Symmetry

Research

12 pages, 289 KiB

Open AccessArticle

Duality on q-Starlike Functions Associated with Fractional q-Integral Operators and Applications

by Ebrahim Amini, Shrideh Al-Omari, Mojtaba Fardi and Kamsing Nonlaopon

Symmetry 2022, 14(10), 2076; https://doi.org/10.3390/sym14102076 - 06 Oct 2022

Cited by 2 | Viewed by 819

Abstract

In this paper, we make use of the Riemann–Liouville fractional q-integral operator to discuss the class

S_{q, δ}^{*} (α)

of univalent functions for

δ > 0, α \in C - {0}

, and [...] Read more.

In this paper, we make use of the Riemann–Liouville fractional q-integral operator to discuss the class

S_{q, δ}^{*} (α)

of univalent functions for

δ > 0, α \in C - {0}

, and

0 < | q | < 1

. Then, we develop convolution results for the given class of univalent functions by utilizing a concept of the fractional q-difference operator. Moreover, we derive the normalized classes

P_{δ, q}^{ζ} (β, γ)

and

P_{δ, q} (β)

(

0 < | q | < 1

,

δ \geq 0, 0 \leq β \leq 1, ζ > 0)

of analytic functions on a unit disc and provide conditions for the parameters

q, δ, ζ, β

, and

γ

so that

P_{δ, q}^{ζ} (β, γ) \subset S_{q, δ}^{*} (α)

and

P_{δ, q} (β) \subset S_{q, δ}^{*} (α)

for

α \in C - {0}

. Finally, we also propose an application to symmetric q-analogues and Ruscheweh’s duality theory. Full article

(This article belongs to the Special Issue Complex Analysis, in Particular Analytic and Univalent Functions)

11 pages, 286 KiB

Open AccessArticle

A Study on Certain Subclasses of Analytic Functions Involving the Jackson q-Difference Operator

by Abdel Moneim Y. Lashin, Abeer O. Badghaish and Badriah Maeed Algethami

Symmetry 2022, 14(7), 1471; https://doi.org/10.3390/sym14071471 - 19 Jul 2022

Cited by 5 | Viewed by 1242

Abstract

We introduce two new subclasses of analytic functions in the open symmetric unit disc using a linear operator associated with the q-binomial theorem. In addition, we discuss inclusion relations and properties preserving integral operators for functions in these classes. This paper generalizes [...] Read more.

We introduce two new subclasses of analytic functions in the open symmetric unit disc using a linear operator associated with the q-binomial theorem. In addition, we discuss inclusion relations and properties preserving integral operators for functions in these classes. This paper generalizes some known results, as well as provides some new ones. Full article

(This article belongs to the Special Issue Complex Analysis, in Particular Analytic and Univalent Functions)

13 pages, 1859 KiB

Open AccessArticle

Subordination Involving Regular Coulomb Wave Functions

by Saiful R. Mondal

Symmetry 2022, 14(5), 1007; https://doi.org/10.3390/sym14051007 - 16 May 2022

Cited by 3 | Viewed by 1364

Abstract

The functions

\sqrt{1 + z}

,

e^{z}

,

1 + A z

,

A \in (0, 1]

map the unit disc

D

to a domain which is symmetric about the x-axis. The Regular Coulomb wave function (RCWF [...] Read more.

The functions

\sqrt{1 + z}

,

e^{z}

,

1 + A z

,

A \in (0, 1]

map the unit disc

D

to a domain which is symmetric about the x-axis. The Regular Coulomb wave function (RCWF)

F_{L, η}

is a function involving two parameters L and

η

, and

F_{L, η}

is symmetric about these. In this article, we derive conditions on the parameter L and

η

for which the normalized form

f_{L}

of

F_{L, η}

are subordinated by

\sqrt{1 + z}

. We also consider the subordination by

e^{z}

and

1 + A z

,

A \in (0, 1]

. A few more subordination properties involving RCWF are discussed, which leads to the star-likeness of normalized Regular Coulomb wave functions. Full article

(This article belongs to the Special Issue Complex Analysis, in Particular Analytic and Univalent Functions)

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13 pages, 527 KiB

Open AccessArticle

A Nonstandard Finite Difference Method for a Generalized Black–Scholes Equation

by Mohammad Mehdizadeh Khalsaraei, Mohammad Mehdi Rashidi, Ali Shokri, Higinio Ramos and Pari Khakzad

Symmetry 2022, 14(1), 141; https://doi.org/10.3390/sym14010141 - 12 Jan 2022

Cited by 3 | Viewed by 2009

Abstract

An implicit finite difference scheme for the numerical solution of a generalized Black–Scholes equation is presented. The method is based on the nonstandard finite difference technique. The positivity property is discussed and it is shown that the proposed method is consistent, stable and [...] Read more.

An implicit finite difference scheme for the numerical solution of a generalized Black–Scholes equation is presented. The method is based on the nonstandard finite difference technique. The positivity property is discussed and it is shown that the proposed method is consistent, stable and also the order of the scheme respect to the space variable is two. As the Black–Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset, the proposed method will be more appropriate for solving such symmetric models. In order to illustrate the efficiency of the new method, we applied it on some test examples. The obtained results confirm the theoretical behavior regarding the order of convergence. Furthermore, the numerical results are in good agreement with the exact solution and are more accurate than other existing results in the literature. Full article

(This article belongs to the Special Issue Complex Analysis, in Particular Analytic and Univalent Functions)

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15 pages, 821 KiB

Open AccessArticle

Multivalent Prestarlike Functions with Respect to Symmetric Points

by Daniel Breaz, Kadhavoor R. Karthikeyan and Alagiriswamy Senguttuvan

Symmetry 2022, 14(1), 20; https://doi.org/10.3390/sym14010020 - 24 Dec 2021

Cited by 11 | Viewed by 2110

Abstract

A class of p-valent functions of complex order is defined with the primary motive of unifying the concept of prestarlike functions with various other classes of multivalent functions. Interesting properties such as inclusion relations, integral representation, coefficient estimates and the solution to [...] Read more.

A class of p-valent functions of complex order is defined with the primary motive of unifying the concept of prestarlike functions with various other classes of multivalent functions. Interesting properties such as inclusion relations, integral representation, coefficient estimates and the solution to the Fekete–Szegő problem are obtained for the defined function class. Further, we extended the results using quantum calculus. Several consequences of our main results are pointed out. Full article

(This article belongs to the Special Issue Complex Analysis, in Particular Analytic and Univalent Functions)

12 pages, 346 KiB

Open AccessArticle

On Convex Functions Associated with Symmetric Cardioid Domain

by Sarfraz Nawaz Malik, Mohsan Raza, Qin Xin, Janusz Sokół, Rabbiya Manzoor and Saira Zainab

Symmetry 2021, 13(12), 2321; https://doi.org/10.3390/sym13122321 - 04 Dec 2021

Cited by 9 | Viewed by 1615

Abstract

The geometry of the image domain plays an important role in the characterization of analytic functions. Therefore, for a comprehensive and detailed study of these functions, a thorough analysis of the geometrical properties of their domains is of prime interest. In this regard, [...] Read more.

The geometry of the image domain plays an important role in the characterization of analytic functions. Therefore, for a comprehensive and detailed study of these functions, a thorough analysis of the geometrical properties of their domains is of prime interest. In this regard, new geometrical structures are introduced and studied as an image domain and then their subsequent analytic functions are defined. Inspired and motivated by ongoing research, Malik et al. introduced a very innovative domain named the cardioid domain, which is symmetric about a real axis. Extending the same work on this symmetric cardioid domain, in this article, we provide a deeper analysis and define and study the convex functions associated with the symmetric cardioid domain, named cardio-convex functions. Full article

(This article belongs to the Special Issue Complex Analysis, in Particular Analytic and Univalent Functions)

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18 pages, 309 KiB

Open AccessArticle

On q-Starlike Functions Defined by q-Ruscheweyh Differential Operator in Symmetric Conic Domain

by Saira Zainab, Mohsan Raza, Qin Xin, Mehwish Jabeen, Sarfraz Nawaz Malik and Sadia Riaz

Symmetry 2021, 13(10), 1947; https://doi.org/10.3390/sym13101947 - 16 Oct 2021

Cited by 11 | Viewed by 1440

Abstract

Motivated by q-analogue theory and symmetric conic domain, we study here the q-version of the Ruscheweyh differential operator by applying it to the starlike functions which are related with the symmetric conic domain. The primary aim of this work is to [...] Read more.

Motivated by q-analogue theory and symmetric conic domain, we study here the q-version of the Ruscheweyh differential operator by applying it to the starlike functions which are related with the symmetric conic domain. The primary aim of this work is to first define and then study a new class of holomorphic functions using the q-Ruscheweyh differential operator. A new class

k - S T_{q}^{τ} [C, D]

of k-Janowski starlike functions associated with the symmetric conic domain, which are defined by the generalized Ruscheweyh derivative operator in the open unit disk, is introduced. The necessary and sufficient condition for a function to be in the class

k - S T_{q}^{τ} [C, D]

is established. In addition, the coefficient bound, partial sums and radii of starlikeness for the functions from the class of k-Janowski starlike functions related with symmetric conic domain are included. Full article

(This article belongs to the Special Issue Complex Analysis, in Particular Analytic and Univalent Functions)

11 pages, 300 KiB

Open AccessArticle

Subclasses of Uniform Univalent Functions Associated with Srivastava and Attiya Operator

by Mohammad Yaseen, Irfan Ali, Sardar Muhammad Hussain and Jong-Suk Ro

Symmetry 2021, 13(8), 1536; https://doi.org/10.3390/sym13081536 - 20 Aug 2021

Viewed by 1376

Abstract

In this paper, we introduce new subclasses

k - S T_{s} (p, β)

and

k - U K_{s} (p, β)

of analytic and univalent functions in the canonical domain associated with the Srivastava and Attiya [...] Read more.

In this paper, we introduce new subclasses

k - S T_{s} (p, β)

and

k - U K_{s} (p, β)

of analytic and univalent functions in the canonical domain associated with the Srivastava and Attiya operator. The radius problems of these subclasses regarding symmetrical points are investigated and compared with previous known results. Certain properties and conditions of these subclasses such as integral representation are also discussed in this work. Full article

(This article belongs to the Special Issue Complex Analysis, in Particular Analytic and Univalent Functions)

15 pages, 288 KiB

Open AccessArticle

Linear Differential Equations on Some Classes of Weighted Function Spaces

by Ahmed El-Sayed Ahmed and Amnah E. Shammaky

Symmetry 2021, 13(7), 1113; https://doi.org/10.3390/sym13071113 - 22 Jun 2021

Viewed by 1082

Abstract

Some weighted-type classes of holomorphic function spaces were introduced in the current study. Moreover, as an application of the new defined classes, the specific growth of certain entire-solutions of a linear-type differential equation by the use of concerned coefficients of certain analytic-type functions, [...] Read more.

Some weighted-type classes of holomorphic function spaces were introduced in the current study. Moreover, as an application of the new defined classes, the specific growth of certain entire-solutions of a linear-type differential equation by the use of concerned coefficients of certain analytic-type functions, that is the equation

h^{(k)} + K_{k - 1} (υ) h^{(k - 1)} + \dots + K_{1} (υ) h^{'} + K_{0} (υ) h = 0

, will be discussed in this current research, whereas the considered coefficients

K_{0} (υ), \dots, K_{k - 1} (υ)

are holomorphic in the disc

Γ_{R} = {υ \in C : | υ | < R}, 0 < R \leq \infty .

In addition, some non-trivial specific examples are illustrated to clear the roles of the obtained results with some sharpness sense. Hence, the obtained results are strengthen to some previous interesting results from the literature. Full article

(This article belongs to the Special Issue Complex Analysis, in Particular Analytic and Univalent Functions)

12 pages, 260 KiB

Open AccessArticle

A Note on Type-Two Degenerate Poly-Changhee Polynomials of the Second Kind

by Dmitry V. Dolgy and Waseem A. Khan

Symmetry 2021, 13(4), 579; https://doi.org/10.3390/sym13040579 - 01 Apr 2021

Cited by 14 | Viewed by 1166

Abstract

In this paper, we first define type-two degenerate poly-Changhee polynomials of the second kind by using modified degenerate polyexponential functions. We derive new identities and relations between type-two degenerate poly-Changhee polynomials of the second kind. Finally, we derive type-two degenerate unipoly-Changhee polynomials of [...] Read more.

In this paper, we first define type-two degenerate poly-Changhee polynomials of the second kind by using modified degenerate polyexponential functions. We derive new identities and relations between type-two degenerate poly-Changhee polynomials of the second kind. Finally, we derive type-two degenerate unipoly-Changhee polynomials of the second kind and discuss some of their identities. Full article

(This article belongs to the Special Issue Complex Analysis, in Particular Analytic and Univalent Functions)

12 pages, 261 KiB

Open AccessArticle

Estimates on Some General Classes of Holomorphic Function Spaces

by Amnah E. Shammaky and Ahmed El-Sayed Ahmed

Symmetry 2021, 13(4), 528; https://doi.org/10.3390/sym13040528 - 24 Mar 2021

Viewed by 1202

Abstract

In this current manuscript, some general classes of weighted analytic function spaces in a unit disc are defined and studied. Special functions significant in both analytic

T (p, q, m, s; Ψ)

norms and analytic

Ψ

-Bloch [...] Read more.

In this current manuscript, some general classes of weighted analytic function spaces in a unit disc are defined and studied. Special functions significant in both analytic

T (p, q, m, s; Ψ)

norms and analytic

Ψ

-Bloch norms serve as a framework for introducing new families of analytic classes. An application in operator theory is provided by establishing important properties of the composition-type operator

C_{ϕ}

such as the boundedness and compactness with the help of the defined new classes. Full article

(This article belongs to the Special Issue Complex Analysis, in Particular Analytic and Univalent Functions)

14 pages, 301 KiB

Open AccessArticle

On the Difference of Inverse Coefficients of Univalent Functions

by Young Jae Sim and Derek Keith Thomas

Symmetry 2020, 12(12), 2040; https://doi.org/10.3390/sym12122040 - 09 Dec 2020

Cited by 9 | Viewed by 1475

Abstract

Let f be analytic in the unit disk

D = {z \in C : | z | < 1}

, and

S

be the subclass of normalized univalent functions with

f (0) = 0

, and [...] Read more.

Let f be analytic in the unit disk

D = {z \in C : | z | < 1}

, and

S

be the subclass of normalized univalent functions with

f (0) = 0

, and

f^{'} (0) = 1 .

Let F be the inverse function of f, given by

F (z) = ω + \sum_{n = 2}^{\infty} A_{n} ω^{n}

for some

| ω | \leq r_{0} (f)

. Let

S^{*} \subset S

be the subset of starlike functions in

D

, and

C

the subset of convex functions in

D .

We show that

- 1 \leq | A_{3} | - | A_{2} | \leq 3

for

f \in S

, the upper bound being sharp, and sharp upper and lower bounds for

| A_{3} | - | A_{2} |

for the more important subclasses of

S^{*}

and

C

, and for some related classes of Bazilevič functions. Full article

(This article belongs to the Special Issue Complex Analysis, in Particular Analytic and Univalent Functions)

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