Mathematics

Research

5 pages, 263 KiB

Open AccessArticle

Two Velichko-like Theorems for C(X)

by Salvador López-Alfonso, Manuel López-Pellicer and Santiago Moll-López

Mathematics 2023, 11(24), 4930; https://doi.org/10.3390/math11244930 - 12 Dec 2023

Viewed by 708

Abstract

This paper provides two new Velichko-like theorems for the weak counterpart of the locally convex space

C_{k} (X)

of all real-valued functions defined on a Tychonoff space X equipped with the compact-open topology

τ_{k}

. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

8 pages, 271 KiB

Open AccessArticle

Generalized Taylor’s Formula and Steffensen’s Inequality

by Asfand Fahad, Saad Ihsaan Butt, Josip Pečarić and Marjan Praljak

Mathematics 2023, 11(16), 3570; https://doi.org/10.3390/math11163570 - 17 Aug 2023

Cited by 1 | Viewed by 995

Abstract

New Steffensen-type inequalities are obtained by combining generalized Taylor expansions, Rabier and Pečarić extensions of Steffensen’s inequality and Faà di Bruno’s formula for higher order derivatives of the composition. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

25 pages, 398 KiB

Open AccessArticle

Estimations of the Jensen Gap and Their Applications Based on 6-Convexity

by Muhammad Adil Khan, Asadullah Sohail, Hidayat Ullah and Tareq Saeed

Mathematics 2023, 11(8), 1957; https://doi.org/10.3390/math11081957 - 20 Apr 2023

Cited by 4 | Viewed by 1661

Abstract

The main purpose of this manuscript is to present some new estimations of the Jensen gap in a discrete sense along with their applications. The proposed estimations for the Jensen gap are provided with the help of the notion of 6-convex functions. Some [...] Read more.

The main purpose of this manuscript is to present some new estimations of the Jensen gap in a discrete sense along with their applications. The proposed estimations for the Jensen gap are provided with the help of the notion of 6-convex functions. Some numerical experiments are performed to determine the significance and correctness of the intended estimates. Several outcomes of the main results are discussed for the Hölder inequality and the power and quasi-arithmetic means. Furthermore, some applications are presented in information theory, which provide some bounds for the divergences, Bhattacharyya coefficient, Shannon entropy, and Zipf–Mandelbrot entropy. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

19 pages, 3415 KiB

Open AccessArticle

A Large Scale Analysis for Testing a Mathematical Model for the Study of Vascular Pathologies

by Arianna Travaglini, Gianluca Vinti, Giovanni Battista Scalera and Michele Scialpi

Mathematics 2023, 11(8), 1831; https://doi.org/10.3390/math11081831 - 12 Apr 2023

Cited by 2 | Viewed by 1423

Abstract

In this paper, we carry out a study developed on 13,677 images from 15 patients affected by moderate/severe atheromatous disease of the abdominal aortic tract. A procedure to extract the pervious lumen of the aorta artery from basal CT images is exploited and [...] Read more.

In this paper, we carry out a study developed on 13,677 images from 15 patients affected by moderate/severe atheromatous disease of the abdominal aortic tract. A procedure to extract the pervious lumen of the aorta artery from basal CT images is exploited and tested on a large scale. In particular, the above method takes advantage of the reconstruction and enhancing properties of the sampling Kantorovich algorithm which allows the information content of images to be increased. The processed image is compared, slice by slice, by superposition, with the corresponding contrast medium reference image. Numerical indices of errors were computed and analyzed in order to test the validity of the proposed method. The results achieved confirm, both from the numerical and clinical point of view, the good performance and accuracy of the proposed method, opening the possibility to perform an assisted diagnosis avoiding the injection of the contrast medium. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

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13 pages, 282 KiB

Open AccessArticle

On Some Inequalities Involving Generalized Distance Functions

by Mohamed Jleli and Bessem Samet

Mathematics 2023, 11(5), 1157; https://doi.org/10.3390/math11051157 - 26 Feb 2023

Viewed by 1088

Abstract

In this paper, a new class of generalized distance functions with respect to a pair of mappings is introduced. Next, some inequalities involving such distance functions are established. Our obtained results generalize and cover some recent results from the literature. Moreover, new distance [...] Read more.

In this paper, a new class of generalized distance functions with respect to a pair of mappings is introduced. Next, some inequalities involving such distance functions are established. Our obtained results generalize and cover some recent results from the literature. Moreover, new distance inequalities for self-crossing polygons are obtained. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

21 pages, 363 KiB

Open AccessArticle

A New Seminorm for d-Tuples of A-Bounded Operators and Their Applications

by Najla Altwaijry, Kais Feki and Nicuşor Minculete

Mathematics 2023, 11(3), 685; https://doi.org/10.3390/math11030685 - 29 Jan 2023

Cited by 6 | Viewed by 1358

Abstract

The aim of this paper was to introduce and investigate a new seminorm of operator tuples on a complex Hilbert space

H

when an additional semi-inner product structure defined by a positive (semi-definite) operator A on

H

is considered. We prove the equality [...] Read more.

The aim of this paper was to introduce and investigate a new seminorm of operator tuples on a complex Hilbert space

H

when an additional semi-inner product structure defined by a positive (semi-definite) operator A on

H

is considered. We prove the equality between this new seminorm and the well-known A-joint seminorm in the case of A-doubly-commuting tuples of A-hyponormal operators. This study is an extension of a well-known result in [Results Math 75, 93(2020)] and allows us to show that the following equalities

r_{A} (T) = ω_{A} (T) = {∥ T ∥}_{A}

hold for every A-doubly-commuting d-tuple of A-hyponormal operators

T = (T_{1}, \dots, T_{d})

. Here,

r_{A} (T), {∥ T ∥}_{A}

, and

ω_{A} (T)

denote the A-joint spectral radius, the A-joint operator seminorm, and the A-joint numerical radius of

T

, respectively. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

11 pages, 275 KiB

Open AccessArticle

Improvement of Furuta’s Inequality with Applications to Numerical Radius

by Mohammad W. Alomari, Mojtaba Bakherad, Monire Hajmohamadi, Christophe Chesneau, Víctor Leiva and Carlos Martin-Barreiro

Mathematics 2023, 11(1), 36; https://doi.org/10.3390/math11010036 - 22 Dec 2022

Cited by 1 | Viewed by 1358

Abstract

In diverse branches of mathematics, several inequalities have been studied and applied. In this article, we improve Furuta’s inequality. Subsequently, we apply this improvement to obtain new radius inequalities that not been reported in the current literature. Numerical examples illustrate the main findings. [...] Read more.

In diverse branches of mathematics, several inequalities have been studied and applied. In this article, we improve Furuta’s inequality. Subsequently, we apply this improvement to obtain new radius inequalities that not been reported in the current literature. Numerical examples illustrate the main findings. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

10 pages, 324 KiB

Open AccessArticle

A Refined Jensen Inequality Connected to an Arbitrary Positive Finite Sequence

by Shanhe Wu, Muhammad Adil Khan, Tareq Saeed and Zaid Mohammed Mohammed Mahdi Sayed

Mathematics 2022, 10(24), 4817; https://doi.org/10.3390/math10244817 - 18 Dec 2022

Cited by 1 | Viewed by 1505

Abstract

The prime purpose of this paper is to provide a refinement of Jensen’s inequality in connection with a positive finite sequence. We deal with the refinement for particular cases and point out the relation between the new result with earlier results of Jensen’s [...] Read more.

The prime purpose of this paper is to provide a refinement of Jensen’s inequality in connection with a positive finite sequence. We deal with the refinement for particular cases and point out the relation between the new result with earlier results of Jensen’s inequality. As results, we obtain refinements of the quasi-arithmetic and power mean inequalities. Finally, several results are obtained in information theory with the help of the main results. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

8 pages, 243 KiB

Open AccessArticle

Two Approximation Formulas for Bateman’s G-Function with Bounded Monotonic Errors

by Mansour Mahmoud and Hanan Almuashi

Mathematics 2022, 10(24), 4787; https://doi.org/10.3390/math10244787 - 16 Dec 2022

Viewed by 981

Abstract

Two new approximation formulas for Bateman’s G-function are presented with strictly monotonic error functions and we deduced their sharp bounds. We also studied the completely monotonic (CM) degrees of two functions involving

G (r)

, deducing two of its inequalities [...] Read more.

Two new approximation formulas for Bateman’s G-function are presented with strictly monotonic error functions and we deduced their sharp bounds. We also studied the completely monotonic (CM) degrees of two functions involving

G (r)

, deducing two of its inequalities and improving some of the recently published results. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

17 pages, 322 KiB

Open AccessArticle

Further Inequalities for the Weighted Numerical Radius of Operators

by Najla Altwaijry, Kais Feki and Nicuşor Minculete

Mathematics 2022, 10(19), 3576; https://doi.org/10.3390/math10193576 - 30 Sep 2022

Cited by 7 | Viewed by 1951

Abstract

This paper deals with the so-called A-numerical radius associated with a positive (semi-definite) bounded linear operator A acting on a complex Hilbert space

H

. Several new inequalities involving this concept are established. In particular, we prove several estimates for [...] Read more.

This paper deals with the so-called A-numerical radius associated with a positive (semi-definite) bounded linear operator A acting on a complex Hilbert space

H

. Several new inequalities involving this concept are established. In particular, we prove several estimates for

2 \times 2

operator matrices whose entries are A-bounded operators. Some of the obtained results cover and extend well-known recent results due to Bani-Domi and Kittaneh. In addition, several improvements of the generalized Kittaneh estimates are obtained. The inequalities given by Feki in his work represent a generalization of the inequalities given by Kittaneh. Some refinements of the inequalities due to Feki are also presented. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

10 pages, 267 KiB

Open AccessArticle

On Several Bounds for Types of Angular Distances

by Augusta Raţiu and Nicuşor Minculete

Mathematics 2022, 10(18), 3303; https://doi.org/10.3390/math10183303 - 12 Sep 2022

Cited by 1 | Viewed by 1253

Abstract

In this study, we introduce the expression

d_{λ} (x, y) : = λ ∥ x ∥ + (1 - λ) ∥ y ∥ - ∥ λ x + (1 - λ) y ∥

on the [...] Read more.

In this study, we introduce the expression

d_{λ} (x, y) : = λ ∥ x ∥ + (1 - λ) ∥ y ∥ - ∥ λ x + (1 - λ) y ∥

on the real normed space

X (X, ∥ \cdot ∥)

, where

x, y \in X

and

λ \in R

. We characterize this expression and find various estimates of it. We also obtain a generalization and some refinements of Maligranda’s inequality. Finally, we give some relations between

d_{λ} (x, y)

and several types of angular distances between two nonzero vectors in a real normed space. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

13 pages, 291 KiB

Open AccessArticle

Convergence of Special Sequences of Semi-Exponential Operators

by Ana Maria Acu, Vijay Gupta, Ioan Raşa and Florin Sofonea

Mathematics 2022, 10(16), 2978; https://doi.org/10.3390/math10162978 - 18 Aug 2022

Cited by 14 | Viewed by 1591

Abstract

Several papers, mainly written by J. de la Call and co-authors, contain modifications of classical sequences of positive linear operators to obtain new sequences converging to limits which are not necessarily the identity operator. Such results were obtained using probabilistic methods. Recently, results [...] Read more.

Several papers, mainly written by J. de la Call and co-authors, contain modifications of classical sequences of positive linear operators to obtain new sequences converging to limits which are not necessarily the identity operator. Such results were obtained using probabilistic methods. Recently, results of this type have been obtained with analytic methods. Semi-exponential operators have also been introduced, extending the theory of exponential operators. We combine these two approaches, applying the semi-exponential operators in a new context and enlarging the list of operators representable as limits of other operators. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

9 pages, 284 KiB

Open AccessArticle

Hyers–Ulam Stability of Order k for Euler Equation and Euler–Poisson Equation in the Calculus of Variations

by Daniela Marian, Sorina Anamaria Ciplea and Nicolaie Lungu

Mathematics 2022, 10(15), 2556; https://doi.org/10.3390/math10152556 - 22 Jul 2022

Viewed by 1359

Abstract

In this paper, we define and study Hyers–Ulam stability of order 1 for Euler’s equation and Hyers–Ulam stability of order

m - 1

for the Euler–Poisson equation in the calculus of variations in two special cases, when these equations have the form [...] Read more.

In this paper, we define and study Hyers–Ulam stability of order 1 for Euler’s equation and Hyers–Ulam stability of order

m - 1

for the Euler–Poisson equation in the calculus of variations in two special cases, when these equations have the form

y^{″} (x) = f (x)

and

y^{(m)} (x) = f (x),

respectively. We prove some estimations for

|J [y (x)] - J [y_{0} (x)]|

, where y is an approximate solution and

y_{0}

is an exact solution of the corresponding Euler and Euler-Poisson equations, respectively. We also give two examples. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

13 pages, 293 KiB

Open AccessArticle

A Reverse Hardy–Hilbert’s Inequality Containing Multiple Parameters and One Partial Sum

by Bicheng Yang, Shanhe Wu and Xingshou Huang

Mathematics 2022, 10(13), 2362; https://doi.org/10.3390/math10132362 - 5 Jul 2022

Viewed by 1705

Abstract

In this work, by introducing multiple parameters and utilizing the Euler–Maclaurin summation formula and Abel’s partial summation formula, we first establish a reverse Hardy–Hilbert’s inequality containing one partial sum as the terms of double series. Then, based on the newly proposed inequality, we [...] Read more.

In this work, by introducing multiple parameters and utilizing the Euler–Maclaurin summation formula and Abel’s partial summation formula, we first establish a reverse Hardy–Hilbert’s inequality containing one partial sum as the terms of double series. Then, based on the newly proposed inequality, we characterize the equivalent conditions of the best possible constant factor associated with several parameters. At the end of the paper, we illustrate that more new inequalities can be generated from the special cases of the reverse Hardy–Hilbert’s inequality. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

15 pages, 309 KiB

Open AccessArticle

Improvement of Some Hayashi–Ostrowski Type Inequalities with Applications in a Probability Setting

by Mohammad W. Alomari, Christophe Chesneau, Víctor Leiva and Carlos Martin-Barreiro

Mathematics 2022, 10(13), 2316; https://doi.org/10.3390/math10132316 - 1 Jul 2022

Cited by 2 | Viewed by 1560

Abstract

Different types of mathematical inequalities have been largely analyzed and employed. In this paper, we introduce improvements to some Ostrowski type inequalities and present their corresponding proofs. The presented proofs are based on applying the celebrated Hayashi inequality to certain functions. We provide [...] Read more.

Different types of mathematical inequalities have been largely analyzed and employed. In this paper, we introduce improvements to some Ostrowski type inequalities and present their corresponding proofs. The presented proofs are based on applying the celebrated Hayashi inequality to certain functions. We provide examples that show these improvements. Illustrations of the obtained results are stated in a probability framework. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

9 pages, 265 KiB

Open AccessFeature PaperArticle

Hyers–Ulam Stability of a System of Hyperbolic Partial Differential Equations

by Daniela Marian, Sorina Anamaria Ciplea and Nicolaie Lungu

Mathematics 2022, 10(13), 2183; https://doi.org/10.3390/math10132183 - 23 Jun 2022

Cited by 2 | Viewed by 1383

Abstract

In this paper, we study Hyers–Ulam and generalized Hyers–Ulam–Rassias stability of a system of hyperbolic partial differential equations using Gronwall’s lemma and Perov’s theorem. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

10 pages, 246 KiB

Open AccessArticle

The Best Ulam Constant of the Fréchet Functional Equation

by Irina Opraie, Dorian Popa and Liana Timboş

Mathematics 2022, 10(10), 1769; https://doi.org/10.3390/math10101769 - 22 May 2022

Viewed by 1592

Abstract

In this paper, we prove the Ulam stability of the Fréchet functional equation [...] Read more.

In this paper, we prove the Ulam stability of the Fréchet functional equation

f (x + y + z) + f (x) + f (y) + f (z) = f (x + y) + f (y + z) + f (z + x)

arising from the characterization of inner product spaces and we determine its best Ulam constant. Using this result, we give a stability result for a pexiderized version of the Fréchet functional equation. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

13 pages, 280 KiB

Open AccessArticle

Approximation of Real Functions by a Generalization of Ismail–May Operator

by Adrian Holhoş

Mathematics 2022, 10(10), 1650; https://doi.org/10.3390/math10101650 - 12 May 2022

Cited by 2 | Viewed by 1472

Abstract

In this paper, we generalize a sequence of positive linear operators introduced by Ismail and May and we study some of their approximation properties for different classes of continuous functions. First, we estimate the error of approximation in terms of the usual modulus [...] Read more.

In this paper, we generalize a sequence of positive linear operators introduced by Ismail and May and we study some of their approximation properties for different classes of continuous functions. First, we estimate the error of approximation in terms of the usual modulus of continuity and the second-order modulus of Ditzian and Totik. Then, we characterize the bounded functions that can be approximated uniformly by these new operators. In the last section, we obtain the most important results of the paper. We give the complete asymptotic expansion for the operators and we deduce a Voronovskaya-type theorem, results that hold true for smooth functions with exponential growth. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

13 pages, 452 KiB

Open AccessArticle

Approximation by Tensor-Product Kind Bivariate Operator of a New Generalization of Bernstein-Type Rational Functions and Its GBS Operator

by Esma Yıldız Özkan and Gözde Aksoy

Mathematics 2022, 10(9), 1418; https://doi.org/10.3390/math10091418 - 22 Apr 2022

Cited by 2 | Viewed by 1403

Abstract

We introduce a tensor-product kind bivariate operator of a new generalization of Bernstein-type rational functions and its GBS (generalized Boolean sum) operator, and we investigate their approximation properties by obtaining their rates of convergence. Moreover, we present some graphical comparisons visualizing the convergence [...] Read more.

We introduce a tensor-product kind bivariate operator of a new generalization of Bernstein-type rational functions and its GBS (generalized Boolean sum) operator, and we investigate their approximation properties by obtaining their rates of convergence. Moreover, we present some graphical comparisons visualizing the convergence of tensor-product kind bivariate operator and its GBS operator. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

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14 pages, 273 KiB

Open AccessArticle

On the Best Ulam Constant of the Linear Differential Operator with Constant Coefficients

by Alina Ramona Baias and Dorian Popa

Mathematics 2022, 10(9), 1412; https://doi.org/10.3390/math10091412 - 22 Apr 2022

Cited by 2 | Viewed by 1372

Abstract

The linear differential operator with constant coefficients

D (y) = y^{(n)} + a_{1} y^{(n - 1)} + \dots + a_{n} y, y \in C^{n} (R, X)

acting in [...] Read more.

The linear differential operator with constant coefficients

D (y) = y^{(n)} + a_{1} y^{(n - 1)} + \dots + a_{n} y, y \in C^{n} (R, X)

acting in a Banach space X is Ulam stable if and only if its characteristic equation has no roots on the imaginary axis. We prove that if the characteristic equation of D has distinct roots

r_{k}

satisfying

Re r_{k} > 0,

1 \leq k \leq n,

then the best Ulam constant of D is

K_{D} = \frac{1}{| V |} \int_{0}^{\infty} | \sum_{k = 1}^{n} {(- 1)}^{k} V_{k} e^{- r_{k} x} | d x,

where

V = V (r_{1}, r_{2}, \dots, r_{n})

and

V_{k} = V (r_{1}, \dots, r_{k - 1}, r_{k + 1}, \dots, r_{n}),

1 \leq k \leq n,

are Vandermonde determinants. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

10 pages, 269 KiB

Open AccessArticle

Algebraic Systems with Positive Coefficients and Positive Solutions

by Ana Maria Acu, Ioan Raşa and Ancuţa Emilia Şteopoaie

Mathematics 2022, 10(8), 1327; https://doi.org/10.3390/math10081327 - 16 Apr 2022

Cited by 1 | Viewed by 1510

Abstract

The paper is devoted to the existence, uniqueness and nonuniqueness of positive solutions to nonlinear algebraic systems of equations with positive coefficients. Such systems appear in large numbers of applications, such as steady-state equations in continuous and discrete dynamical models, Dirichlet problems, difference [...] Read more.

The paper is devoted to the existence, uniqueness and nonuniqueness of positive solutions to nonlinear algebraic systems of equations with positive coefficients. Such systems appear in large numbers of applications, such as steady-state equations in continuous and discrete dynamical models, Dirichlet problems, difference equations, boundary value problems, periodic solutions and numerical solutions for differential equations. We apply Brouwer’s fixed point theorem, Krasnoselskii’s fixed point theorem and monotone iterative methods in order to extend some known results and to obtain new results. We relax some hypotheses used in the literature concerning the strict monotonicity of the involved functions. We show that, in some cases, the unique positive solution can be obtained by a monotone increasing iterative method or by a monotone decreasing iterative method. As a consequence of one of our results, we recover the existence of a non-negative solution of the Leontief system and describe a monotone iterative method to find it. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

13 pages, 376 KiB

Open AccessArticle

On a New Generalization of Bernstein-Type Rational Functions and Its Approximation

by Esma Yıldız Özkan and Gözde Aksoy

Mathematics 2022, 10(6), 973; https://doi.org/10.3390/math10060973 - 18 Mar 2022

Cited by 3 | Viewed by 1667

Abstract

In this study, we introduce a new generalization of a Bernstein-type rational function possessing better estimates than the classical Bernstein-type rational function. We investigate its error of approximation globally and locally in terms of the first and second modulus of continuity and a [...] Read more.

In this study, we introduce a new generalization of a Bernstein-type rational function possessing better estimates than the classical Bernstein-type rational function. We investigate its error of approximation globally and locally in terms of the first and second modulus of continuity and a class of Lipschitz-type functions. We present graphical comparisons of its approximation with illustrative examples. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

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16 pages, 351 KiB

Open AccessArticle

A Note on New Construction of Meyer-König and Zeller Operators and Its Approximation Properties

by Qing-Bo Cai, Khursheed J. Ansari and Fuat Usta

Mathematics 2021, 9(24), 3275; https://doi.org/10.3390/math9243275 - 16 Dec 2021

Cited by 3 | Viewed by 1897

Abstract

The topic of approximation with positive linear operators in contemporary functional analysis and theory of functions has emerged in the last century. One of these operators is Meyer–König and Zeller operators and in this study a generalization of Meyer–König and Zeller type operators [...] Read more.

The topic of approximation with positive linear operators in contemporary functional analysis and theory of functions has emerged in the last century. One of these operators is Meyer–König and Zeller operators and in this study a generalization of Meyer–König and Zeller type operators based on a function

τ

by using two sequences of functions will be presented. The most significant point is that the newly introduced operator preserves

{1, τ, τ^{2}}

instead of classical Korovkin test functions. Then asymptotic type formula, quantitative results, and local approximation properties of the introduced operators are given. Finally a numerical example performed by MATLAB is given to visualize the provided theoretical results. Full article

(This article belongs to the Special Issue Mathematical Inequalities, Models and Applications)

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