Research

19 pages, 301 KiB

Open AccessArticle

Some New Generalizations of Reverse Hilbert-Type Inequalities via Supermultiplicative Functions

by Haytham M. Rezk, Ahmed I. Saied, Ghada AlNemer and Mohammed Zakarya

Symmetry 2022, 14(10), 2043; https://doi.org/10.3390/sym14102043 - 30 Sep 2022

Viewed by 716

Our work in this paper is based on the reverse Hölder-type dynamic inequalities illustrated by El-Deeb in 2018 and the reverse Hilbert-type dynamic inequalities illustrated by Rezk in 2021 and 2022. With the help of Specht’s ratio, the concept of supermultiplicative functions, chain [...] Read more.

Our work in this paper is based on the reverse Hölder-type dynamic inequalities illustrated by El-Deeb in 2018 and the reverse Hilbert-type dynamic inequalities illustrated by Rezk in 2021 and 2022. With the help of Specht’s ratio, the concept of supermultiplicative functions, chain rule, and Jensen’s inequality on time scales, we can establish some comprehensive and generalize a number of classical reverse Hilbert-type inequalities to a general time scale space. In time scale calculus, results are unified and extended. At the same time, the theory of time scale calculus is applied to unify discrete and continuous analysis and to combine them in one comprehensive form. This hybrid theory is also widely applied on symmetrical properties which play an essential role in determining the correct methods to solve inequalities. As a special case of our results when the supermultiplicative function represents the identity map, we obtain some results that have been recently published. Full article

(This article belongs to the Special Issue Functional Equations and Inequalities 2021)

13 pages, 296 KiB

Open AccessArticle

A New Accelerated Fixed-Point Algorithm for Classification and Convex Minimization Problems in Hilbert Spaces with Directed Graphs

by Kobkoon Janngam and Rattanakorn Wattanataweekul

Symmetry 2022, 14(5), 1059; https://doi.org/10.3390/sym14051059 - 21 May 2022

Cited by 2 | Viewed by 1305

Abstract

A new accelerated algorithm for approximating the common fixed points of a countable family of G-nonexpansive mappings is proposed, and the weak convergence theorem based on our main results is established in the setting of Hilbert spaces with a symmetric directed graph [...] Read more.

A new accelerated algorithm for approximating the common fixed points of a countable family of G-nonexpansive mappings is proposed, and the weak convergence theorem based on our main results is established in the setting of Hilbert spaces with a symmetric directed graph G. As an application, we apply our results for solving classification and convex minimization problems. We also apply our proposed algorithm to estimate the weight connecting the hidden layer and output layer in a regularized extreme learning machine. For numerical experiments, the proposed algorithm gives a higher performance of accuracy of the testing set than that of FISTA-S, FISTA, and nAGA. Full article

(This article belongs to the Special Issue Functional Equations and Inequalities 2021)

18 pages, 317 KiB

Open AccessArticle

Inequalities for Approximation of New Defined Fuzzy Post-Quantum Bernstein Polynomials via Interval-Valued Fuzzy Numbers

by Esma Yıldız Özkan

Symmetry 2022, 14(4), 696; https://doi.org/10.3390/sym14040696 - 28 Mar 2022

Cited by 3 | Viewed by 1163

Abstract

In this study, we introduce new defined fuzzy post-quantum Bernstein polynomials and present examples illustrating their existence. We investigate their approximation properties via interval-valued fuzzy numbers. We obtain a fuzzy Korovkin-type approximation result, and we obtain inequalities estimating the rate of fuzzy convergence [...] Read more.

In this study, we introduce new defined fuzzy post-quantum Bernstein polynomials and present examples illustrating their existence. We investigate their approximation properties via interval-valued fuzzy numbers. We obtain a fuzzy Korovkin-type approximation result, and we obtain inequalities estimating the rate of fuzzy convergence for these polynomials by means of the fuzzy modulus of continuity and Lipschitz-type fuzzy functions. Lastly, we present a Voronovskaja type asymptotic result for fuzzy post-quantum Bernstein polynomials. The methods in this paper are crucial and symmetric in terms of extending the approximation results of these polynomials from the real function space to the fuzzy function space and the applicability to the other operators. Full article

(This article belongs to the Special Issue Functional Equations and Inequalities 2021)

11 pages, 276 KiB

Open AccessArticle

Some Geometric Constants Related to the Midline of Equilateral Triangles in Banach Spaces

by Bingren Chen, Zhijian Yang, Qi Liu and Yongjin Li

Symmetry 2022, 14(2), 348; https://doi.org/10.3390/sym14020348 - 09 Feb 2022

Cited by 1 | Viewed by 931

Abstract

We will introduce some new geometric constants based on the constant

H (X)

proposed by Gao and the constant

A_{2} (X)

proposed by M. Baronti et al. We first provide a study of a new constant [...] Read more.

We will introduce some new geometric constants based on the constant

H (X)

proposed by Gao and the constant

A_{2} (X)

proposed by M. Baronti et al. We first provide a study of a new constant

M_{1} (X)

closely related to the midlines of equilateral triangles, including a discussion of some of its properties and the connections with other parameters of the sphere. Next, we focus on a new constant

M_{2} (X)

and its generalized form

M_{2} (X, p, q)

, along with some of their basic properties. Finally, we concentrate on a new constant

M_{3} (X)

and discuss some of its properties. Full article

(This article belongs to the Special Issue Functional Equations and Inequalities 2021)

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6 pages, 226 KiB

Open AccessArticle

Symmetry of Syzygies of a System of Functional Equations Defining a Ring Homomorphism

by Roman Ger

Symmetry 2021, 13(12), 2343; https://doi.org/10.3390/sym13122343 - 06 Dec 2021

Cited by 1 | Viewed by 1479

Abstract

I deal with an alienation problem for the system of two fundamental Cauchy functional equations with an unknown function f mapping a ring X into an integral domain Y and preserving binary operations of addition and multiplication, respectively. The resulting syzygies obtained by [...] Read more.

I deal with an alienation problem for the system of two fundamental Cauchy functional equations with an unknown function f mapping a ring X into an integral domain Y and preserving binary operations of addition and multiplication, respectively. The resulting syzygies obtained by adding (resp. multiplying) these two equations side by side are discussed. The first of these two syzygies was first examined by Jean Dhombres in 1988 who proved that under some additional conditions concering the domain and range rings it forces f to be a ring homomorphism (alienation phenomenon). The novelty of the present paper is to look for sufficient conditions upon f solving the other syzygy to be alien. Full article

(This article belongs to the Special Issue Functional Equations and Inequalities 2021)

15 pages, 303 KiB

Open AccessArticle

The Injectivity Theorem on a Non-Compact Kähler Manifold

by Jingcao Wu

Symmetry 2021, 13(11), 2222; https://doi.org/10.3390/sym13112222 - 20 Nov 2021

Viewed by 1141

Abstract

In this paper, we establish an injectivity theorem on a weakly pseudoconvex Kähler manifold X with negative sectional curvature. For this purpose, we develop the harmonic theory in this circumstance. The negative sectional curvature condition is usually satisfied by the manifolds with hyperbolicity, [...] Read more.

In this paper, we establish an injectivity theorem on a weakly pseudoconvex Kähler manifold X with negative sectional curvature. For this purpose, we develop the harmonic theory in this circumstance. The negative sectional curvature condition is usually satisfied by the manifolds with hyperbolicity, such as symmetric spaces, bounded symmetric domains in

C^{n}

, hyperconvex bounded domains, and so on. Full article

(This article belongs to the Special Issue Functional Equations and Inequalities 2021)

16 pages, 310 KiB

Open AccessEditor’s ChoiceArticle

On the Fekete–Szegö Problem for Meromorphic Functions Associated with p,q-Wright Type Hypergeometric Function

by Adriana Cătaş

Symmetry 2021, 13(11), 2143; https://doi.org/10.3390/sym13112143 - 10 Nov 2021

Cited by 15 | Viewed by 1770

Abstract

Making use of a post-quantum derivative operator, we define two classes of meromorphic analytic functions. For the considered family of functions, we aim to investigate the sharp bounds’ values in the case of the Fekete–Szegö problem. The study of the well-known Fekete–Szegö functional [...] Read more.

Making use of a post-quantum derivative operator, we define two classes of meromorphic analytic functions. For the considered family of functions, we aim to investigate the sharp bounds’ values in the case of the Fekete–Szegö problem. The study of the well-known Fekete–Szegö functional in the post-quantum calculus case for meromorphic functions provides new outcomes for research in the field. With the extended

p, q

-operator, we establish certain inequalities’ relations concerning meromorphic functions. In the final part of the paper, a new

p, q

-analogue of the q-Wright type hypergeometric function is introduced. This function generalizes the classical and symmetrical Gauss hypergeometric function. All the obtained results are sharp. Full article

(This article belongs to the Special Issue Functional Equations and Inequalities 2021)

10 pages, 274 KiB

Open AccessArticle

Geometric Constants in Banach Spaces Related to the Inscribed Quadrilateral of Unit Balls

by Asif Ahmad, Qi Liu and Yongjin Li

Symmetry 2021, 13(7), 1294; https://doi.org/10.3390/sym13071294 - 19 Jul 2021

Cited by 2 | Viewed by 1802

Abstract

We introduce a new geometric constant

J i n (X)

based on a generalization of the parallelogram law, which is symmetric and related to the length of the inscribed quadrilateral side of the unit ball. We first investigate some basic properties [...] Read more.

We introduce a new geometric constant

J i n (X)

based on a generalization of the parallelogram law, which is symmetric and related to the length of the inscribed quadrilateral side of the unit ball. We first investigate some basic properties of this new coefficient. Next, it is shown that, for a Banach space,

J i n (X)

becomes 16 if and only if the norm is induced by an inner product. Moreover, its properties and some relations between other well-known geometric constants are studied. Finally, a sufficient condition which implies normal structure is presented. Full article

(This article belongs to the Special Issue Functional Equations and Inequalities 2021)

13 pages, 311 KiB

Open AccessArticle

Some Properties Concerning the J_L(X) and Y_J(X) Which Related to Some Special Inscribed Triangles of Unit Ball

by Asif Ahmad, Yuankang Fu and Yongjin Li

Symmetry 2021, 13(7), 1285; https://doi.org/10.3390/sym13071285 - 16 Jul 2021

Cited by 2 | Viewed by 1318

Abstract

In this paper, we will make some further discussions on the

J_{L} (X)

and

Y_{J} (X)

which are symmetric and related to the side lengths of some special inscribed triangles of the unit ball, and also introduce [...] Read more.

In this paper, we will make some further discussions on the

J_{L} (X)

and

Y_{J} (X)

which are symmetric and related to the side lengths of some special inscribed triangles of the unit ball, and also introduce two new geometric constants

L_{1} (X, ▵)

,

L_{2} (X, ▵)

which related to the perimeters of some special inscribed triangles of the unit ball. Firstly, we discuss the relations among

J_{L} (X)

,

Y_{J} (X)

and some geometric properties of Banach spaces, including uniformly non-square and uniformly convex. It is worth noting that we point out that uniform non-square spaces can be characterized by the side lengths of some special inscribed triangles of unit ball. Secondly, we establish some inequalities for

J_{L} (X)

,

Y_{J} (X)

and some significant geometric constants, including the James constant

J (X)

and the von Neumann-Jordan constant

C_{NJ} (X)

. Finally, we introduce the two new geometric constants

L_{1} (X, ▵)

,

L_{2} (X, ▵)

, and calculate the bounds of

L_{1} (X, ▵)

and

L_{2} (X, ▵)

as well as the values of

L_{1} (X, ▵)

and

L_{2} (X, ▵)

for two Banach spaces. Full article

(This article belongs to the Special Issue Functional Equations and Inequalities 2021)

19 pages, 335 KiB

Open AccessArticle

Hermite–Hadamard Inclusions for Co-Ordinated Interval-Valued Functions via Post-Quantum Calculus

by Jessada Tariboon, Muhammad Aamir Ali, Hüseyin Budak and Sotiris K. Ntouyas

Symmetry 2021, 13(7), 1216; https://doi.org/10.3390/sym13071216 - 07 Jul 2021

Cited by 7 | Viewed by 1289

Abstract

In this paper, the notions of post-quantum integrals for two-variable interval-valued functions are presented. The newly described integrals are then used to prove some new Hermite–Hadamard inclusions for co-ordinated convex interval-valued functions. Many of the findings in this paper are important extensions of [...] Read more.

In this paper, the notions of post-quantum integrals for two-variable interval-valued functions are presented. The newly described integrals are then used to prove some new Hermite–Hadamard inclusions for co-ordinated convex interval-valued functions. Many of the findings in this paper are important extensions of previous findings in the literature. Finally, we present a few examples of our new findings. Analytic inequalities of this nature and especially the techniques involved have applications in various areas in which symmetry plays a prominent role. Full article

(This article belongs to the Special Issue Functional Equations and Inequalities 2021)

11 pages, 292 KiB

Open AccessArticle

Continuous Wavelet Transform of Schwartz Distributions in

D_{L^{2}}^{'} (R^{n})

, n ≥ 1

by Jagdish N. Pandey

Symmetry 2021, 13(7), 1106; https://doi.org/10.3390/sym13071106 - 22 Jun 2021

Cited by 2 | Viewed by 1190

Abstract

We define a testing function space

D_{L^{2}} (R^{n})

consisting of a class of

C^{\infty}

functions defined on

R^{n}

,

n \geq 1

whose every derivtive is

L^{2} (R^{n})

integrable and equip it [...] Read more.

We define a testing function space

D_{L^{2}} (R^{n})

consisting of a class of

C^{\infty}

functions defined on

R^{n}

,

n \geq 1

whose every derivtive is

L^{2} (R^{n})

integrable and equip it with a topology generated by a separating collection of seminorms

{γ_{k}}_{| k | = 0}^{\infty}

on

D_{L^{2}} (R^{n})

, where

| k | = 0, 1, 2, \dots

and

γ_{k} (ϕ) = {∥ ϕ^{(k)} ∥}_{2}, ϕ \in D_{L^{2}} (R^{n}) .

We then extend the continuous wavelet transform to distributions in

D_{L^{2}}^{'} (R^{n})

,

n \geq 1

and derive the corresponding wavelet inversion formula interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is defined by an element

ψ (x)

of

D_{L^{2}} (R^{n}) \cap D_{L^{1}} (R^{n})

,

n \geq 1

which, when integrated along each of the real axes

X_{1}, X_{2}, \dots X_{n}

vanishes, but none of its moments

\int_{R^{n}} x^{m} ψ (x) d x

is zero; here

x^{m} = x_{1}^{m_{1}} x_{2}^{m_{2}} \dots x_{n}^{m_{n}}

,

d x = d x_{1} d x_{2} \dots d x_{n}

and

m = (m_{1}, m_{2}, \dots m_{n})

and each of

m_{1}, m_{2}, \dots m_{n}

is

\geq 1

. The set of such wavelets will be denoted by

D_{M} (R^{n})

. Full article

(This article belongs to the Special Issue Functional Equations and Inequalities 2021)

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