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13 pages, 762 KB

Open AccessArticle

Starlike Functions with Respect to (ℓ, κ)-Symmetric Points Associated with the Vertical Domain

by Daniel Breaz, Kadhavoor R. Karthikeyan and Dharmaraj Mohankumar

Symmetry 2025, 17(6), 933; https://doi.org/10.3390/sym17060933 - 12 Jun 2025

Viewed by 460

The study of various subclasses of univalent functions involving the solutions to various differential equations is not totally new, but studies of analytic functions with respect to

(ℓ, κ)

-symmetric points are rarely conducted. Here, using a differential operator which [...] Read more.

The study of various subclasses of univalent functions involving the solutions to various differential equations is not totally new, but studies of analytic functions with respect to

(ℓ, κ)

-symmetric points are rarely conducted. Here, using a differential operator which was defined using the Hadamard product of Mittag–Leffler function and general analytic function, we introduce a new class of starlike functions with respect to

(ℓ, κ)

-symmetric points associated with the vertical domain. To define the function class, we use a Carathéodory function which was recently introduced to study the impact of various conic regions on the vertical domain. We obtain several results concerned with integral representations and coefficient inequalities for functions belonging to this class. The results obtained by us here not only unify the recent studies associated with the vertical domain but also provide essential improvements of the corresponding results. Full article

(This article belongs to the Special Issue Symmetry in Mathematical Analysis and Applications, 2nd Edition)

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14 pages, 310 KB

Open AccessArticle

Norm Estimates of the Pre-Schwarzian Derivatives for Functions with Conic-like Domains

by Sidra Zafar, Abbas Kareem Wanas, Mohamed Abdalla and Syed Zakar Hussain Bukhari

Mathematics 2023, 11(11), 2490; https://doi.org/10.3390/math11112490 - 29 May 2023

Viewed by 2190

Abstract

The pre-Schwarzianand Schwarzian derivatives of analytic functions f are defined in

U

, where

U

is the open unit disk. The pre-Schwarzian as well as Schwarzian derivatives are popular tools for studying the geometric properties of analytic mappings. These can also be used [...] Read more.

The pre-Schwarzianand Schwarzian derivatives of analytic functions f are defined in

U

, where

U

is the open unit disk. The pre-Schwarzian as well as Schwarzian derivatives are popular tools for studying the geometric properties of analytic mappings. These can also be used to obtain either necessary or sufficient conditions for the univalence of a function f. Because of the computational difficulty, the pre-Schwarzian norm has received more attention than the Schwarzian norm. It has applications in the theory of hypergeometric functions, conformal mappings, Teichmüller spaces, and univalent functions. In this paper, we find sharp norm estimates of the pre-Schwarzian derivatives of certain subfamilies of analytic functions involving some conic-like image domains. These results may also be extended to the families of strongly starlike, convex, as well as to functions with symmetric and conjugate symmetric points. Full article

(This article belongs to the Special Issue Complex Analysis and Geometric Function Theory, 2nd Edition)

20 pages, 332 KB

Open AccessArticle

Applications of Symmetric Conic Domains to a Subclass of q-Starlike Functions

by Shahid Khan, Nazar Khan, Aftab Hussain, Serkan Araci, Bilal Khan and Hamed H. Al-Sulami

Symmetry 2022, 14(4), 803; https://doi.org/10.3390/sym14040803 - 12 Apr 2022

Cited by 11 | Viewed by 2190

Abstract

In this paper, the theory of symmetric q-calculus and conic regions are used to define a new subclass of q-starlike functions involving a certain conic domain. By means of this newly defined domain, a new subclass of normalized analytic functions in [...] Read more.

In this paper, the theory of symmetric q-calculus and conic regions are used to define a new subclass of q-starlike functions involving a certain conic domain. By means of this newly defined domain, a new subclass of normalized analytic functions in the open unit disk E is given. Certain properties of this subclass, such as its structural formula, necessary and sufficient conditions, coefficient estimates, Fekete–Szegö problem, distortion inequalities, closure theorem and subordination results, are investigated. Some new and known consequences of our main results as corollaries are also highlighted. Full article

(This article belongs to the Special Issue Symmetry in Pure Mathematics and Real and Complex Analysis)

16 pages, 298 KB

Open AccessArticle

Coefficient Estimates for a Family of Starlike Functions Endowed with Quasi Subordination on Conic Domain

by Arzu Akgül and Luminita-Ioana Cotîrlă

Symmetry 2022, 14(3), 582; https://doi.org/10.3390/sym14030582 - 16 Mar 2022

Cited by 4 | Viewed by 2018

Abstract

In 1999, for

(0 \leq k < \infty)

, the concept of conic domain by defining k-uniform convex functions were introduced by Kanas and Wisniowska and then in 2000, they defined related k-starlike functions denoted by [...] Read more.

In 1999, for

(0 \leq k < \infty)

, the concept of conic domain by defining k-uniform convex functions were introduced by Kanas and Wisniowska and then in 2000, they defined related k-starlike functions denoted by

k - U C V

and

k - S T

respectively. Motivated by their studies, in our work, we define the class of k-parabolic starlike functions, denoted

k - S_{H_{m}, q}

, by using quasi-subordination for m-fold symmetric analytic functions, making use of conic domain

Ω_{k}

. We determine the coefficient bounds and estimate Fekete–Szegö functional by the help of m-th root transform and quasi subordination for functions belonging the class

k - S_{H_{m}, q}

. Full article

(This article belongs to the Special Issue Symmetry in Pure Mathematics and Real and Complex Analysis)

18 pages, 309 KB

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 18 | Viewed by 2278

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)

12 pages, 281 KB

Open AccessArticle

A Subclass of q-Starlike Functions Defined by Using a Symmetric q-Derivative Operator and Related with Generalized Symmetric Conic Domains

by Shahid Khan, Saqib Hussain, Muhammad Naeem, Maslina Darus and Akhter Rasheed

Mathematics 2021, 9(9), 917; https://doi.org/10.3390/math9090917 - 21 Apr 2021

Cited by 23 | Viewed by 2547

Abstract

In this paper, the concepts of symmetric q-calculus and conic regions are used to define a new domain

\tilde{Ω_{k, q, α}},

which generalizes the symmetric conic domains. By using the domain [...] Read more.

In this paper, the concepts of symmetric q-calculus and conic regions are used to define a new domain

\tilde{Ω_{k, q, α}},

which generalizes the symmetric conic domains. By using the domain

\tilde{Ω_{k, q, α}}

, we define a new subclass of analytic and q-starlike functions in the open unit disk U and establish some new results for functions of this class. We also investigate a number of useful properties and characteristics of this subclass, such as coefficients estimates, structural formulas, distortion inequalities, necessary and sufficient conditions, closure and subordination results. The proposed approach is also compared with some existing methods to show the reliability and effectiveness of the proposed methods. Full article

(This article belongs to the Special Issue Complex Analysis and Geometric Function Theory)

18 pages, 331 KB

Open AccessArticle

Fekete-Szegö Type Problems and Their Applications for a Subclass of q-Starlike Functions with Respect to Symmetrical Points

by Hari Mohan Srivastava, Nazar Khan, Maslina Darus, Shahid Khan, Qazi Zahoor Ahmad and Saqib Hussain

Mathematics 2020, 8(5), 842; https://doi.org/10.3390/math8050842 - 22 May 2020

Cited by 45 | Viewed by 3835

Abstract

In this article, by using the concept of the quantum (or q-) calculus and a general conic domain

Ω_{k, q}

, we study a new subclass of normalized analytic functions with respect to symmetrical points in an open unit disk. [...] Read more.

In this article, by using the concept of the quantum (or q-) calculus and a general conic domain

Ω_{k, q}

, we study a new subclass of normalized analytic functions with respect to symmetrical points in an open unit disk. We solve the Fekete-Szegö type problems for this newly-defined subclass of analytic functions. We also discuss some applications of the main results by using a q-Bernardi integral operator. Full article

(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)

15 pages, 3558 KB

Open AccessArticle

Focal Conic Stacking in Smectic A Liquid Crystals: Smectic Flower and Apollonius Tiling

by Claire Meyer, Loic Le Cunff, Malika Belloul and Guillaume Foyart

Materials 2009, 2(2), 499-513; https://doi.org/10.3390/ma2020499 - 22 Apr 2009

Cited by 34 | Viewed by 18767

Abstract

We investigate two different textures of smectic A liquid crystals. These textures are particularly symmetric when they are observed at crossed polars optical microscopy. For both textures, a model has been made in order to examine the link between the defective macroscopic texture [...] Read more.

We investigate two different textures of smectic A liquid crystals. These textures are particularly symmetric when they are observed at crossed polars optical microscopy. For both textures, a model has been made in order to examine the link between the defective macroscopic texture and the microscopic disposition of the layers. We present in particular in the case of some hexagonal tiling of circles (similar to the Apollonius tiling) some numeric simulation in order to visualize the smectic layers. We discuss of the nature of the smectic layers, which permit to assure their continuity from one focal conic domain to another adjacent one. Full article

(This article belongs to the Special Issue Liquid Crystals)

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