unified Generalized Fractional Integral Operators with Extended Unified Mittag–Leffler Functions

by Wengui Yang

Fractal Fract. 2023, 7(8), 613; https://doi.org/10.3390/fractalfract7080613 - 9 Aug 2023

Cited by 6 | Viewed by 1714

Abstract

In this article, we obtain certain novel reverse Hölder- and Minkowski-type inequalities for modified unified generalized fractional integral operators (FIOs) with extended unified Mittag–Leffler functions (MLFs). The predominant results of this article generalize and extend the existing fractional Hölder- and Minkowski-type integral inequalities [...] Read more.

In this article, we obtain certain novel reverse Hölder- and Minkowski-type inequalities for modified unified generalized fractional integral operators (FIOs) with extended unified Mittag–Leffler functions (MLFs). The predominant results of this article generalize and extend the existing fractional Hölder- and Minkowski-type integral inequalities in the literature. As applications, the reverse versions of weighted Radon-, Jensen- and power mean-type inequalities for modified unified generalized FIOs with extended unified MLFs are also investigated. Full article

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

26 pages, 401 KB

Open AccessArticle

Results on Minkowski-Type Inequalities for Weighted Fractional Integral Operators

by Hari Mohan Srivastava, Soubhagya Kumar Sahoo, Pshtiwan Othman Mohammed, Artion Kashuri and Nejmeddine Chorfi

Symmetry 2023, 15(8), 1522; https://doi.org/10.3390/sym15081522 - 2 Aug 2023

Cited by 9 | Viewed by 2397

Abstract

This article considers a general family of weighted fractional integral operators and utilizes this general operator to establish numerous reverse Minkowski inequalities. When it comes to understanding and investigating convexity and inequality, symmetry is crucial. It provides insightful explanations, clearer explanations, and useful [...] Read more.

This article considers a general family of weighted fractional integral operators and utilizes this general operator to establish numerous reverse Minkowski inequalities. When it comes to understanding and investigating convexity and inequality, symmetry is crucial. It provides insightful explanations, clearer explanations, and useful methods to help with the learning of key mathematical ideas. The kernel of the general family of weighted fractional integral operators is related to a wide variety of extensions and generalizations of the Mittag-Leffler function and the Hurwitz-Lerch zeta function. It delves into the applications of fractional-order integral and derivative operators in mathematical and engineering sciences. Furthermore, this article derives specific cases for selected functions and presents various applications to illustrate the obtained results. Additionally, novel applications involving the Digamma function are introduced. Full article

(This article belongs to the Special Issue Asymmetric and Symmetric Study on Number Theory and Cryptography)

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