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23 pages, 5446 KiB

Open AccessArticle

A Parametric Method for Proving Some Analytic Inequalities

by Branko Malešević, Miloš Mićović and Bojana Mihailović

Axioms 2024, 13(8), 520; https://doi.org/10.3390/axioms13080520 - 1 Aug 2024

Cited by 1 | Viewed by 1020

In this paper, a parametric method for proving inequalities is described. The method is based on associating a considered inequality with the corresponding stratified family of functions. Many inequalities from the theory of analytic inequalities can be interpreted using families of functions that [...] Read more.

In this paper, a parametric method for proving inequalities is described. The method is based on associating a considered inequality with the corresponding stratified family of functions. Many inequalities from the theory of analytic inequalities can be interpreted using families of functions that are stratified with respect to some parameter. By discussing the sign of the functions from the family by the parameter according to which the family is stratified, inequalities are obtained that contain the best possible constants, if they exist. The application of this method is demonstrated for four inequalities: the Cusa–Huygens inequality, the Wilker-type inequality and the two Mitrinović–Adamović-type inequalities. Significantly simpler proofs and improvements of all these inequalities are provided. Full article

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

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12 pages, 259 KiB

Open AccessArticle

Some Refinements and Generalizations of Bohr’s Inequality

by Salma Aljawi, Cristian Conde and Kais Feki

Axioms 2024, 13(7), 436; https://doi.org/10.3390/axioms13070436 - 28 Jun 2024

Viewed by 674

Abstract

In this article, we delve into the classic Bohr inequality for complex numbers, a fundamental result in complex analysis with broad mathematical applications. We offer refinements and generalizations of Bohr’s inequality, expanding on the established inequalities of N. G. de Bruijn and Radon, [...] Read more.

In this article, we delve into the classic Bohr inequality for complex numbers, a fundamental result in complex analysis with broad mathematical applications. We offer refinements and generalizations of Bohr’s inequality, expanding on the established inequalities of N. G. de Bruijn and Radon, as well as leveraging the class of functions defined by the Daykin–Eliezer–Carlitz inequality. Our novel contribution lies in demonstrating that Bohr’s and Bergström’s inequalities can be derived from one another, revealing a deeper interconnectedness between these results. Furthermore, we present several new generalizations of Bohr’s inequality, along with other notable inequalities from the literature, and discuss their various implications. By providing more comprehensive and verifiable conditions, our work extends previous research and enhances the understanding and applicability of Bohr’s inequality in mathematical studies. Full article

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

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 1 | Viewed by 1089

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)

25 pages, 336 KiB

Open AccessArticle

Fundamental Properties of Muckenhoupt and Gehring Weights on Time Scales

by Ravi P. Agarwal, Mohamed Abdalla Darwish, Hamdi Ali Elshamy and Samir H. Saker

Axioms 2024, 13(2), 98; https://doi.org/10.3390/axioms13020098 - 31 Jan 2024

Cited by 2 | Viewed by 1727

Abstract

Some fundamental properties of the Muckenhoupt class

A_{p}

of weights and the Gehring class

G_{q}

of weights on time scales and some relations between them will be proved in this paper. To prove the main results, we will apply an approach [...] Read more.

Some fundamental properties of the Muckenhoupt class

A_{p}

of weights and the Gehring class

G_{q}

of weights on time scales and some relations between them will be proved in this paper. To prove the main results, we will apply an approach based on proving some properties of integral operators on time scales with powers and certain mathematical relations connecting the norms of Muckenhoupt and Gehring classes. The results as special cases cover the results for functions following David Cruz-Uribe, C. J. Neugebauer, and A. Popoli, and when the time scale equals the positive integers, the results for sequences are essentially new. Full article

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

14 pages, 298 KiB

Open AccessArticle

Improvements of Integral Majorization Inequality with Applications to Divergences

by Abdul Basir, Muhammad Adil Khan, Hidayat Ullah, Yahya Almalki, Chanisara Metpattarahiran and Thanin Sitthiwirattham

Axioms 2024, 13(1), 21; https://doi.org/10.3390/axioms13010021 - 28 Dec 2023

Cited by 2 | Viewed by 1563

Abstract

Within the recent wave of research advancements, mathematical inequalities and their practical applications play a notably significant role across various domains. In this regard, inequalities offer a captivating arena for scholarly endeavors and investigational pursuits. This research work aims to present new improvements [...] Read more.

Within the recent wave of research advancements, mathematical inequalities and their practical applications play a notably significant role across various domains. In this regard, inequalities offer a captivating arena for scholarly endeavors and investigational pursuits. This research work aims to present new improvements for the integral majorization inequalities using an interesting aproach. Certain previous improvements have been achieved for the Jensen inequality as direct outcomes of the main results. Additionally, estimates for the Csiszár divergence and its cases are provided as applications of the main results. The circumstances under which the principal outcomes offer enhanced estimations for majorization differences are also underscored and emphasized. Full article

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

13 pages, 286 KiB

Open AccessArticle

Generalized Refinements of Reversed AM-GM Operator Inequalities for Positive Linear Maps

by Yonghui Ren

Axioms 2023, 12(10), 977; https://doi.org/10.3390/axioms12100977 - 17 Oct 2023

Viewed by 1256

Abstract

We shall present some more generalized and further refinements of reversed AM-GM operator inequalities for positive linear maps due to Xue’s and Ali’s publications. Full article

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

13 pages, 284 KiB

Open AccessArticle

Equivalent Statements of Two Multidimensional Hilbert-Type Integral Inequalities with Parameters

by Yiyuan Li, Yanru Zhong and Bicheng Yang

Axioms 2023, 12(10), 956; https://doi.org/10.3390/axioms12100956 - 10 Oct 2023

Cited by 1 | Viewed by 1133

Abstract

By means of the weight functions, the idea of introduced parameters and the transfer formulas, two multidimensional Hilbert-type integral inequalities with the general nonhomogeneous kernel as [...] Read more.

By means of the weight functions, the idea of introduced parameters and the transfer formulas, two multidimensional Hilbert-type integral inequalities with the general nonhomogeneous kernel as

H (| | x | |_{α}^{λ_{1}} | | y | |_{β}^{λ_{2}})

(λ_{1}, λ_{2} \neq 0)

are given, which are some extensions of the Hilbert-type integral inequalities in the two-dimensional case. Some equivalent conditions of the best value and several parameters related to the new inequalities are provided. Two corollaries regarding the kernel, represented as

k_{λ} {(| | x | |_{α}^{λ_{1}}, | | y | |_{β}^{λ_{2}})}^{} (λ_{1}, λ_{2} \neq 0)

, are given, and a few new inequalities for the particular parameters are obtained. Full article

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

20 pages, 346 KiB

Open AccessArticle

Innovative Interpolating Polynomial Approach to Fractional Integral Inequalities and Real-World Implementations

by Muhammad Samraiz, Saima Naheed, Ayesha Gul, Gauhar Rahman and Miguel Vivas-Cortez

Axioms 2023, 12(10), 914; https://doi.org/10.3390/axioms12100914 - 26 Sep 2023

Cited by 2 | Viewed by 1637

Abstract

Our paper explores Hermite–Hadamard inequalities through the application of Abel–Gontscharoff Green’s function methodology, which involves interpolating polynomials and Riemann-type generalized fractional integrals. While establishing our main results, we explore new identities. These identities are used to estimate novel findings for functions, such that [...] Read more.

Our paper explores Hermite–Hadamard inequalities through the application of Abel–Gontscharoff Green’s function methodology, which involves interpolating polynomials and Riemann-type generalized fractional integrals. While establishing our main results, we explore new identities. These identities are used to estimate novel findings for functions, such that the second derivative of the functions is monotone, absolutely convex, and concave. A section relating the results of exploration to generalized means and trapezoid formulas is included in the applications. We anticipate that the method presented in this study will inspire further research in this field. Full article

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

17 pages, 346 KiB

Open AccessArticle

Derivation of Bounds for Majorization Differences by a Novel Method and Its Applications in Information Theory

by Abdul Basir, Muhammad Adil Khan, Hidayat Ullah, Yahya Almalki, Saowaluck Chasreechai and Thanin Sitthiwirattham

Axioms 2023, 12(9), 885; https://doi.org/10.3390/axioms12090885 - 16 Sep 2023

Cited by 3 | Viewed by 1529

Abstract

In the recent era of research developments, mathematical inequalities and their applications perform a very consequential role in different aspects, and they provide an engaging area for research activities. In this paper, we propose a new approach for the improvement of the classical [...] Read more.

In the recent era of research developments, mathematical inequalities and their applications perform a very consequential role in different aspects, and they provide an engaging area for research activities. In this paper, we propose a new approach for the improvement of the classical majorization inequality and its weighted versions in a discrete sense. The proposed improvements give several estimates for the majorization differences. Some earlier improvements of the Jensen and Slater inequalities are deduced as direct consequences of the obtained results. We also discuss the conditions under which the main results give better estimates for the majorization differences. Applications of the acquired results are also presented in information theory. Full article

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

20 pages, 321 KiB

Open AccessArticle

On Further Refinements of Numerical Radius Inequalities

by Ayman Hazaymeh, Ahmad Qazza, Raed Hatamleh, Mohammad W. Alomari and Rania Saadeh

Axioms 2023, 12(9), 807; https://doi.org/10.3390/axioms12090807 - 22 Aug 2023

Cited by 26 | Viewed by 1152

Abstract

This paper introduces several generalized extensions of some recent numerical radius inequalities of Hilbert space operators. More preciously, these inequalities refine the recent inequalities that were proved in literature. It has already been demonstrated that some inequalities can be improved or restored by [...] Read more.

This paper introduces several generalized extensions of some recent numerical radius inequalities of Hilbert space operators. More preciously, these inequalities refine the recent inequalities that were proved in literature. It has already been demonstrated that some inequalities can be improved or restored by concatenating some into one inequality. The main idea of this paper is to extend the existing numerical radius inequalities by providing a unified framework. We also present a numerical example to demonstrate the effectiveness of the proposed approach. Roughly, our approach combines the existing inequalities, proved in literature, into a single inequality that can be used to obtain improved or restored results. This unified approach allows us to extend the existing numerical radius inequalities and show their effectiveness through numerical experiments. Full article

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

11 pages, 294 KiB

Open AccessArticle

A Generalized Convexity and Inequalities Involving the Unified Mittag–Leffler Function

by Ghulam Farid, Hafsa Tariq, Ferdous M. O. Tawfiq, Jong-Suk Ro and Saira Zainab

Axioms 2023, 12(8), 795; https://doi.org/10.3390/axioms12080795 - 17 Aug 2023

Cited by 1 | Viewed by 1256

Abstract

This article aims to obtain inequalities containing the unified Mittag–Leffler function which give bounds of integral operators for a generalized convexity. These findings provide generalizations and refinements of many inequalities. By setting values of monotone functions, it is possible to reproduce results for [...] Read more.

This article aims to obtain inequalities containing the unified Mittag–Leffler function which give bounds of integral operators for a generalized convexity. These findings provide generalizations and refinements of many inequalities. By setting values of monotone functions, it is possible to reproduce results for classical convexities. The Hadamard-type inequalities for several classes related to convex functions are identified in remarks, and some of them are also presented in last section. Full article

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

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