Symmetry

Research

17 pages, 528 KiB

Open AccessArticle

Applications of a q-Integral Operator to a Certain Class of Analytic Functions Associated with a Symmetric Domain

by Adeel Ahmad, Hanen Louati, Akhter Rasheed, Asad Ali, Saqib Hussain, Shreefa O. Hilali and Afrah Y. Al-Rezami

Symmetry 2024, 16(11), 1443; https://doi.org/10.3390/sym16111443 - 31 Oct 2024

Viewed by 55

Abstract

In this article, our objective is to define and study a new subclass of analytic functions associated with the q-analogue of the sine function, operating in conjunction with a convolution operator. By manipulating the parameter q, we observe that the image [...] Read more.

In this article, our objective is to define and study a new subclass of analytic functions associated with the q-analogue of the sine function, operating in conjunction with a convolution operator. By manipulating the parameter q, we observe that the image of the unit disc under the q-sine function exhibits a visually appealing resemblance to a figure-eight shape that is symmetric about the real axis. Additionally, we investigate some important geometrical problems like necessary and sufficient conditions, coefficient bounds, Fekete-Szegö inequality, and partial sum results for the functions belonging to this newly defined subclass. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

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11 pages, 395 KiB

Open AccessArticle

On λ-Pseudo Bi-Starlike Functions Related to Second Einstein Function

by Alaa H. El-Qadeem, Gangadharan Murugusundaramoorthy, Borhen Halouani, Ibrahim S. Elshazly, Kaliappan Vijaya and Mohamed A. Mamon

Symmetry 2024, 16(11), 1429; https://doi.org/10.3390/sym16111429 - 27 Oct 2024

Viewed by 352

Abstract

A new class

B_{Σ}^{λ} (γ, κ)

of bi-starlike

λ

-pseudo functions related to the second Einstein function is presented in this paper.

c_{2}

and

c_{3}

indicate the initial Taylor coefficients of [...] Read more.

A new class

B_{Σ}^{λ} (γ, κ)

of bi-starlike

λ

-pseudo functions related to the second Einstein function is presented in this paper.

c_{2}

and

c_{3}

indicate the initial Taylor coefficients of

ϕ \in B_{Σ}^{λ} (γ, κ),

and the bounds for

| c_{2} |

and

| c_{3} |

are obtained. Additionally, for

ϕ \in B_{Σ}^{λ} (γ, κ)

, we calculate the Fekete–Szegö functional. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

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14 pages, 280 KiB

Open AccessArticle

Attributes of Subordination of a Specific Subclass of p-Valent Meromorphic Functions Connected to a Linear Operator

by Rabha M. El-Ashwah, Alaa Hassan El-Qadeem, Gangadharan Murugusundaramoorthy, Ibrahim S. Elshazly and Borhen Halouani

Symmetry 2024, 16(10), 1338; https://doi.org/10.3390/sym16101338 - 10 Oct 2024

Viewed by 832

Abstract

This work examines subordination conclusions for a specific subclass of p-valent meromorphic functions on the punctured unit disc of the complex plane where the function has a pole of order p. A new linear operator is used to define the subclass that is [...] Read more.

This work examines subordination conclusions for a specific subclass of p-valent meromorphic functions on the punctured unit disc of the complex plane where the function has a pole of order p. A new linear operator is used to define the subclass that is being studied. Furthermore, we present several corollaries with intriguing specific situations of the results. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

13 pages, 286 KiB

Open AccessArticle

Hankel Determinants of Normalized Analytic Functions Associated with Hyperbolic Secant Function

by Sushil Kumar, Daniel Breaz, Luminita-Ioana Cotîrlă and Asena Çetinkaya

Symmetry 2024, 16(10), 1303; https://doi.org/10.3390/sym16101303 - 3 Oct 2024

Cited by 2 | Viewed by 833

Abstract

In this paper, we consider a subclass of normalized analytic functions associated with the hyperbolic secant function. We compute the sharp bounds on third- and fourth-order Hermitian–Toeplitz determinants for functions in this class. Moreover, we determine the bounds on second- and third-order Hankel [...] Read more.

In this paper, we consider a subclass of normalized analytic functions associated with the hyperbolic secant function. We compute the sharp bounds on third- and fourth-order Hermitian–Toeplitz determinants for functions in this class. Moreover, we determine the bounds on second- and third-order Hankel determinants, as well as on the generalized Zalcman conjecture. We examine a Briot–Bouquet-type differential subordination involving the Bernardi integral operator. Finally, we obtain a univalent solution to the Briot–Bouquet differential equation, and discuss the majorization property for such function classes. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

17 pages, 301 KiB

Open AccessArticle

Sharp Results for a New Class of Analytic Functions Associated with the q-Differential Operator and the Symmetric Balloon-Shaped Domain

by Adeel Ahmad, Jianhua Gong, Akhter Rasheed, Saqib Hussain, Asad Ali and Zeinebou Cheikh

Symmetry 2024, 16(9), 1134; https://doi.org/10.3390/sym16091134 - 2 Sep 2024

Cited by 1 | Viewed by 677

Abstract

In our current study, we apply differential subordination and quantum calculus to introduce and investigate a new class of analytic functions associated with the q-differential operator and the symmetric balloon-shaped domain. We obtain sharp results concerning the Maclaurin coefficients the second and third-order [...] Read more.

In our current study, we apply differential subordination and quantum calculus to introduce and investigate a new class of analytic functions associated with the q-differential operator and the symmetric balloon-shaped domain. We obtain sharp results concerning the Maclaurin coefficients the second and third-order Hankel determinants, the Zalcman conjecture, and its generalized conjecture for this newly defined class of q-starlike functions with respect to symmetric points. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

15 pages, 570 KiB

Open AccessArticle

Inverse Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytic Functions

by Sergey K. Sekatskii

Symmetry 2024, 16(9), 1100; https://doi.org/10.3390/sym16091100 - 23 Aug 2024

Viewed by 675

Abstract

Recently, we established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. The same theorem was subsequently applied to calculate certain infinite sums and study the properties of [...] Read more.

Recently, we established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. The same theorem was subsequently applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this article, we discuss what are, in a sense, inverse applications of this theorem. We first prove a Lemma that if two meromorphic on the whole complex plane functions f(z) and g(z) have the same zeroes and poles, taking into account their orders, and have appropriate asymptotic for large |z|, then for some integer n,

\frac{d^{n} \ln (f (z))}{d z^{n}} = \frac{d^{n} \ln (g (z))}{d z^{n}}

. The use of this Lemma enables proofs of many identities between elliptic functions, their transformation and n-tuple product rules. In particular, we show how exactly for any complex number a, ℘(z)-a, where ℘(z) is the Weierstrass ℘ function, can be presented as a product and ratio of three elliptic

θ_{1}

functions of certain arguments. We also establish n-tuple rules for some elliptic theta functions. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

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13 pages, 284 KiB

Open AccessArticle

On the Fekete–Szegö Problem for Certain Classes of (γ,δ)-Starlike and (γ,δ)-Convex Functions Related to Quasi-Subordinations

by Norah Saud Almutairi, Awatef Shahen, Adriana Cătaş and Hanan Darwish

Symmetry 2024, 16(8), 1043; https://doi.org/10.3390/sym16081043 - 14 Aug 2024

Viewed by 759

Abstract

In the present paper, we propose new generalized classes of (p,q)-starlike and (p,q)-convex functions. These classes are introduced by making use of a (p,q)-derivative operator. There are established Fekete–Szegö estimates

| a_{3} - μ a_{2}^{2} |

for functions belonging to [...] Read more.

In the present paper, we propose new generalized classes of (p,q)-starlike and (p,q)-convex functions. These classes are introduced by making use of a (p,q)-derivative operator. There are established Fekete–Szegö estimates

| a_{3} - μ a_{2}^{2} |

for functions belonging to the newly introduced subclasses. Certain subclasses of analytic univalent functions associated with quasi-subordination are defined. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

13 pages, 256 KiB

Open AccessArticle

Maximum and Minimum Results for the Green’s Functions in Delta Fractional Difference Settings

by Pshtiwan Othman Mohammed, Carlos Lizama, Alina Alb Lupas, Eman Al-Sarairah and Mohamed Abdelwahed

Symmetry 2024, 16(8), 991; https://doi.org/10.3390/sym16080991 - 5 Aug 2024

Viewed by 772

Abstract

The present paper is dedicated to the examination of maximum and minimum results based on Green’s functions via delta fractional differences for a class of fractional boundary problems. For such a purpose, we built the corresponding Green’s functions based on the falling factorial [...] Read more.

The present paper is dedicated to the examination of maximum and minimum results based on Green’s functions via delta fractional differences for a class of fractional boundary problems. For such a purpose, we built the corresponding Green’s functions based on the falling factorial functions. In addition, using the constructed Green’s function, the positivity of the function and its corresponding delta function are presented. We also verified the occurrence of two distinct functions with the same Green’s function. The maximality and minimality of the Green’s function show a good qualitative agreement. Finally, we considered some special examples to explain the obtained results. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

13 pages, 266 KiB

Open AccessArticle

A General and Comprehensive Subclass of Univalent Functions Associated with Certain Geometric Functions

by Tariq Al-Hawary, Basem Frasin and Ibtisam Aldawish

Symmetry 2024, 16(8), 982; https://doi.org/10.3390/sym16080982 - 2 Aug 2024

Viewed by 881

Abstract

In this paper, taking into account the intriguing recent results of Rabotnov functions, Poisson functions, Bessel functions and Wright functions, we consider a new comprehensive subclass

O_{μ} (Δ_{1}, Δ_{2}, Δ_{3}, Δ_{4})

of univalent [...] Read more.

In this paper, taking into account the intriguing recent results of Rabotnov functions, Poisson functions, Bessel functions and Wright functions, we consider a new comprehensive subclass

O_{μ} (Δ_{1}, Δ_{2}, Δ_{3}, Δ_{4})

of univalent functions defined in the unit disk

Λ = {τ \in C : |τ| < 1}

. More specifically, we investigate some sufficient conditions for Rabotnov functions, Poisson functions, Bessel functions and Wright functions to be in this subclass. Some corollaries of our main results are given. The novelty and the advantage of this research could inspire researchers of further studies to find new sufficient conditions to be in the subclass

O_{μ} (Δ_{1}, Δ_{2}, Δ_{3}, Δ_{4})

not only for the aforementioned special functions but for different types of special functions, especially for hypergeometric functions, Dini functions, Sturve functions and others. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

16 pages, 277 KiB

Open AccessArticle

Initial Coefficient Bounds for Certain New Subclasses of Bi-Univalent Functions Involving Mittag–Leffler Function with Bounded Boundary Rotation

by Ibtisam Aldawish, Prathviraj Sharma, Sheza M. El-Deeb, Mariam R. Almutiri and Srikandan Sivasubramanian

Symmetry 2024, 16(8), 971; https://doi.org/10.3390/sym16080971 - 31 Jul 2024

Viewed by 854

Abstract

By using the Mittag–Leffler function associated with functions of bounded boundary rotation, the authors introduce a few new subclasses of bi-univalent functions involving the Mittag–Leffler function with bounded boundary rotation in the open unit disk

D .

For these new classes, the authors [...] Read more.

By using the Mittag–Leffler function associated with functions of bounded boundary rotation, the authors introduce a few new subclasses of bi-univalent functions involving the Mittag–Leffler function with bounded boundary rotation in the open unit disk

D .

For these new classes, the authors establish initial coefficient bounds of

| a_{2} |

and

| a_{3} |

. Furthermore, the famous Fekete–Szegö coefficient inequality is also obtained for these new classes of functions. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

16 pages, 558 KiB

Open AccessArticle

On Ozaki Close-to-Convex Functions with Bounded Boundary Rotation

by Prathviraj Sharma, Asma Alharbi, Srikandan Sivasubramanian and Sheza M. El-Deeb

Symmetry 2024, 16(7), 839; https://doi.org/10.3390/sym16070839 - 3 Jul 2024

Cited by 1 | Viewed by 1028

Abstract

In the present investigation, we introduce a new subclass of univalent functions

F (u, λ)

and a subclass of bi-univalent function

F_{o, Σ} (u, λ)

with bounded boundary and bounded radius rotation. Some examples of [...] Read more.

In the present investigation, we introduce a new subclass of univalent functions

F (u, λ)

and a subclass of bi-univalent function

F_{o, Σ} (u, λ)

with bounded boundary and bounded radius rotation. Some examples of the functions belonging to the classes

F (u, λ)

are also derived. For these new classes, the authors derive many interesting relations between these classes and the existing familiar subclasses in the literature. Furthermore, the authors establish new coefficient estimates for these classes. Apart from the above, the first two initial coefficient bounds for the class

F_{o, Σ} (u, λ)

are established. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

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10 pages, 254 KiB

Open AccessArticle

On Quasi-Subordination for Bi-Univalency Involving Generalized Distribution Series

by Sunday Olufemi Olatunji, Matthew Olanrewaju Oluwayemi, Saurabh Porwal and Alina Alb Lupas

Symmetry 2024, 16(6), 773; https://doi.org/10.3390/sym16060773 - 20 Jun 2024

Viewed by 756

Abstract

Various researchers have considered different forms of bi-univalent functions in recent times, and this has continued to gain more attention in Geometric Function Theory (GFT), but not much study has been conducted in the area of application of the certain probability concept in [...] Read more.

Various researchers have considered different forms of bi-univalent functions in recent times, and this has continued to gain more attention in Geometric Function Theory (GFT), but not much study has been conducted in the area of application of the certain probability concept in geometric functions. In this manuscript, our motivation is the application of analytic and bi-univalent functions. In particular, the researchers examine bi-univalency of a generalized distribution series related to Bell numbers as a family of Caratheodory functions. Some coefficients of the class of the function are obtained. The results are new as far work on bi-univalency is concerned. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

19 pages, 787 KiB

Open AccessArticle

Nonexpansiveness and Fractal Maps in Hilbert Spaces

by María A. Navascués

Symmetry 2024, 16(6), 738; https://doi.org/10.3390/sym16060738 - 13 Jun 2024

Viewed by 645

Abstract

Picard iteration is on the basis of a great number of numerical methods and applications of mathematics. However, it has been known since the 1950s that this method of fixed-point approximation may not converge in the case of nonexpansive mappings. In this paper, [...] Read more.

Picard iteration is on the basis of a great number of numerical methods and applications of mathematics. However, it has been known since the 1950s that this method of fixed-point approximation may not converge in the case of nonexpansive mappings. In this paper, an extension of the concept of nonexpansiveness is presented in the first place. Unlike the classical case, the new maps may be discontinuous, adding an element of generality to the model. Some properties of the set of fixed points of the new maps are studied. Afterwards, two iterative methods of fixed-point approximation are analyzed, in the frameworks of b-metric and Hilbert spaces. In the latter case, it is proved that the symmetrically averaged iterative procedures perform well in the sense of convergence with the least number of operations at each step. As an application, the second part of the article is devoted to the study of fractal mappings on Hilbert spaces defined by means of nonexpansive operators. The paper considers fractal mappings coming from

φ

-contractions as well. In particular, the new operators are useful for the definition of an extension of the concept of

α

-fractal function, enlarging its scope to more abstract spaces and procedures. The fractal maps studied here have quasi-symmetry, in the sense that their graphs are composed of transformed copies of itself. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

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15 pages, 279 KiB

Open AccessArticle

Results for Analytic Function Associated with Briot–Bouquet Differential Subordinations and Linear Fractional Integral Operators

by Ebrahim Amini, Wael Salameh, Shrideh Al-Omari and Hamzeh Zureigat

Symmetry 2024, 16(6), 711; https://doi.org/10.3390/sym16060711 - 7 Jun 2024

Cited by 1 | Viewed by 547

Abstract

In this paper, we present a new class of linear fractional differential operators that are based on classical Gaussian hypergeometric functions. Then, we utilize the new operators and the concept of differential subordination to construct a convex set of analytic functions. Moreover, through [...] Read more.

In this paper, we present a new class of linear fractional differential operators that are based on classical Gaussian hypergeometric functions. Then, we utilize the new operators and the concept of differential subordination to construct a convex set of analytic functions. Moreover, through an examination of a certain operator, we establish several notable results related to differential subordination. In addition, we derive inclusion relation results by employing Briot–Bouquet differential subordinations. We also introduce a perspective study for developing subordination results using Gaussian hypergeometric functions and provide certain properties for further research in complex dynamical systems. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

18 pages, 319 KiB

Open AccessArticle

A Singular Tempered Sub-Diffusion Fractional Model Involving a Non-Symmetrically Quasi-Homogeneous Operator

by Xinguang Zhang, Peng Chen, Lishuang Li and Yonghong Wu

Symmetry 2024, 16(6), 671; https://doi.org/10.3390/sym16060671 - 30 May 2024

Viewed by 406

Abstract

In this paper, we focus on the existence of positive solutions for a singular tempered sub-diffusion fractional model involving a quasi-homogeneous nonlinear operator. By using the spectrum theory and computing the fixed point index, some new sufficient conditions for the existence of positive [...] Read more.

In this paper, we focus on the existence of positive solutions for a singular tempered sub-diffusion fractional model involving a quasi-homogeneous nonlinear operator. By using the spectrum theory and computing the fixed point index, some new sufficient conditions for the existence of positive solutions are derived. It is worth pointing out that the nonlinearity of the equation contains a tempered fractional sub-diffusion term, and is allowed to possess strong singularities in time and space variables. In particular, the quasi-homogeneous operator is a nonlinear and non-symmetrical operator. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

11 pages, 323 KiB

Open AccessArticle

Sharp Bounds on Toeplitz Determinants for Starlike and Convex Functions Associated with Bilinear Transformations

by Pishtiwan Othman Sabir

Symmetry 2024, 16(5), 595; https://doi.org/10.3390/sym16050595 - 11 May 2024

Cited by 1 | Viewed by 1008

Abstract

Starlike and convex functions have gained increased prominence in both academic literature and practical applications over the past decade. Concurrently, logarithmic coefficients play a pivotal role in estimating diverse properties within the realm of analytic functions, whether they are univalent or nonunivalent. In [...] Read more.

Starlike and convex functions have gained increased prominence in both academic literature and practical applications over the past decade. Concurrently, logarithmic coefficients play a pivotal role in estimating diverse properties within the realm of analytic functions, whether they are univalent or nonunivalent. In this paper, we rigorously derive bounds for specific Toeplitz determinants involving logarithmic coefficients pertaining to classes of convex and starlike functions concerning symmetric points. Furthermore, we present illustrative examples showcasing the sharpness of these established bounds. Our findings represent a substantial contribution to the advancement of our understanding of logarithmic coefficients and their profound implications across diverse mathematical contexts. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

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18 pages, 355 KiB

Open AccessArticle

On Uniformly Starlike Functions with Respect to Symmetrical Points Involving the Mittag-Leffler Function and the Lambert Series

by Jamal Salah

Symmetry 2024, 16(5), 580; https://doi.org/10.3390/sym16050580 - 8 May 2024

Cited by 1 | Viewed by 765

Abstract

The aim of this paper is to define the linear operator based on the generalized Mittag-Leffler function and the Lambert series. By using this operator, we introduce a new subclass of β-uniformly starlike functions

Τ_{J} (α_{i})

. Further, we [...] Read more.

The aim of this paper is to define the linear operator based on the generalized Mittag-Leffler function and the Lambert series. By using this operator, we introduce a new subclass of β-uniformly starlike functions

Τ_{J} (α_{i})

. Further, we obtain coefficient estimates, convex linear combinations, and radii of close-to-convexity, starlikeness, and convexity for functions

{f \in Τ}_{J} (α_{i})

. In addition, we investigate the inclusion conditions of the Hadamard product and the integral transform. Finally, we determine the second Hankel inequality for functions belonging to this subclass. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

16 pages, 2559 KiB

Open AccessArticle

On Multiple-Type Wave Solutions for the Nonlinear Coupled Time-Fractional Schrödinger Model

by Pshtiwan Othman Mohammed, Ravi P. Agarwal, Iver Brevik, Mohamed Abdelwahed, Artion Kashuri and Majeed A. Yousif

Symmetry 2024, 16(5), 553; https://doi.org/10.3390/sym16050553 - 3 May 2024

Cited by 4 | Viewed by 1632

Abstract

Recently, nonlinear fractional models have become increasingly important for describing phenomena occurring in science and engineering fields, especially those including symmetric kernels. In the current article, we examine two reliable methods for solving fractional coupled nonlinear Schrödinger models. These methods are known as [...] Read more.

Recently, nonlinear fractional models have become increasingly important for describing phenomena occurring in science and engineering fields, especially those including symmetric kernels. In the current article, we examine two reliable methods for solving fractional coupled nonlinear Schrödinger models. These methods are known as the Sardar-subequation technique (SSET) and the improved generalized tanh-function technique (IGTHFT). Numerous novel soliton solutions are computed using different formats, such as periodic, bell-shaped, dark, and combination single bright along with kink, periodic, and single soliton solutions. Additionally, single solitary wave, multi-wave, and periodic kink combined solutions are evaluated. The behavioral traits of the retrieved solutions are illustrated by certain distinctive two-dimensional, three-dimensional, and contour graphs. The results are encouraging, since they show that the suggested methods are trustworthy, consistent, and efficient in finding accurate solutions to the various challenging nonlinear problems that have recently surfaced in applied sciences, engineering, and nonlinear optics. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

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20 pages, 315 KiB

Open AccessArticle

Differential Subordination and Superordination Using an Integral Operator for Certain Subclasses of p-Valent Functions

by Norah Saud Almutairi, Awatef Shahen and Hanan Darwish

Symmetry 2024, 16(4), 501; https://doi.org/10.3390/sym16040501 - 21 Apr 2024

Viewed by 718

Abstract

This work presents a novel investigation that utilizes the integral operator

I_{p, λ}^{n}

in the field of geometric function theory, with a specific focus on sandwich theorems. We obtained findings about the differential subordination and superordination of a novel formula [...] Read more.

This work presents a novel investigation that utilizes the integral operator

I_{p, λ}^{n}

in the field of geometric function theory, with a specific focus on sandwich theorems. We obtained findings about the differential subordination and superordination of a novel formula for a generalized integral operator. Additionally, certain sandwich theorems were discovered. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

16 pages, 316 KiB

Open AccessArticle

On the Analytic Extension of Lauricella–Saran’s Hypergeometric Function F_K to Symmetric Domains

by Roman Dmytryshyn and Vitaliy Goran

Symmetry 2024, 16(2), 220; https://doi.org/10.3390/sym16020220 - 11 Feb 2024

Cited by 5 | Viewed by 997

Abstract

In this paper, we consider the representation and extension of the analytic functions of three variables by special families of functions, namely branched continued fractions. In particular, we establish new symmetric domains of the analytical continuation of Lauricella–Saran’s hypergeometric function

F_{K}

with [...] Read more.

In this paper, we consider the representation and extension of the analytic functions of three variables by special families of functions, namely branched continued fractions. In particular, we establish new symmetric domains of the analytical continuation of Lauricella–Saran’s hypergeometric function

F_{K}

with certain conditions on real and complex parameters using their branched continued fraction representations. We use a technique that extends the convergence, which is already known for a small domain, to a larger domain to obtain domains of convergence of branched continued fractions and the PC method to prove that they are also domains of analytical continuation. In addition, we discuss some applicable special cases and vital remarks. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

13 pages, 316 KiB

Open AccessArticle

Coefficient Bounds for Two Subclasses of Analytic Functions Involving a Limacon-Shaped Domain

by Daniel Breaz, Trailokya Panigrahi, Sheza M. El-Deeb, Eureka Pattnayak and Srikandan Sivasubramanian

Symmetry 2024, 16(2), 183; https://doi.org/10.3390/sym16020183 - 3 Feb 2024

Cited by 1 | Viewed by 1147

Abstract

In the current exploration, we defined new subclasses of analytic functions, namely

R_{l i m} (l, ν)

and

C_{l i m} (l, ν)

, defined by subordination linked with a Limacon-shaped domain. We found a [...] Read more.

In the current exploration, we defined new subclasses of analytic functions, namely

R_{l i m} (l, ν)

and

C_{l i m} (l, ν)

, defined by subordination linked with a Limacon-shaped domain. We found a few initial coefficient bounds and Fekete–Szegő inequalities for the functions in the above-stated new classes. The corresponding results have been derived for the function

h^{- 1}

. Additionally, we discuss the Poisson distribution as an application of our consequences. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

21 pages, 381 KiB

Open AccessArticle

Geometric Nature of Special Functions on Domain Enclosed by Nephroid and Leminscate Curve

by Reem Alzahrani and Saiful R. Mondal

Symmetry 2024, 16(1), 19; https://doi.org/10.3390/sym16010019 - 22 Dec 2023

Cited by 1 | Viewed by 1646

Abstract

In this work, the geometric nature of solutions to two second-order differential equations,

z y^{''} (z) + a (z) y^{'} (z) + b (z) y (z) = 0

and [...] Read more.

In this work, the geometric nature of solutions to two second-order differential equations,

z y^{''} (z) + a (z) y^{'} (z) + b (z) y (z) = 0

and

z^{2} y^{''} (z) + a (z) y^{'} (z) + b (z) y (z) = d (z)

, is studied. Here,

a (z)

,

b (z)

, and

d (z)

are analytic functions defined on the unit disc. Using differential subordination, we established that the normalized solution

F (z)

(with F(0) = 1) of above differential equations maps the unit disc to the domain bounded by the leminscate curve

\sqrt{1 + z}

. We construct several examples by the judicious choice of

a (z)

,

b (z)

, and

d (z)

. The examples include Bessel functions, Struve functions, the Bessel–Sturve kernel, confluent hypergeometric functions, and many other special functions. We also established a connection with the nephroid domain. Directly using subordination, we construct functions that are subordinated by a nephroid function. Two open problems are also suggested in the conclusion. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

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12 pages, 1815 KiB

Open AccessArticle

Certain Results on Subclasses of Analytic and Bi-Univalent Functions Associated with Coefficient Estimates and Quasi-Subordination

by Elaf Ibrahim Badiwi, Waggas Galib Atshan, Ameera N. Alkiffai and Alina Alb Lupas

Symmetry 2023, 15(12), 2208; https://doi.org/10.3390/sym15122208 - 17 Dec 2023

Viewed by 1200

Abstract

The purpose of the present paper is to introduce and investigate new subclasses of analytic function class of bi-univalent functions defined in open unit disks connected with a linear q-convolution operator, which are associated with quasi-subordination. We find coefficient estimates of [...] Read more.

The purpose of the present paper is to introduce and investigate new subclasses of analytic function class of bi-univalent functions defined in open unit disks connected with a linear q-convolution operator, which are associated with quasi-subordination. We find coefficient estimates of

|h_{2}|, |h_{3}|

for functions in these subclasses. Several known and new consequences of these results are also pointed out. There is symmetry between the results of the subclass

f_{q, \sum}^{μ} (ζ, n, ρ, σ, ϑ, γ, δ, φ)

and the results of the subclass

ℵ_{\sum}^{q, δ} (λ, ζ, n, ρ, σ, ϑ, φ)

. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

16 pages, 333 KiB

Open AccessArticle

Binomial Series-Confluent Hypergeometric Distribution and Its Applications on Subclasses of Multivalent Functions

by Ibtisam Aldawish, Sheza M. El-Deeb and Gangadharan Murugusundaramoorthy

Symmetry 2023, 15(12), 2186; https://doi.org/10.3390/sym15122186 - 11 Dec 2023

Cited by 1 | Viewed by 1055

Abstract

Over the past ten years, analytical functions’ reputation in the literature and their application have grown. We study some practical issues pertaining to multivalent functions with bounded boundary rotation that associate with the combination of confluent hypergeometric functions and binomial series in this [...] Read more.

Over the past ten years, analytical functions’ reputation in the literature and their application have grown. We study some practical issues pertaining to multivalent functions with bounded boundary rotation that associate with the combination of confluent hypergeometric functions and binomial series in this research. A novel subset of multivalent functions is established through the use of convolution products and specific inclusion properties are examined through the application of second order differential inequalities in the complex plane. Furthermore, for multivalent functions, we examined inclusion findings using Bernardi integral operators. Moreover, we will demonstrate how the class proposed in this study, in conjunction with the acquired results, generalizes other well-known (or recently discovered) works that are called out as exceptions in the literature. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

14 pages, 310 KiB

Open AccessArticle

An Application of Touchard Polynomials on Subclasses of Analytic Functions

by Ekram E. Ali, Waffa Y. Kota, Rabha M. El-Ashwah, Abeer M. Albalahi, Fatma E. Mansour and R. A. Tahira

Symmetry 2023, 15(12), 2125; https://doi.org/10.3390/sym15122125 - 29 Nov 2023

Viewed by 1039

Abstract

The aim of this work is to discuss some conditions for Touchard polynomials to be in the classes

{TB}_{b} (ρ, σ)

and

{TK}_{b} (ρ, σ)

. Also, we obtain some connection between [...] Read more.

The aim of this work is to discuss some conditions for Touchard polynomials to be in the classes

{TB}_{b} (ρ, σ)

and

{TK}_{b} (ρ, σ)

. Also, we obtain some connection between

R_{η} (D, E)

and

{TK}_{b} (ρ, σ)

. Also, we investigate several mapping properties involving these subclasses. Further, we discuss the geometric properties of an integral operator related to the Touchard polynomial. Additionally, briefly mentioned are specific instances of our primary results. Also, several particular examples are presented. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

25 pages, 1038 KiB

Open AccessArticle

New Versions of Fuzzy-Valued Integral Inclusion over p-Convex Fuzzy Number-Valued Mappings and Related Fuzzy Aumman’s Integral Inequalities

by Nasser Aedh Alreshidi, Muhammad Bilal Khan, Daniel Breaz and Luminita-Ioana Cotirla

Symmetry 2023, 15(12), 2123; https://doi.org/10.3390/sym15122123 - 28 Nov 2023

Cited by 1 | Viewed by 860

Abstract

It is well known that both concepts of symmetry and convexity are directly connected. Similarly, in fuzzy theory, both ideas behave alike. It is important to note that real and interval-valued mappings are exceptional cases of fuzzy number-valued mappings ( [...] Read more.

It is well known that both concepts of symmetry and convexity are directly connected. Similarly, in fuzzy theory, both ideas behave alike. It is important to note that real and interval-valued mappings are exceptional cases of fuzzy number-valued mappings (

F N V M

s) because fuzzy theory depends upon the unit interval that make a significant contribution to overcoming the issues that arise in the theory of interval analysis and fuzzy number theory. In this paper, the new class of

p

-convexity over up and down (

U D

) fuzzy relation has been introduced which is known as

U D

-

p

-convex fuzzy number-valued mappings (

U D

-

p

-convex

F N V M

s). We offer a thorough analysis of Hermite–Hadamard-type inequalities for

F N V M

s that are

U D

-

p

-convex using the fuzzy Aumann integral. Some previous results from the literature are expanded upon and broadly applied in our study. Additionally, we offer precise justifications for the key theorems that Kunt and İşcan first deduced in their article titled “Hermite–Hadamard–Fejer type inequalities for p-convex functions”. Some new and classical exceptional cases are also discussed. Finally, we illustrate our findings with well-defined examples. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

15 pages, 329 KiB

Open AccessArticle

Analytic Invariants of Semidirect Products of Symmetric Groups on Banach Spaces

by Nataliia Baziv and Andriy Zagorodnyuk

Symmetry 2023, 15(12), 2117; https://doi.org/10.3390/sym15122117 - 27 Nov 2023

Cited by 1 | Viewed by 841

Abstract

We consider algebras of polynomials and analytic functions that are invariant with respect to semidirect products of groups of bounded operators on Banach spaces with symmetric bases. In particular, we consider algebras of so-called block-symmetric and double-symmetric analytic functions on Banach spaces [...] Read more.

We consider algebras of polynomials and analytic functions that are invariant with respect to semidirect products of groups of bounded operators on Banach spaces with symmetric bases. In particular, we consider algebras of so-called block-symmetric and double-symmetric analytic functions on Banach spaces

ℓ_{p} (C^{n})

and the homomorphisms of these algebras. In addition, we describe an algebraic basis in the algebra of double-symmetric polynomials and discuss a structure of the spectrum of the algebra of double-symmetric analytic functions on

ℓ_{p} (C^{n}) .

Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

16 pages, 323 KiB

Open AccessArticle

New Subclass of Close-to-Convex Functions Defined by Quantum Difference Operator and Related to Generalized Janowski Function

by Suha B. Al-Shaikh, Mohammad Faisal Khan, Mustafa Kamal and Naeem Ahmad

Symmetry 2023, 15(11), 1974; https://doi.org/10.3390/sym15111974 - 25 Oct 2023

Cited by 1 | Viewed by 1011

Abstract

This work begins with a discussion of the quantum calculus operator theory and proceeds to develop and investigate a new family of close-to-convex functions in an open unit disk. Considering the quantum difference operator, we define and study a new subclass of close-to-convex [...] Read more.

This work begins with a discussion of the quantum calculus operator theory and proceeds to develop and investigate a new family of close-to-convex functions in an open unit disk. Considering the quantum difference operator, we define and study a new subclass of close-to-convex functions connected with generalized Janowski functions. We prove the necessary and sufficient conditions for functions that belong to newly defined classes, including the inclusion relations and estimations of the coefficients. The Fekete–Szegő problem for a more general class is also discussed. The results of this investigation expand upon those of the previous study. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

26 pages, 378 KiB

Open AccessArticle

Exploring a Special Class of Bi-Univalent Functions: q-Bernoulli Polynomial, q-Convolution, and q-Exponential Perspective

by Timilehin Gideon Shaba, Serkan Araci, Babatunde Olufemi Adebesin and Ayhan Esi

Symmetry 2023, 15(10), 1928; https://doi.org/10.3390/sym15101928 - 17 Oct 2023

Cited by 5 | Viewed by 1267

Abstract

This research article introduces a novel operator termed q-convolution, strategically integrated with foundational principles of q-calculus. Leveraging this innovative operator alongside q-Bernoulli polynomials, a distinctive class of functions emerges, characterized by both analyticity and bi-univalence. The determination of initial coefficients [...] Read more.

This research article introduces a novel operator termed q-convolution, strategically integrated with foundational principles of q-calculus. Leveraging this innovative operator alongside q-Bernoulli polynomials, a distinctive class of functions emerges, characterized by both analyticity and bi-univalence. The determination of initial coefficients within the Taylor-Maclaurin series for this function class is accomplished, showcasing precise bounds. Additionally, explicit computation of the second Hankel determinant is provided. These pivotal findings, accompanied by their corollaries and implications, not only enrich but also extend previously published results. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

14 pages, 301 KiB

Open AccessArticle

The Backward Shift and Two Infinite-Dimension Phenomena in Banach Spaces

by Zoriana Novosad and Andriy Zagorodnyuk

Symmetry 2023, 15(10), 1855; https://doi.org/10.3390/sym15101855 - 2 Oct 2023

Viewed by 989

Abstract

We consider the backward shift operator on a sequence Banach space in the context of two infinite-dimensional phenomena: the existence of topologically transitive operators, and the existence of entire analytic functions of the unbounded type. It is well known that the weighted backward [...] Read more.

We consider the backward shift operator on a sequence Banach space in the context of two infinite-dimensional phenomena: the existence of topologically transitive operators, and the existence of entire analytic functions of the unbounded type. It is well known that the weighted backward shift (for an appropriated weight) is topologically transitive on

1 \leq p < \infty

and on

c_{0} .

We construct some generalizations of the weighted backward shift for non-separable Banach spaces, which remains topologically transitive. Also, we show that the backward shift, in some sense, generates analytic functions of the unbounded type. We introduce the notion of a generator of analytic functions of the unbounded type on a Banach space and investigate its properties. In addition, we show that, using this operator, one can obtain a quasi-extension operator of analytic functions in a germ of zero for entire analytic functions. The results are supported by examples. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

26 pages, 875 KiB

Open AccessArticle

Analysis of Coefficient-Related Problems for Starlike Functions with Symmetric Points Connected with a Three-Leaf-Shaped Domain

by Huo Tang, Muhammad Arif, Muhammad Abbas, Ferdous M. O. Tawfiq and Sarfraz Nawaz Malik

Symmetry 2023, 15(10), 1837; https://doi.org/10.3390/sym15101837 - 28 Sep 2023

Cited by 1 | Viewed by 1310

Abstract

The basic aspect of the research on coefficient problems for numerous families of univalent functions is to describe the coefficients of functions in a specific family by the coefficients of the Carathéodory functions. Thus, in utilizing the inequalities that are known for the [...] Read more.

The basic aspect of the research on coefficient problems for numerous families of univalent functions is to describe the coefficients of functions in a specific family by the coefficients of the Carathéodory functions. Thus, in utilizing the inequalities that are known for the class of Carathéodory functions, coefficient functionals may be examined. Several coefficient problems will be addressed in this study by utilizing the methodology for the abovementioned functions’ family. The family of starlike functions with respect to symmetric points connected to a three-leaf-shaped image domain is the topic of our investigation. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

9 pages, 267 KiB

Open AccessArticle

Coefficient Inequalities and Fekete–Szegö-Type Problems for Family of Bi-Univalent Functions

by Tariq Al-Hawary, Ala Amourah, Hasan Almutairi and Basem Frasin

Symmetry 2023, 15(9), 1747; https://doi.org/10.3390/sym15091747 - 12 Sep 2023

Cited by 4 | Viewed by 842

Abstract

In this study, we present a novel family of holomorphic and bi-univalent functions, denoted as

E_{Ω} (η, ε; Ϝ

). We establish the coefficient bounds for this family by utilizing the generalized telephone numbers. Additionally, we solve the Fekete–Szegö [...] Read more.

In this study, we present a novel family of holomorphic and bi-univalent functions, denoted as

E_{Ω} (η, ε; Ϝ

). We establish the coefficient bounds for this family by utilizing the generalized telephone numbers. Additionally, we solve the Fekete–Szegö functional for functions that belong to this family within the open unit disk. Moreover, our results have several consequences. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

13 pages, 287 KiB

Open AccessArticle

On Hankel and Inverse Hankel Determinants of Order Two for Some Subclasses of Analytic Functions

by Nehad Ali Shah, Naseer Bin Turki, Sang-Ro Lee, Seonhui Kang and Jae Dong Chung

Symmetry 2023, 15(9), 1674; https://doi.org/10.3390/sym15091674 - 30 Aug 2023

Viewed by 849

Abstract

In view of the subclass

S L^{*} (β),

which for

β = 0

reduces to the class

S L^{*},

two more subclasses

C L (β)

and

G L (β)

are introduced. For all [...] Read more.

In view of the subclass

S L^{*} (β),

which for

β = 0

reduces to the class

S L^{*},

two more subclasses

C L (β)

and

G L (β)

are introduced. For all these three subclasses, we investigate upper bounds of second Hankel and second inverse Hankel determinates. In most of the cases, the results are sharp. Full article

(This article belongs to the Special Issue Symmetry in Geometric Theory of Analytic Functions)

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