Editorial

3 pages, 161 KiB

Open AccessEditorial

Introduction to the Special Issue in Axioms Titled Current Research on Mathematical Inequalities

by Christophe Chesneau

Axioms 2023, 12(2), 109; https://doi.org/10.3390/axioms12020109 - 19 Jan 2023

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The importance of inequalities in Mathematics is beautifully summarized in a citation attributed to Respected Professor Andrey Nikolaevich Kolmogorov: [...] Full article

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

Research

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17 pages, 314 KiB

Open AccessArticle

Some New Estimates for the Berezin Number of Hilbert Space Operators

by Najla Altwaijry, Kais Feki and Nicuşor Minculete

Axioms 2022, 11(12), 683; https://doi.org/10.3390/axioms11120683 - 29 Nov 2022

Cited by 1 | Viewed by 1088

Abstract

In this paper, we have developed new estimates of some estimates involving the Berezin norm and Berezin number of bounded linear operators defined on a reproducing kernel Hilbert space

H_{Ω}

. The uniqueness or novelty of this article consists of new estimates [...] Read more.

In this paper, we have developed new estimates of some estimates involving the Berezin norm and Berezin number of bounded linear operators defined on a reproducing kernel Hilbert space

H_{Ω}

. The uniqueness or novelty of this article consists of new estimates of Berezin numbers for different types of operators. These estimates improve the upper bounds of the Berezin numbers obtained by other similar papers. We give several upper bounds for

{ber}^{r} (S^{*} T)

, where

T, S \in B (H_{Ω})

and

r \geq 1

. We also present an estimation of

{ber}^{2 r} (\sum_{i = 1}^{d} T_{i})

where

T_{i} \in B (H_{Ω})

,

i = \bar{1, d}

and

r \geq 1

. Some of the obtained inequalities represent improvements to earlier ones. In this work, the ideas and methodologies presented may serve as a starting point for future investigation in this field. Full article

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

20 pages, 351 KiB

Open AccessArticle

Operator Jensen’s Inequality for Operator Superquadratic Functions

by Mohammad W. Alomari, Christophe Chesneau and Ahmad Al-Khasawneh

Axioms 2022, 11(11), 617; https://doi.org/10.3390/axioms11110617 - 06 Nov 2022

Cited by 2 | Viewed by 1108

Abstract

In this work, an operator superquadratic function (in the operator sense) for positive Hilbert space operators is defined. Several examples with some important properties together with some observations which are related to the operator convexity are pointed out. A general Bohr’s inequality for [...] Read more.

In this work, an operator superquadratic function (in the operator sense) for positive Hilbert space operators is defined. Several examples with some important properties together with some observations which are related to the operator convexity are pointed out. A general Bohr’s inequality for positive operators is thus deduced. A Jensen-type inequality is proved. Equivalent statements of a non-commutative version of Jensen’s inequality for operator superquadratic function are also established. Finally, several trace inequalities for superquadratic functions (in the ordinary sense) are provided as well. Full article

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

12 pages, 307 KiB

Open AccessArticle

On Symmetrized Stochastic Harmonically Convexity and Hermite–Hadamard Type Inequalities

by Muhammad Amer Latif

Axioms 2022, 11(10), 570; https://doi.org/10.3390/axioms11100570 - 20 Oct 2022

Cited by 2 | Viewed by 1140

Abstract

Throughout this study, the concept of symmetrized harmonically convex stochastic processes will be discussed in further detail. Some certain characterizations for symmetrized harmonically convex stochastic processes are discussed that use Hermite–Hadamard-type inequalities. A Hyers–Ulam-type stability result for harmonically convex stochastic processes is given [...] Read more.

Throughout this study, the concept of symmetrized harmonically convex stochastic processes will be discussed in further detail. Some certain characterizations for symmetrized harmonically convex stochastic processes are discussed that use Hermite–Hadamard-type inequalities. A Hyers–Ulam-type stability result for harmonically convex stochastic processes is given as well. Full article

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15 pages, 291 KiB

Open AccessArticle

Weighted Integral Inequalities for Harmonic Convex Functions in Connection with Fejér’s Result

by Muhammad Amer Latif

Axioms 2022, 11(10), 564; https://doi.org/10.3390/axioms11100564 - 18 Oct 2022

Cited by 4 | Viewed by 961

Abstract

In this study, on the subject of harmonic convex functions, we introduce some new functionals linked with weighted integral inequalities for harmonic convex functions. In addition, certain new inequalities of the Fejér type are discovered. Full article

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11 pages, 773 KiB

Open AccessArticle

Univariate and Multivariate Ostrowski-Type Inequalities Using Atangana–Baleanu Caputo Fractional Derivative

by Henok Desalegn Desta, Deepak B. Pachpatte, Jebessa B. Mijena and Tadesse Abdi

Axioms 2022, 11(9), 482; https://doi.org/10.3390/axioms11090482 - 19 Sep 2022

Cited by 3 | Viewed by 1133

Abstract

In this paper, we obtain some univariate and multivariate Ostrowski-type inequalities using the Atangana–Baleanu fractional derivative in the sense of Liouville–Caputo (ABC). The results obtained for both left and right ABC fractional derivatives can be applied to study further fractional inequalities and estimate [...] Read more.

In this paper, we obtain some univariate and multivariate Ostrowski-type inequalities using the Atangana–Baleanu fractional derivative in the sense of Liouville–Caputo (ABC). The results obtained for both left and right ABC fractional derivatives can be applied to study further fractional inequalities and estimate various non-local function problems since the operator consists of a non-singular kernel. The obtained results are more generalized in nature. Full article

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

16 pages, 356 KiB

Open AccessArticle

Interval Fejér-Type Inequalities for Left and Right-λ-Preinvex Functions in Interval-Valued Settings

by Tareq Saeed, Muhammad Bilal Khan, Savin Treanțǎ, Hamed H. Alsulami and Mohammed Sh. Alhodaly

Axioms 2022, 11(8), 368; https://doi.org/10.3390/axioms11080368 - 27 Jul 2022

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Abstract

For left and right λ-preinvex interval-valued functions (left and right λ-preinvex IVFs) in interval-valued Riemann operator settings, we create Hermite–Hadamard (H-H) type inequalities in the current study. Additionally, we create Hermite–Hadamard–Fejér (H-H-Fejér)-type inequalities [...] Read more.

For left and right λ-preinvex interval-valued functions (left and right λ-preinvex IVFs) in interval-valued Riemann operator settings, we create Hermite–Hadamard (H-H) type inequalities in the current study. Additionally, we create Hermite–Hadamard–Fejér (H-H-Fejér)-type inequalities for preinvex functions of the left and right interval-valued type under some mild conditions. Moreover, some exceptional new and classical cases are also obtained. Some useful examples are also presented to prove the validity of the results. Full article

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

16 pages, 334 KiB

Open AccessArticle

New Diamond-α Steffensen-Type Inequalities for Convex Functions over General Time Scale Measure Spaces

by Ksenija Smoljak Kalamir

Axioms 2022, 11(7), 323; https://doi.org/10.3390/axioms11070323 - 01 Jul 2022

Cited by 4 | Viewed by 1032

Abstract

In this paper, we extend some Steffensen-type inequalities to time scales by using the diamond-

α

-dynamic integral. Further, we prove some new Steffensen-type inequalities for convex functions utilizing positive

σ

-finite measures in time scale calculus. Moreover, as a special case, we [...] Read more.

In this paper, we extend some Steffensen-type inequalities to time scales by using the diamond-

α

-dynamic integral. Further, we prove some new Steffensen-type inequalities for convex functions utilizing positive

σ

-finite measures in time scale calculus. Moreover, as a special case, we obtain these inequalities for the delta and the nabla integral. By using the relation between calculus on time scales

T

and differential calculus on

R

, we obtain already-known Steffensen-type inequalities. Full article

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

16 pages, 293 KiB

Open AccessArticle

Some Generalized Euclidean Operator Radius Inequalities

by Mohammad W. Alomari, Khalid Shebrawi and Christophe Chesneau

Axioms 2022, 11(6), 285; https://doi.org/10.3390/axioms11060285 - 13 Jun 2022

Cited by 8 | Viewed by 1505

Abstract

In this work, some generalized Euclidean operator radius inequalities are established. Refinements of some well-known results are provided. Among others, some bounds in terms of the Cartesian decomposition of a given Hilbert space operator are proven. Full article

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

13 pages, 319 KiB

Open AccessArticle

Bounds for Quotients of Inverse Trigonometric and Inverse Hyperbolic Functions

by Sumedh B. Thool, Yogesh J. Bagul, Ramkrishna M. Dhaigude and Christophe Chesneau

Axioms 2022, 11(6), 262; https://doi.org/10.3390/axioms11060262 - 30 May 2022

Cited by 2 | Viewed by 1387

Abstract

We establish new simple bounds for the quotients of inverse trigonometric and inverse hyperbolic functions such as

\frac{{sin}^{- 1} x}{{sinh}^{- 1} x}

and

\frac{{tanh}^{- 1} x}{{tan}^{- 1} x}

. The main results provide polynomial bounds using even [...] Read more.

We establish new simple bounds for the quotients of inverse trigonometric and inverse hyperbolic functions such as

\frac{{sin}^{- 1} x}{{sinh}^{- 1} x}

and

\frac{{tanh}^{- 1} x}{{tan}^{- 1} x}

. The main results provide polynomial bounds using even quadratic functions and exponential bounds under the form

e^{a x^{2}} .

Graph validation is also performed. Full article

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

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17 pages, 321 KiB

Open AccessArticle

Some Generalizations of the Jensen-Type Inequalities with Applications

by Mirna Rodić

Axioms 2022, 11(5), 227; https://doi.org/10.3390/axioms11050227 - 13 May 2022

Cited by 4 | Viewed by 1737

Abstract

Motivated by some results about reverses of the Jensen inequality for positive measure, in this paper we give generalizations of those results for real Stieltjes measure

d λ

which is not necessarily positive using several Green functions. Utilizing these results we define some [...] Read more.

Motivated by some results about reverses of the Jensen inequality for positive measure, in this paper we give generalizations of those results for real Stieltjes measure

d λ

which is not necessarily positive using several Green functions. Utilizing these results we define some new mean value theorems of Lagrange and Cauchy types, and derive some new Cauchy-type means. Full article

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

8 pages, 280 KiB

Open AccessArticle

Coefficient Estimates and Fekete–Szegö Functional Inequalities for a Certain Subclass of Analytic and Bi-Univalent Functions

by Mohamed Illafe, Ala Amourah and Maisarah Haji Mohd

Axioms 2022, 11(4), 147; https://doi.org/10.3390/axioms11040147 - 24 Mar 2022

Cited by 11 | Viewed by 2022

Abstract

The present paper introduces a new class of bi-univalent functions defined on a symmetric domain using Gegenbauer polynomials. For functions in this class, we have derived the estimates of the Taylor–Maclaurin coefficients,

|a_{2}|

and

|a_{3}|

, and the Fekete-Szegö functional. Several [...] Read more.

The present paper introduces a new class of bi-univalent functions defined on a symmetric domain using Gegenbauer polynomials. For functions in this class, we have derived the estimates of the Taylor–Maclaurin coefficients,

|a_{2}|

and

|a_{3}|

, and the Fekete-Szegö functional. Several new results follow upon specializing the parameters involved in our main results. Full article

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

9 pages, 251 KiB

Open AccessArticle

On Some New Ostrowski–Mercer-Type Inequalities for Differentiable Functions

by Ifra Bashir Sial, Nichaphat Patanarapeelert, Muhammad Aamir Ali, Hüseyin Budak and Thanin Sitthiwirattham

Axioms 2022, 11(3), 132; https://doi.org/10.3390/axioms11030132 - 14 Mar 2022

Cited by 9 | Viewed by 1801

Abstract

In this paper, we establish a new integral identity involving differentiable functions, and then we use the newly established identity to prove some Ostrowski–Mercer-type inequalities for differentiable convex functions. It is also demonstrated that the newly established inequalities are generalizations of some of [...] Read more.

In this paper, we establish a new integral identity involving differentiable functions, and then we use the newly established identity to prove some Ostrowski–Mercer-type inequalities for differentiable convex functions. It is also demonstrated that the newly established inequalities are generalizations of some of the Ostrowski inequalities established inside the literature. There are also some applications to the special means of real numbers given. Full article

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

17 pages, 651 KiB

Open AccessArticle

K-Nearest Neighbor Estimation of Functional Nonparametric Regression Model under NA Samples

by Xueping Hu, Jingya Wang, Liuliu Wang and Keming Yu

Axioms 2022, 11(3), 102; https://doi.org/10.3390/axioms11030102 - 25 Feb 2022

Cited by 4 | Viewed by 2727

Abstract

Functional data, which provides information about curves, surfaces or anything else varying over a continuum, has become a commonly encountered type of data. The k-nearest neighbor (kNN) method, as a nonparametric method, has become one of the most popular supervised machine learning algorithms [...] Read more.

Functional data, which provides information about curves, surfaces or anything else varying over a continuum, has become a commonly encountered type of data. The k-nearest neighbor (kNN) method, as a nonparametric method, has become one of the most popular supervised machine learning algorithms used to solve both classification and regression problems. This paper is devoted to the k-nearest neighbor (kNN) estimators of the nonparametric functional regression model when the observed variables take values from negatively associated (NA) sequences. The consistent and complete convergence rate for the proposed kNN estimator is first provided. Then, numerical assessments, including simulation study and real data analysis, are conducted to evaluate the performance of the proposed method and compare it with the standard nonparametric kernel approach. Full article

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

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8 pages, 255 KiB

Open AccessArticle

Pólya–Szegö Integral Inequalities Using the Caputo–Fabrizio Approach

by Asha B. Nale, Vaijanath L. Chinchane, Satish K. Panchal and Christophe Chesneau

Axioms 2022, 11(2), 79; https://doi.org/10.3390/axioms11020079 - 17 Feb 2022

Cited by 2 | Viewed by 1374

Abstract

In this article, we establish some of the Pólya–Szegö and Minkowsky-type fractional integral inequalities by considering the Caputo–Fabrizio fractional integral. Moreover, we give some special cases of Pólya–Szegö inequalities. Full article

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

9 pages, 378 KiB

Open AccessArticle

Perturbation of One-Dimensional Time-Independent Schrödinger Equation with a Near-Hyperbolic Potential

by Byungbae Kim and Soon-Mo Jung

Axioms 2022, 11(2), 63; https://doi.org/10.3390/axioms11020063 - 02 Feb 2022

Cited by 2 | Viewed by 1712

Abstract

The authors have recently investigated a type of Hyers–Ulam stability of one-dimensional time-independent Schrödinger equation with a symmetric parabolic potential wall. In this paper, we investigate a type of Hyers–Ulam stability of the Schrödinger equation with a near-hyperbolic potential. Full article

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18 pages, 305 KiB

Open AccessArticle

Post-Quantum Midpoint-Type Inequalities Associated with Twice-Differentiable Functions

by Thanin Sitthiwirattham, Ghulam Murtaza, Muhammad Aamir Ali, Chanon Promsakon, Ifra Bashir Sial and Praveen Agarwal

Axioms 2022, 11(2), 46; https://doi.org/10.3390/axioms11020046 - 23 Jan 2022

Cited by 6 | Viewed by 1973

Abstract

In this study, first we establish a

(p, q)

-integral identity involving the second

(p, q)

-derivative, and then, we use this result to prove some new midpoint-type inequalities for twice-

(p, q)

-differentiable convex functions. It is also shown [...] Read more.

In this study, first we establish a

(p, q)

-integral identity involving the second

(p, q)

-derivative, and then, we use this result to prove some new midpoint-type inequalities for twice-

(p, q)

-differentiable convex functions. It is also shown that the newly established results are the refinements of the comparable results in the literature. Full article

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

8 pages, 256 KiB

Open AccessArticle

A Note on Hermite–Hadamard–Fejer Type Inequalities for Functions Whose n-th Derivatives Are m-Convex or (α,m)-Convex Functions

by Sanja Kovač

Axioms 2022, 11(1), 16; https://doi.org/10.3390/axioms11010016 - 31 Dec 2021

Cited by 1 | Viewed by 1109

Abstract

In this paper, we develop some Hermite–Hadamard–Fejér type inequalities for n-times differentiable functions whose absolute values of

n

-th derivatives are

(α, m)

-convex function. The results obtained in this paper are extensions and generalizations of the existing ones. [...] Read more.

In this paper, we develop some Hermite–Hadamard–Fejér type inequalities for n-times differentiable functions whose absolute values of

n

-th derivatives are

(α, m)

-convex function. The results obtained in this paper are extensions and generalizations of the existing ones. As a special case, the generalization of the remainder term of the midpoint and trapezoidal quadrature formulas are obtained. Full article

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

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