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A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 May 2021) | Viewed by 33165

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Dr. Daciana Alina Alb Lupas

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Department of Mathematics and Computer Science, Faculty of Informatics and Sciences, University of Oradea, Universitatii Street, 410087 Oradea, Romania
Interests: topological algebra; geometric function theory; inequalities
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Special Issue Information

Dear Colleagues,

The field of functional equations is an ever-growing branch of mathematics with far-reaching applications; it is increasingly used to investigate problems in mathematical analysis, combinatorics, biology, information theory, statistics, physics, the behavioral sciences, and engineering.

Inequalities play a significant role in all fields of mathematics and present a very active and attractive field of research.

This Special Issue promotes an exchange of ideas between eminent mathematicians from many parts of the world, dedicated to the Functional Equations and Analytic Inequalities. It is intended to boost cooperation among mathematicians working on a broad variety of pure and applied mathematical areas.

This volume of ideas and mathematical methods will include a wide area of applications in which the equations, inequalities, and computational techniques relevant to their solutions play an important role. These ideas and methods have a significant effect on everyday life, as new tools have been developed and achieved revolutionary research results, bringing scientists even closer to exact sciences, encouraging the emergence of new approaches, techniques, and perspectives in functional equations, analytical inequalities, etc. Please note that all submitted papers should be within the scope of the journal.

Prof. Alina Alb Lupas
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

Functional equations in several variables
difference equations
stability problems
set-valued functional equations
iterative functional equations and iteration theory
chaos
dynamical systems
iterative roots
iteration groups and semigroups
functional inequalities
convexity
inclusions for multivalued functions
differential and difference inequalities
means
applications in ODEs
PDEs
functional analysis
operator theory
approximation theory
number theory
actuarial mathematics
social sciences and others

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Published Papers (15 papers)

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Research

13 pages, 284 KiB

Open AccessArticle

Some Results of New Subclasses for Bi-Univalent Functions Using Quasi-Subordination

by Waggas Galib Atshan, Ibtihal Abdul Ridha Rahman and Alina Alb Lupaş

Symmetry 2021, 13(9), 1653; https://doi.org/10.3390/sym13091653 - 8 Sep 2021

Cited by 19 | Viewed by 1969

Abstract

In this paper, we introduce new subclasses

R_{Σ, b, c}^{μ, α} (λ, δ, τ, Φ)

and

K_{Σ, b, c}^{μ, α} (λ, δ, η, Φ)

of bi-univalent functions [...] Read more.

In this paper, we introduce new subclasses

R_{Σ, b, c}^{μ, α} (λ, δ, τ, Φ)

and

K_{Σ, b, c}^{μ, α} (λ, δ, η, Φ)

of bi-univalent functions in the open unit disk U by using quasi-subordination conditions and determine estimates of the coefficients

|a_{2}|

and

|a_{3}|

for functions of these subclasses. We discuss the improved results for the associated classes involving many of the new and well-known consequences. We notice that there is symmetry in the results obtained for the new subclasses

R_{Σ, b, c}^{μ, α} (λ, δ, τ, Φ)

and

K_{Σ, b, c}^{μ, α} (λ, δ, η, Φ)

, as there is a symmetry for the estimations of the coefficients

a_{2}

and

a_{3}

for all the subclasses defind in our this paper. Full article

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

14 pages, 329 KiB

Open AccessArticle

Applications of the q-Srivastava-Attiya Operator Involving a Certain Family of Bi-Univalent Functions Associated with the Horadam Polynomials

by Hari Mohan Srivastava, Abbas Kareem Wanas and Rekha Srivastava

Symmetry 2021, 13(7), 1230; https://doi.org/10.3390/sym13071230 - 8 Jul 2021

Cited by 53 | Viewed by 3325

Abstract

In this article, by making use of the q-Srivastava-Attiya operator, we introduce and investigate a new family

{SW}_{Σ} (δ, γ, λ, s, t, q, r)

of normalized holomorphic and bi-univalent functions in the [...] Read more.

In this article, by making use of the q-Srivastava-Attiya operator, we introduce and investigate a new family

{SW}_{Σ} (δ, γ, λ, s, t, q, r)

of normalized holomorphic and bi-univalent functions in the open unit disk

U

, which are associated with the Bazilevič functions and the

λ

-pseudo-starlike functions as well as the Horadam polynomials. We estimate the second and the third coefficients in the Taylor-Maclaurin expansions of functions belonging to the holomorphic and bi-univalent function class, which we introduce here. Furthermore, we establish the Fekete-Szegö inequality for functions in the family

{SW}_{Σ} (δ, γ, λ, s, t, q, r)

. Relevant connections of some of the special cases of the main results with those in several earlier works are also pointed out. Our usage here of the basic or quantum (or q-) extension of the familiar Hurwitz-Lerch zeta function

Φ (z, s, a)

is justified by the fact that several members of this family of zeta functions possess properties with local or non-local symmetries. Our study of the applications of such quantum (or q-) extensions in this paper is also motivated by the symmetric nature of quantum calculus itself. Full article

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

15 pages, 297 KiB

Open AccessArticle

Some Bounds for the Complex Čebyšev Functional of Functions of Bounded Variation

by Silvestru Sever Dragomir

Symmetry 2021, 13(6), 990; https://doi.org/10.3390/sym13060990 - 2 Jun 2021

Viewed by 1624

Abstract

In this paper, we provide several bounds for the modulus of the complex Čebyšev functional. Applications to the trapezoid and mid-point inequalities, that are symmetric inequalities, are also provided. Full article

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

14 pages, 339 KiB

Open AccessArticle

On Markov Moment Problem and Related Results

by Octav Olteanu

Symmetry 2021, 13(6), 986; https://doi.org/10.3390/sym13060986 - 1 Jun 2021

Cited by 12 | Viewed by 2326

Abstract

We prove new results and complete our recently published theorems on the vector-valued Markov moment problem, by means of polynomial approximation on unbounded subsets, also applying an extension of the positive linear operators’ result. The domain is the Banach lattice of continuous real-valued [...] Read more.

L_{ν}^{1}

space, where

ν

is a positive moment determinate measure on a closed unbounded set. The existence and uniqueness of the operator solution are proved. Our solutions satisfy the interpolation moment conditions and are between two given linear operators on the positive cone of the domain space. The norm controlling of the solution is emphasized. The most part of the results are stated and proved in terms of quadratic forms. This type of result represents the first aim of the paper. Secondly, we construct a polynomial solution for a truncated multidimensional moment problem. Full article

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

11 pages, 260 KiB

Open AccessArticle

New Geometric Constants in Banach Spaces Related to the Inscribed Equilateral Triangles of Unit Balls

by Yuankang Fu, Qi Liu and Yongjin Li

Symmetry 2021, 13(6), 951; https://doi.org/10.3390/sym13060951 - 27 May 2021

Cited by 3 | Viewed by 2162

Abstract

Geometric constant is one of the important tools to study geometric properties of Banach spaces. In this paper, we will introduce two new geometric constants

J_{L} (X)

and

Y_{J} (X)

in Banach spaces, which are symmetric and [...] Read more.

Geometric constant is one of the important tools to study geometric properties of Banach spaces. In this paper, we will introduce two new geometric constants

J_{L} (X)

and

Y_{J} (X)

in Banach spaces, which are symmetric and related to the side lengths of inscribed equilateral triangles of unit balls. The upper and lower bounds of

J_{L} (X)

and

Y_{J} (X)

as well as the values of

J_{L} (X)

and

Y_{J} (X)

for Hilbert spaces and some common Banach spaces will be calculated. In addition, some inequalities for

J_{L} (X)

Y_{J} (X)

and some significant geometric constants will be presented. Furthermore, the sufficient conditions for uniformly non-square and normal structure, and the necessary conditions for uniformly non-square and uniformly convex will be established. Full article

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

15 pages, 272 KiB

Open AccessArticle

Asymptotic Properties of Solutions to Discrete Volterra Monotone Type Equations

by Janusz Migda, Małgorzata Migda and Ewa Schmeidel

Symmetry 2021, 13(6), 918; https://doi.org/10.3390/sym13060918 - 21 May 2021

Cited by 2 | Viewed by 1621

Abstract

We investigate the higher order nonlinear discrete Volterra equations. We study solutions with prescribed asymptotic behavior. For example, we establish sufficient conditions for the existence of asymptotically polynomial, asymptotically periodic or asymptotically symmetric solutions. On the other hand, we are dealing with the problem of approximation of solutions. Among others, we present conditions under which any bounded solution is asymptotically periodic. Using our techniques, based on the iterated remainder operator, we can control the degree of approximation. In this paper we choose a positive non-increasing sequence u and use

o (u_{n})

as a measure of approximation. Full article

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

10 pages, 238 KiB

Open AccessArticle

Univalence Conditions for Gaussian Hypergeometric Function Involving Differential Inequalities

by Georgia Irina Oros

Symmetry 2021, 13(5), 904; https://doi.org/10.3390/sym13050904 - 19 May 2021

Cited by 7 | Viewed by 1683

Abstract

In their paper published in 1990, Miller and Mocanu have investigated the special function Gaussian hypergeometric function in view of its relation to the theory of analytic functions, stating conditions for this function to be univalent using [...] Read more.

a, b, c \in ℝ, c \neq 0, - 1, - 2, \dots

. The study done in this paper extends the results on the univalence of the considered function taking

a, b, c \in ℂ,

with

c \neq 0, - 1, - 2, \dots

two criteria being stated in the corollaries of the proved theorems. An interpretation of the univalence results from the sets inclusion view is also given, underlining the geometrical properties of the outcomes. Examples showing how the univalence results can be applied are also included. Full article

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

16 pages, 283 KiB

Open AccessArticle

Boundary Value Problems of Hadamard Fractional Differential Equations of Variable Order

by Snezhana Hristova, Amar Benkerrouche, Mohammed Said Souid and Ali Hakem

Symmetry 2021, 13(5), 896; https://doi.org/10.3390/sym13050896 - 18 May 2021

Cited by 24 | Viewed by 2452

Abstract

A boundary value problem for Hadamard fractional differential equations of variable order is studied. Note the symmetry of a transformation of a system of differential equations is connected with the locally solvability which is the same as the existence of solutions. It leads to the necessity of obtaining existence criteria for a boundary value problem for Hadamard fractional differential equations of variable order. Also, the stability in the sense of Ulam–Hyers–Rassias is investigated. The results are obtained based on the Kuratowski measure of noncompactness. An example illustrates the validity of the observed results. Full article

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

13 pages, 285 KiB

Open AccessArticle

Invariant Solutions of Black–Scholes Equation with Ornstein–Uhlenbeck Process

by Maba Boniface Matadi and Phumlani Lawrence Zondi

Symmetry 2021, 13(5), 847; https://doi.org/10.3390/sym13050847 - 11 May 2021

Cited by 2 | Viewed by 1752

Abstract

This paper analyses the model of Black–Scholes option pricing from the point of view of the group theoretic approach. The study identified new independent variables that lead to the transformation of the Black–Scholes equation. Furthermore, corresponding determining equations were constructed and new symmetries were found. As a result, the findings of the study demonstrate of the integrability of the model to present an invariant solution for the Ornstein–Uhlenbeck stochastic process. Full article

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

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21 pages, 379 KiB

Open AccessArticle

New Hermite–Hadamard Inequalities in Fuzzy-Interval Fractional Calculus and Related Inequalities

by Muhammad Bilal Khan, Pshtiwan Othman Mohammed, Muhammad Aslam Noor and Y. S. Hamed

Symmetry 2021, 13(4), 673; https://doi.org/10.3390/sym13040673 - 13 Apr 2021

Cited by 54 | Viewed by 2520

Abstract

It is a familiar fact that inequalities have become a very popular method using fractional integrals, and that this method has been the driving force behind many studies in recent years. Many forms of inequality have been studied, resulting in the introduction of new trend in inequality theory. The aim of this paper is to use a fuzzy order relation to introduce various types of inequalities. On the fuzzy interval space, this fuzzy order relation is defined level by level. With the help of this relation, firstly, we derive some discrete Jensen and Schur inequalities for convex fuzzy interval-valued functions (convex fuzzy-IVF), and then, we present Hermite–Hadamard inequalities (

H H

-inequalities) for convex fuzzy-IVF via fuzzy interval Riemann–Liouville fractional integrals. These outcomes are a generalization of a number of previously known results, and many new outcomes can be deduced as a result of appropriate parameter “

γ

” and real valued function “

Ω

” selections. We hope that our fuzzy order relations results can be used to evaluate a number of mathematical problems related to real-world applications. Full article

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

18 pages, 307 KiB

Open AccessArticle

Hilfer-Polya, ψ-Hilfer Ostrowski and ψ-Hilfer-Hilbert-Pachpatte Fractional Inequalities

by George A. Anastassiou

Symmetry 2021, 13(3), 463; https://doi.org/10.3390/sym13030463 - 12 Mar 2021

Cited by 2 | Viewed by 1379

Abstract

Here we present Hilfer-Polya,

ψ

-Hilfer Ostrowski and

ψ

-Hilfer-Hilbert-Pachpatte types fractional inequalities. They are univariate inequalities involving left and right Hilfer and

ψ

-Hilfer fractional derivatives. All estimates are with respect to norms

{∥\cdot∥}_{p}

1 \leq p \leq \infty

. [...] Read more.

Here we present Hilfer-Polya,

ψ

-Hilfer Ostrowski and

ψ

-Hilfer-Hilbert-Pachpatte types fractional inequalities. They are univariate inequalities involving left and right Hilfer and

ψ

-Hilfer fractional derivatives. All estimates are with respect to norms

{∥\cdot∥}_{p}

1 \leq p \leq \infty

. At the end we provide applications. Full article

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

16 pages, 276 KiB

Open AccessArticle

A Hilbert-Type Integral Inequality in the Whole Plane Related to the Arc Tangent Function

by Michael Th. Rassias, Bicheng Yang and Andrei Raigorodskii

Symmetry 2021, 13(2), 351; https://doi.org/10.3390/sym13020351 - 21 Feb 2021

Cited by 3 | Viewed by 1981

Abstract

In this work we establish a few equivalent statements of a Hilbert-type integral inequality in the whole plane related to the kernel of the arc tangent function. We prove that the constant factor, which is associated with the cosine function, is optimal. Some [...] Read more.

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

14 pages, 273 KiB

Open AccessArticle

About the Cauchy–Bunyakovsky–Schwarz Inequality for Hilbert Space Operators

by Nicuşor Minculete

Symmetry 2021, 13(2), 305; https://doi.org/10.3390/sym13020305 - 11 Feb 2021

Cited by 4 | Viewed by 2170

Abstract

The symmetric shape of some inequalities between two sequences of real numbers generates inequalities of the same shape in operator theory. In this paper, we study a new refinement of the Cauchy–Bunyakovsky–Schwarz inequality for Euclidean spaces and several inequalities for two bounded linear operators on a Hilbert space, where we mention Bohr’s inequality and Bergström’s inequality for operators. We present an inequality of the Cauchy–Bunyakovsky–Schwarz type for bounded linear operators, by the technique of the monotony of a sequence. We also prove a refinement of the Aczél inequality for bounded linear operators on a Hilbert space. Finally, we present several applications of some identities for Hermitian operators. Full article

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

11 pages, 268 KiB

Open AccessArticle

Applications of Inequalities in the Complex Plane Associated with Confluent Hypergeometric Function

by Georgia Irina Oros

Symmetry 2021, 13(2), 259; https://doi.org/10.3390/sym13020259 - 4 Feb 2021

Cited by 8 | Viewed by 1714

Abstract

The idea of inequality has been extended from the real plane to the complex plane through the notion of subordination introduced by Professors Miller and Mocanu in two papers published in 1978 and 1981. With this notion came a whole new theory called the theory of differential subordination or admissible functions theory. Later, in 2003, a particular form of inequality in the complex plane was also defined by them as dual notion for subordination, the notion of differential superordination and with it, the theory of differential superordination appeared. In this paper, the theory of differential superordination is applied to confluent hypergeometric function. Hypergeometric functions are intensely studied nowadays, the interest on the applications of those functions in complex analysis being renewed by their use in the proof of Bieberbach’s conjecture given by de Branges in 1985. Using the theory of differential superodination, best subordinants of certain differential superordinations involving confluent (Kummer) hypergeometric function are stated in the theorems and relation with previously obtained results are highlighted in corollaries using particular functions and in a sandwich-type theorem. An example is also enclosed in order to show how the theoretical findings can be applied. Full article

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

15 pages, 336 KiB

Open AccessArticle

A Modified PRP-CG Type Derivative-Free Algorithm with Optimal Choices for Solving Large-Scale Nonlinear Symmetric Equations

by Jamilu Sabi’u, Kanikar Muangchoo, Abdullah Shah, Auwal Bala Abubakar and Lateef Olakunle Jolaoso

Symmetry 2021, 13(2), 234; https://doi.org/10.3390/sym13020234 - 30 Jan 2021

Cited by 7 | Viewed by 2235

Abstract

Inspired by the large number of applications for symmetric nonlinear equations, this article will suggest two optimal choices for the modified Polak–Ribiére–Polyak (PRP) conjugate gradient (CG) method by minimizing the measure function of the search direction matrix and combining the proposed direction with the default Newton direction. In addition, the corresponding PRP parameters are incorporated with the Li and Fukushima approximate gradient to propose two robust CG-type algorithms for finding solutions for large-scale systems of symmetric nonlinear equations. We have also demonstrated the global convergence of the suggested algorithms using some classical assumptions. Finally, we demonstrated the numerical advantages of the proposed algorithms compared to some of the existing methods for nonlinear symmetric equations. Full article

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

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