Editorial

5 pages, 203 KiB

Open AccessEditorial

Symmetry in Functional Equations and Analytic Inequalities II

by Alina Alb Lupas

Symmetry 2022, 14(2), 268; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14020268 - 29 Jan 2022

Cited by 1 | Viewed by 1495

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 [...] Full article

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

Research

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11 pages, 278 KiB

Open AccessArticle

Application of Lie Symmetry to a Mathematical Model that Describes a Cancer Sub-Network

by Maba Boniface Matadi

Symmetry 2022, 14(2), 400; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14020400 - 17 Feb 2022

Cited by 3 | Viewed by 1300

Abstract

In this paper, a mathematical model of a cancer sub-network is analysed from the view point of Lie symmetry methods. This model discusses a human cancer cell which is developed due to the dysfunction of some genes at the R-checkpoint during the [...] Read more.

In this paper, a mathematical model of a cancer sub-network is analysed from the view point of Lie symmetry methods. This model discusses a human cancer cell which is developed due to the dysfunction of some genes at the R-checkpoint during the cell cycle. The primary purpose of this paper is to apply the techniques of Lie symmetry to the model and present some approximated solutions for the three-dimensional system of first-order ordinary differential equations describing a cancer sub-network. The result shows that the phosphatase gene (Cdc25A) regulates the cyclin-dependent kinases inhibitor (

P 27^{K i p 1}

). Furthermore, this research discovered that the activity that reverses the inhibitory effects on cell cycle progression at the R-checkpoint initiates a pathway. Full article

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

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8 pages, 775 KiB

Open AccessArticle

Exploiting the Pascal Distribution Series and Gegenbauer Polynomials to Construct and Study a New Subclass of Analytic Bi-Univalent Functions

by Ala Amourah, Basem Aref Frasin, Morad Ahmad and Feras Yousef

Symmetry 2022, 14(1), 147; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14010147 - 13 Jan 2022

Cited by 28 | Viewed by 1706

Abstract

In the present analysis, we aim to construct a new subclass of analytic bi-univalent functions defined on symmetric domain by means of the Pascal distribution series and Gegenbauer polynomials. Thereafter, we provide estimates of Taylor–Maclaurin coefficients

|a_{2}|

and

|a_{3}|

for functions [...] Read more.

In the present analysis, we aim to construct a new subclass of analytic bi-univalent functions defined on symmetric domain by means of the Pascal distribution series and Gegenbauer polynomials. Thereafter, we provide estimates of Taylor–Maclaurin coefficients

|a_{2}|

and

|a_{3}|

for functions in the aforementioned class, and next, we solve the Fekete–Szegö functional problem. Moreover, some interesting findings for new subclasses of analytic bi-univalent functions will emerge by reducing the parameters in our main results. Full article

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

9 pages, 276 KiB

Open AccessArticle

Inscribed Triangles in the Unit Sphere and a New Class of Geometric Constants

by Bingren Chen, Qi Liu and Yongjin Li

Symmetry 2022, 14(1), 72; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14010072 - 04 Jan 2022

Cited by 2 | Viewed by 1179

Abstract

In this paper, we firstly investigate the constant

H (X)

proposed by Gao further by discussing several properties of it that have not yet been discovered. Secondly, we focus on a new constant

G_{L} (X)

closely related to [...] Read more.

In this paper, we firstly investigate the constant

H (X)

proposed by Gao further by discussing several properties of it that have not yet been discovered. Secondly, we focus on a new constant

G_{L} (X)

closely related to

H (X)

, along with a variety of geometric properties. In addition, we show several relations among it and the several basic geometric constants via a few inequalities. Finally, we manage to characterize the geometric properties of its generalized forms

G_{L} (X, p)

and

C_{L} (X)

explicitly. Full article

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

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12 pages, 274 KiB

Open AccessEditor’s ChoiceArticle

Best Dominants and Subordinants for Certain Sandwich-Type Theorems

by Adriana Cătaş, Emilia Borşa and Loredana Iambor

Symmetry 2022, 14(1), 62; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14010062 - 03 Jan 2022

Cited by 3 | Viewed by 1249

Abstract

In this paper, we aim to present a survey on subordination and superordination theorems related to the class of analytic functions defined in a symmetric domain, which is the open unit disc. The results were deduced by making use of a new differential [...] Read more.

In this paper, we aim to present a survey on subordination and superordination theorems related to the class of analytic functions defined in a symmetric domain, which is the open unit disc. The results were deduced by making use of a new differential operator. We present two properties of this operator from which we constructed the final results. Moreover, based on the obtained outcomes, we give two sandwich-type theorems. Some interesting further consequences are also taken into consideration. Full article

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

11 pages, 286 KiB

Open AccessArticle

A Generalized Class of Functions Defined by the q-Difference Operator

by Loriana Andrei and Vasile-Aurel Caus

Symmetry 2021, 13(12), 2361; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122361 - 08 Dec 2021

Cited by 4 | Viewed by 1454

Abstract

The goal of the present investigation is to introduce a new class of analytic functions (

K_{t, q}

), defined in the open unit disk, by means of the q-difference operator, which may have symmetric or assymetric properties, and to [...] Read more.

The goal of the present investigation is to introduce a new class of analytic functions (

K_{t, q}

), defined in the open unit disk, by means of the q-difference operator, which may have symmetric or assymetric properties, and to establish the relationship between the new defined class and appropriate subordination. We derived relationships of this class and obtained sufficient conditions for an analytic function to be

K_{t, q}

. Finally, in the concluding section, we have taken the decision to restate the clearly-proved fact that any attempt to create the rather simple

(p, q)

-variations of the results, which we have provided in this paper, will be a rather inconsequential and trivial work, simply because the added parameter p is obviously redundant. Full article

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

19 pages, 380 KiB

Open AccessArticle

Integral Inequalities for Generalized Harmonically Convex Functions in Fuzzy-Interval-Valued Settings

by Muhammad Bilal Khan, Pshtiwan Othman Mohammed, José António Tenreiro Machado and Juan L. G. Guirao

Symmetry 2021, 13(12), 2352; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122352 - 07 Dec 2021

Cited by 11 | Viewed by 1745

Abstract

It is a well-known fact that convex and non-convex fuzzy mappings play a critical role in the study of fuzzy optimization. Due to the behavior of its definition, the idea of convexity also plays a significant role in the subject of inequalities. The [...] Read more.

It is a well-known fact that convex and non-convex fuzzy mappings play a critical role in the study of fuzzy optimization. Due to the behavior of its definition, the idea of convexity also plays a significant role in the subject of inequalities. The concepts of convexity and symmetry have a tight connection. We may use whatever we learn from both the concepts, owing to the significant correlation that has developed between both in recent years. In this paper, we introduce a new class of harmonically convex fuzzy-interval-valued functions which is known as harmonically h-convex fuzzy-interval-valued functions (abbreviated as harmonically h-convex F-I-V-Fs) by means of fuzzy order relation. This fuzzy order relation is defined level-wise through Kulisch–Miranker order relation defined on interval space. Some properties of this class are investigated. BY using fuzzy order relation and h-convex F-I-V-Fs, Hermite–Hadamard type inequalities for harmonically are developed via fuzzy Riemann integral. We have also obtained some new inequalities for the product of harmonically h-convex F-I-V-Fs. Moreover, we establish Hermite–Hadamard–Fej’er inequality for harmonically h-convex F-I-V-Fs via fuzzy Riemann integral. These outcomes are a generalization of a number of previously known results, as well as many new outcomes can be deduced as a result of appropriate parameter

“ θ ”

and real valued function

“ \nabla ”

selections. For the validation of the main results, we have added some nontrivial examples. We hope that the concepts and techniques of this study may open new directions for research. Full article

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

12 pages, 287 KiB

Open AccessArticle

Approximations of Symmetric Functions on Banach Spaces with Symmetric Bases

by Mariia Martsinkiv and Andriy Zagorodnyuk

Symmetry 2021, 13(12), 2318; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122318 - 04 Dec 2021

Cited by 5 | Viewed by 1281

Abstract

This paper is devoted to studying approximations of symmetric continuous functions by symmetric analytic functions on a Banach space X with a symmetric basis. We obtain some positive results for the case when X admits a separating polynomial using a symmetrization operator. However, [...] Read more.

This paper is devoted to studying approximations of symmetric continuous functions by symmetric analytic functions on a Banach space X with a symmetric basis. We obtain some positive results for the case when X admits a separating polynomial using a symmetrization operator. However, even in this case, there is a counter-example because the symmetrization operator is well defined only on a narrow, proper subspace of the space of analytic functions on

X

. For

X = c_{0}

, we introduce

ε

-slice G-analytic functions that have a behavior similar to G-analytic functions at points

x \in c_{0}

such that all coordinates of x are greater than

ε

, and we prove a theorem on approximations of uniformly continuous functions on

c_{0}

by

ε

-slice G-analytic functions. Full article

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

14 pages, 281 KiB

Open AccessArticle

Starlikeness of New General Differential Operators Associated with q-Bessel Functions

by Loriana Andrei and Vasile-Aurel Caus

Symmetry 2021, 13(12), 2310; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13122310 - 03 Dec 2021

Cited by 4 | Viewed by 903

Abstract

Owning to the importance and great interest of differential operators, two generalized differential operators, which may be symmetric or assymetric, are newly introduced in the present paper. Motivated by the familiar Jackson’s second and third Bessel functions, we derive necessary and sufficient conditions [...] Read more.

Owning to the importance and great interest of differential operators, two generalized differential operators, which may be symmetric or assymetric, are newly introduced in the present paper. Motivated by the familiar Jackson’s second and third Bessel functions, we derive necessary and sufficient conditions for which the new generalized operators belong to the class of q-starlike functions of order alpha. Several corollaries and consequences of the main results are also pointed out. Full article

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

19 pages, 326 KiB

Open AccessArticle

Some Hermite–Hadamard-Type Fractional Integral Inequalities Involving Twice-Differentiable Mappings

by Soubhagya Kumar Sahoo, Muhammad Tariq, Hijaz Ahmad, Ayman A. Aly, Bassem F. Felemban and Phatiphat Thounthong

Symmetry 2021, 13(11), 2209; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13112209 - 19 Nov 2021

Cited by 9 | Viewed by 1365

Abstract

The theory of fractional analysis has been a focal point of fascination for scientists in mathematical science, given its essential definitions, properties, and applications in handling real-life problems. In the last few decades, many mathematicians have shown their considerable interest in the theory [...] Read more.

The theory of fractional analysis has been a focal point of fascination for scientists in mathematical science, given its essential definitions, properties, and applications in handling real-life problems. In the last few decades, many mathematicians have shown their considerable interest in the theory of fractional calculus and convexity due to their wide range of applications in almost all branches of applied sciences, especially in numerical analysis, physics, and engineering. The objective of this article is to establish Hermite-Hadamard type integral inequalities by employing the k-Riemann-Liouville fractional operator and its refinements, whose absolute values are twice-differentiable h-convex functions. Moreover, we also present some special cases of our presented results for different types of convexities. Moreover, we also study how q-digamma functions can be applied to address the newly investigated results. Mathematical integral inequalities of this class and the arrangements associated have applications in diverse domains in which symmetry presents a salient role. Full article

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

17 pages, 370 KiB

Open AccessArticle

Initial Coefficient Estimates and Fekete–Szegö Inequalities for New Families of Bi-Univalent Functions Governed by (p − q)-Wanas Operator

by Abbas Kareem Wanas and Luminiţa-Ioana Cotîrlǎ

Symmetry 2021, 13(11), 2118; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13112118 - 08 Nov 2021

Cited by 24 | Viewed by 1800

Abstract

The motivation of the present article is to define the

(p - q)

-Wanas operator in geometric function theory by the symmetric nature of quantum calculus. We also initiate and explore certain new families of holormorphic and bi-univalent functions [...] Read more.

The motivation of the present article is to define the

(p - q)

-Wanas operator in geometric function theory by the symmetric nature of quantum calculus. We also initiate and explore certain new families of holormorphic and bi-univalent functions

A_{E} (λ, σ, δ, s, t, p, q; ϑ)

and

S_{E} (μ, γ, σ, δ, s, t, p, q; ϑ)

which are defined in the unit disk U associated with the

(p - q)

-Wanas operator. The upper bounds for the initial Taylor–Maclaurin coefficients and Fekete–Szegö-type inequalities for the functions in these families are obtained. Furthermore, several consequences of our results are pointed out based on the various special choices of the involved parameters. Full article

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

14 pages, 285 KiB

Open AccessArticle

On Existence Theorems to Symmetric Functional Set-Valued Differential Equations

by Marek T. Malinowski

Symmetry 2021, 13(7), 1219; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13071219 - 07 Jul 2021

Cited by 4 | Viewed by 1167

Abstract

In this paper, we consider functional set-valued differential equations in their integral representations that possess integrals symmetrically on both sides of the equations. The solutions have values that are the nonempty compact and convex subsets. The main results contain a Peano type theorem [...] Read more.

In this paper, we consider functional set-valued differential equations in their integral representations that possess integrals symmetrically on both sides of the equations. The solutions have values that are the nonempty compact and convex subsets. The main results contain a Peano type theorem on the existence of the solution and a Picard type theorem on the existence and uniqueness of the solution to such equations. The proofs are based on sequences of approximations that are constructed with appropriate Hukuhara differences of sets. An estimate of the magnitude of the solution’s values is provided as well. We show the closeness of the unique solutions when the equations differ slightly. Full article

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

15 pages, 312 KiB

Open AccessArticle

Fuzzy Differential Subordinations Based upon the Mittag-Leffler Type Borel Distribution

by Hari Mohan Srivastava and Sheza M. El-Deeb

Symmetry 2021, 13(6), 1023; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13061023 - 07 Jun 2021

Cited by 45 | Viewed by 1965

Abstract

In this paper, we investigate several fuzzy differential subordinations that are connected with the Borel distribution series

B (λ, α, β) (z)

of the Mittag-Leffler type, which involves the two-parameter Mittag-Leffler function

E_{α, β} (z)

[...] Read more.

In this paper, we investigate several fuzzy differential subordinations that are connected with the Borel distribution series

B (λ, α, β) (z)

of the Mittag-Leffler type, which involves the two-parameter Mittag-Leffler function

E_{α, β} (z)

. Using the above-mentioned operator

B (λ, α, β),

we also introduce and study a class

M_{λ, α, β}^{F} (η)

of holomorphic and univalent functions in the open unit disk

Δ

. The Mittag-Leffler-type functions, which we have used in the present investigation, belong to the significantly wider family of the Fox-Wright function

_{p} Ψ_{q} (z)

, whose p numerator parameters and q denominator parameters possess a kind of symmetry behavior in the sense that it remains invariant (or unchanged) when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. Here, in this article, we have used such special functions in our study of a general Borel-type probability distribution, which may be symmetric or asymmetric. As symmetry is generally present in most works involving fuzzy sets and fuzzy systems, our usages here of fuzzy subordinations and fuzzy membership functions potentially possess local or non-local symmetry features. Full article

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

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