Symmetry

Research

15 pages, 305 KiB

Open AccessArticle

Sharp Coefficient and Hankel Problems Related to a Symmetric Domain

by Huo Tang, Adeel Ahmad, Akhter Rasheed, Asad Ali, Saqib Hussain and Saima Noor

Symmetry 2023, 15(10), 1865; https://doi.org/10.3390/sym15101865 - 4 Oct 2023

Cited by 1 | Viewed by 866

In the current article, we utilize the concept of subordination to establish a new subclass of analytic functions associated with a bounded domain that is symmetric about the real axis. By applying the convolution technique, we derive the necessary and sufficient condition, the [...] Read more.

In the current article, we utilize the concept of subordination to establish a new subclass of analytic functions associated with a bounded domain that is symmetric about the real axis. By applying the convolution technique, we derive the necessary and sufficient condition, the radius of convexity for this recently introduced class. Furthermore, we prove the sharp upper bounds for the second-order Hankel determinants

| H_{2, 1} ξ |, | H_{2, 2} ξ |

and third-order Hankel determinant

| H_{3, 1} ξ |

for the functions

ξ

belonging to the newly defined class. Full article

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

11 pages, 314 KiB

Open AccessArticle

Certain Class of Close-to-Convex Univalent Functions

by Shatha S. Alhily and Alina Alb Lupas

Symmetry 2023, 15(9), 1789; https://doi.org/10.3390/sym15091789 - 19 Sep 2023

Cited by 1 | Viewed by 1086

Abstract

The purpose of this paper was to define a new class of close-to-convex function, denoted by

C V (δ, α)

, which is a subclass of all functions that are univalent in

D

and have positive coefficients normalized by the [...] Read more.

The purpose of this paper was to define a new class of close-to-convex function, denoted by

C V (δ, α)

, which is a subclass of all functions that are univalent in

D

and have positive coefficients normalized by the conditions

f (0) = 0, f^{'} (0) = 1

, if it satisfies such a condition that is dependent on positive real part. Furthermore, we proved how the power series distribution is essential for determining the sufficient and necessary condition on any function

f

in class

C V (δ, α)

. Full article

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

15 pages, 914 KiB

Open AccessArticle

Estimation of the Bounds of Some Classes of Harmonic Functions with Symmetric Conjugate Points

by Lina Ma, Shuhai Li and Huo Tang

Symmetry 2023, 15(9), 1639; https://doi.org/10.3390/sym15091639 - 25 Aug 2023

Cited by 1 | Viewed by 858

Abstract

In this paper, we introduce some classes of univalent harmonic functions with respect to the symmetric conjugate points by means of subordination, the analytic parts of which are reciprocal starlike (or convex) functions. Further, by combining with the graph of the function, we [...] Read more.

In this paper, we introduce some classes of univalent harmonic functions with respect to the symmetric conjugate points by means of subordination, the analytic parts of which are reciprocal starlike (or convex) functions. Further, by combining with the graph of the function, we discuss the bound of the Bloch constant and the norm of the pre-Schwarzian derivative for the classes. Full article

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

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9 pages, 265 KiB

Open AccessArticle

New Results about Fuzzy Differential Subordinations Associated with Pascal Distribution

by Sheza M. El-Deeb and Luminiţa-Ioana Cotîrlă

Symmetry 2023, 15(8), 1589; https://doi.org/10.3390/sym15081589 - 15 Aug 2023

Cited by 2 | Viewed by 871

Abstract

Based upon the Pascal distribution series [...] Read more.

Based upon the Pascal distribution series

N_{q, λ}^{r, m} Υ (ζ) : = ζ + \sum_{j = m + 1}^{\infty} (\binom{j + r - 2}{r - 1}) [1 + λ (j - 1)] q^{j - 1} {(1 - q)}^{r} a_{j} ζ^{j}

, we can obtain a set of fuzzy differential subordinations in this investigation. We also newly obtain class

P_{q, λ}^{F, r, m} (η)

of univalent analytic functions defined by the operator

N_{q, λ}^{r, m}

, give certain properties for the class

P_{q, λ}^{F, r, m} (η)

and also obtain some applications connected with a special case for the operator. New research directions can be taken on fuzzy differential subordinations associated with symmetry operators. Full article

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

10 pages, 266 KiB

Open AccessArticle

Convex Families of q-Derivative Meromorphic Functions Involving the Polylogarithm Function

by Khadeejah Rasheed Alhindi

Symmetry 2023, 15(7), 1388; https://doi.org/10.3390/sym15071388 - 10 Jul 2023

Cited by 3 | Viewed by 1003

Abstract

The aim of this research study is to establish a novel subclass of meromorphic functions in the mean of q-derivatives in combination with the well-known polylogarithm function. Two additional subfamilies for this class are also defined. Furthermore, the coefficient inequality and distortion [...] Read more.

The aim of this research study is to establish a novel subclass of meromorphic functions in the mean of q-derivatives in combination with the well-known polylogarithm function. Two additional subfamilies for this class are also defined. Furthermore, the coefficient inequality and distortion bounds are highlighted. Finally, the convex families and related set structures are thoroughly investigated. Full article

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

15 pages, 329 KiB

Open AccessArticle

Results on Second-Order Hankel Determinants for Convex Functions with Symmetric Points

by Khalil Ullah, Isra Al-Shbeil, Muhammad Imran Faisal, Muhammad Arif and Huda Alsaud

Symmetry 2023, 15(4), 939; https://doi.org/10.3390/sym15040939 - 19 Apr 2023

Cited by 9 | Viewed by 1803

Abstract

One of the most important problems in the study of geometric function theory is knowing how to obtain the sharp bounds of the coefficients that appear in the Taylor–Maclaurin series of univalent functions. In the present investigation, our aim is to calculate some [...] Read more.

One of the most important problems in the study of geometric function theory is knowing how to obtain the sharp bounds of the coefficients that appear in the Taylor–Maclaurin series of univalent functions. In the present investigation, our aim is to calculate some sharp estimates of problems involving coefficients for the family of convex functions with respect to symmetric points and associated with a hyperbolic tangent function. These problems include the first four initial coefficients, the Fekete–Szegö and Zalcman inequalities, and the second-order Hankel determinant. Additionally, the inverse and logarithmic coefficients of the functions belonging to the defined class are also studied in relation to the current problems. Full article

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

18 pages, 373 KiB

Open AccessArticle

Bi-Starlike Function of Complex Order Involving Mathieu-Type Series Associated with Telephone Numbers

by Kaliappan Vijaya and Gangadharan Murugusundaramoorthy

Symmetry 2023, 15(3), 638; https://doi.org/10.3390/sym15030638 - 3 Mar 2023

Cited by 8 | Viewed by 1526

Abstract

For the first time, we attempted to define two new sub-classes of bi-univalent functions in the open unit disc of the complex order involving Mathieu-type series, associated with generalized telephone numbers. The initial coefficients of functions in these classes were obtained. Moreover, we [...] Read more.

For the first time, we attempted to define two new sub-classes of bi-univalent functions in the open unit disc of the complex order involving Mathieu-type series, associated with generalized telephone numbers. The initial coefficients of functions in these classes were obtained. Moreover, we also determined the Fekete–Szegö inequalities for function in these and several related corollaries. Full article

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

13 pages, 299 KiB

Open AccessArticle

Faber Polynomial Coefficient Estimates for Janowski Type bi-Close-to-Convex and bi-Quasi-Convex Functions

by Shahid Khan, Şahsene Altınkaya, Qin Xin, Fairouz Tchier, Sarfraz Nawaz Malik and Nazar Khan

Symmetry 2023, 15(3), 604; https://doi.org/10.3390/sym15030604 - 27 Feb 2023

Cited by 10 | Viewed by 1638

Abstract

Motivated by the recent work on symmetric analytic functions by using the concept of Faber polynomials, this article introduces and studies two new subclasses of bi-close-to-convex and quasi-close-to-convex functions associated with Janowski functions. By using the Faber polynomial expansion method, it determines the [...] Read more.

Motivated by the recent work on symmetric analytic functions by using the concept of Faber polynomials, this article introduces and studies two new subclasses of bi-close-to-convex and quasi-close-to-convex functions associated with Janowski functions. By using the Faber polynomial expansion method, it determines the general coefficient bounds for the functions belonging to these classes. It also finds initial coefficients of bi-close-to-convex and bi-quasi-convex functions by using Janowski functions. Some known consequences of the main results are also highlighted. Full article

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

13 pages, 298 KiB

Open AccessArticle

Some Properties of Certain Classes of Meromorphic Multivalent Functions Defined by Subordination

by Tamer M. Seoudy and Amnah E. Shammaky

Symmetry 2023, 15(2), 347; https://doi.org/10.3390/sym15020347 - 27 Jan 2023

Cited by 3 | Viewed by 1340

Abstract

In this paper, we define two classes of meromorphic multivalent functions in the punctured disc

U^{*} = \{w \in C : 0 < | w | < 1\}

by using the principle of subordination. We investigate a number of useful results including [...] Read more.

In this paper, we define two classes of meromorphic multivalent functions in the punctured disc

U^{*} = \{w \in C : 0 < | w | < 1\}

by using the principle of subordination. We investigate a number of useful results including subordination results, some connections with a certain integral operator, sandwich properties, an inclusion relationship, and Fekete-Szegö inequalities for the functions belonging these classes. Our results are connected with those in several earlier works, which are related to this field of Geometric Function Theory (GFT) of Complex Analysis. Full article

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

10 pages, 270 KiB

Open AccessArticle

On a New Subclass of q-Starlike Functions Defined in q-Symmetric Calculus

by Asima Razzaque, Saima Noor and Saqib Hussain

Symmetry 2023, 15(2), 334; https://doi.org/10.3390/sym15020334 - 25 Jan 2023

Viewed by 1312

Abstract

Geometric function theory combines geometric tools and their applications for information and communication analysis. It is also successfully used in the field of advanced signals, image processing, machine learning, speech and sound recognition. Various new subclasses of analytic functions have been defined using [...] Read more.

Geometric function theory combines geometric tools and their applications for information and communication analysis. It is also successfully used in the field of advanced signals, image processing, machine learning, speech and sound recognition. Various new subclasses of analytic functions have been defined using quantum calculus to investigate many interesting properties of these subclasses. This article defines a new class of q-starlike functions in the open symmetric unit disc ∇ using symmetric quantum calculus. Extreme points for this class as well as coefficient estimates and closure theorems have been investigated. By fixing several coefficients finitely, all results were generalized into families of analytic functions. Full article

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

20 pages, 333 KiB

Open AccessArticle

Weighted (E_λ, q)(C_λ, 1) Statistical Convergence and Some Results Related to This Type of Convergence

by Ekrem Aljimi, Penpark Sirimark, Astrit Ramizi and Agon Mahmuti

Symmetry 2022, 14(11), 2363; https://doi.org/10.3390/sym14112363 - 9 Nov 2022

Viewed by 1618

Abstract

In this paper, we defined weighted

(E_{λ}, q) (C_{λ}, 1)

statistical convergence. We also proved some properties of this type of statistical convergence by applying [...] Read more.

In this paper, we defined weighted

(E_{λ}, q) (C_{λ}, 1)

statistical convergence. We also proved some properties of this type of statistical convergence by applying

(E_{λ}, q) (C_{λ}, 1)

summability method. Moreover, we used

(E_{λ}, q) (C_{λ}, 1)

summability theorem to prove Korovkin’s type approximation theorem for functions on general and symmetric intervals. We also investigated some of the results of the rate of weighted

(E_{λ}, q) (C_{λ}, 1)

statistical convergence and studied some sequences spaces defined by Orlicz functions. Full article

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

14 pages, 298 KiB

Open AccessArticle

Application of a Multiplier Transformation to Libera Integral Operator Associated with Generalized Distribution

by Jamiu Olusegun Hamzat, Abiodun Tinuoye Oladipo and Georgia Irina Oros

Symmetry 2022, 14(9), 1934; https://doi.org/10.3390/sym14091934 - 16 Sep 2022

Cited by 2 | Viewed by 1283

Abstract

The research presented in this paper deals with analytic p-valent functions related to the generalized probability distribution in the open unit disk U. Using the Hadamard product or convolution, function

f_{s} (z)

is defined as involving an analytic [...] Read more.

The research presented in this paper deals with analytic p-valent functions related to the generalized probability distribution in the open unit disk U. Using the Hadamard product or convolution, function

f_{s} (z)

is defined as involving an analytic p-valent function and generalized distribution expressed in terms of analytic p-valent functions. Neighborhood properties for functions

f_{s} (z)

are established. Further, by applying a previously introduced linear transformation to

f_{s} (z)

and using an extended Libera integral operator, a new generalized Libera-type operator is defined. Moreover, using the same linear transformation, subclasses of starlike, convex, close-to-convex and spiralike functions are defined and investigated in order to obtain geometrical properties that characterize the new generalized Libera-type operator. Symmetry properties are due to the involvement of the Libera integral operator and convolution transform. Full article

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

12 pages, 279 KiB

Open AccessArticle

Symmetric Toeplitz Matrices for a New Family of Prestarlike Functions

by Luminiţa-Ioana Cotîrlă and Abbas Kareem Wanas

Symmetry 2022, 14(7), 1413; https://doi.org/10.3390/sym14071413 - 9 Jul 2022

Cited by 2 | Viewed by 1452

Abstract

By making use of prestarlike functions, we introduce in this paper a certain family of normalized holomorphic functions defined in the open unit disk, and we establish coefficient estimates for the first four determinants of the symmetric Toeplitz matrices [...] Read more.

By making use of prestarlike functions, we introduce in this paper a certain family of normalized holomorphic functions defined in the open unit disk, and we establish coefficient estimates for the first four determinants of the symmetric Toeplitz matrices

T_{2} (2)

,

T_{2} (3)

,

T_{3} (2)

and

T_{3} (1)

for the functions belonging to this family. We also mention some known and new results that follow as special cases of our results. Full article

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

15 pages, 286 KiB

Open AccessArticle

Properties of q-Starlike Functions Associated with the q-Cosine Function

by Mohammad Faisal Khan

Symmetry 2022, 14(6), 1117; https://doi.org/10.3390/sym14061117 - 29 May 2022

Cited by 3 | Viewed by 1494

Abstract

In this paper, our main focus is to define a new subfamily of q-analogue of analytic functions associated with the q-cosine function. Furthermore, we investigate some useful results such as the necessary and sufficient condition based on the convolution idea, growth [...] Read more.

In this paper, our main focus is to define a new subfamily of q-analogue of analytic functions associated with the q-cosine function. Furthermore, we investigate some useful results such as the necessary and sufficient condition based on the convolution idea, growth and distortion bounds, closure theorem, convex combination, radii of starlikeness, extreme point theorem and partial sums results for the newly-defined functions class. Full article

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

17 pages, 341 KiB

Open AccessArticle

Certain Subclasses of Analytic Functions Associated with Generalized Telephone Numbers

by Gangadharan Murugusundaramoorthy and Kaliappan Vijaya

Symmetry 2022, 14(5), 1053; https://doi.org/10.3390/sym14051053 - 20 May 2022

Cited by 14 | Viewed by 1961

Abstract

The goal of this article is to contemplate coefficient estimates for a new class of analytic functions f associated with generalized telephone numbers to originate certain initial Taylor coefficient estimates and Fekete–Szegö inequality for f in the new function class. Comparable results have [...] Read more.

The goal of this article is to contemplate coefficient estimates for a new class of analytic functions f associated with generalized telephone numbers to originate certain initial Taylor coefficient estimates and Fekete–Szegö inequality for f in the new function class. Comparable results have been attained for the function

f^{- 1} .

Further application of our outcomes to certain functions demarcated by convolution products with certain normalized analytic functions in the open unit disk are specified, and we obtain Fekete–Szegö variations for this new function class defined over Poisson and Borel distribution series. Full article

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

15 pages, 291 KiB

Open AccessArticle

On a Certain Subclass of Analytic Functions Defined by Touchard Polynomials

by Bolenini Venkateswarlu, Pinninti Thirupathi Reddy, Şahsene Altınkaya, Nattakan Boonsatit, Porpattama Hammachukiattikul and Vaishnavy Sujatha

Symmetry 2022, 14(4), 838; https://doi.org/10.3390/sym14040838 - 18 Apr 2022

Cited by 4 | Viewed by 2426

Abstract

This paper focuses on the establishment of a new subfamily of analytic functions including Touchard polynomials. Then, we attempt to obtain geometric properties such as coefficient inequalities, distortion properties, extreme points, radii of starlikeness and convexity, partial sums, neighbourhood results and integral means’ [...] Read more.

This paper focuses on the establishment of a new subfamily of analytic functions including Touchard polynomials. Then, we attempt to obtain geometric properties such as coefficient inequalities, distortion properties, extreme points, radii of starlikeness and convexity, partial sums, neighbourhood results and integral means’ inequality for this class. The symmetry properties of the subfamily of functions established in the current paper may be examined as future research directions. Full article

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

20 pages, 332 KiB

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 7 | Viewed by 1829

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)

11 pages, 266 KiB

Open AccessArticle

Quasi-Hadamard Product and Partial Sums for Sakaguchi-Type Function Classes Involving q-Difference Operator

by Asena Çetinkaya and Luminiţa-Ioana Cotîrlă

Symmetry 2022, 14(4), 709; https://doi.org/10.3390/sym14040709 - 31 Mar 2022

Cited by 4 | Viewed by 1778

Abstract

We create two Sakaguchi-type function classes that are starlike and convex with respect to their symmetric points, including a q-difference operator, which may have symmetric or assymetric properties, in the open unit disc. We first obtain sufficient coefficient bounds for these functions. [...] Read more.

We create two Sakaguchi-type function classes that are starlike and convex with respect to their symmetric points, including a q-difference operator, which may have symmetric or assymetric properties, in the open unit disc. We first obtain sufficient coefficient bounds for these functions. In view of these bounds, we obtain quasi-Hadamard products and several partial sums for these function classes. Moreover, the special values of the parameters provided the corresponding consequences of the partial sums. Full article

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

10 pages, 272 KiB

Open AccessArticle

q-Analogue of a New Subclass of Harmonic Univalent Functions Associated with Subordination

by Hasan Bayram

Symmetry 2022, 14(4), 708; https://doi.org/10.3390/sym14040708 - 31 Mar 2022

Cited by 5 | Viewed by 1866

Abstract

In this article, we introduce and investigate the q-analogue of a new subclass of harmonic univalent functions defined by subordination. We first obtain a coefficient characterization of these functions. We give compactness and extreme points, distortion bounds, necessary and sufficient convolution conditions for [...] Read more.

In this article, we introduce and investigate the q-analogue of a new subclass of harmonic univalent functions defined by subordination. We first obtain a coefficient characterization of these functions. We give compactness and extreme points, distortion bounds, necessary and sufficient convolution conditions for this subclass of harmonic univalent functions with negative coefficients. The symmetry properties and other properties of the q-analogue subclass of functions presented in this paper shed light on future studies. Full article

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

17 pages, 301 KiB

Open AccessArticle

Fourth Hankel Determinant Problem Based on Certain Analytic Functions

by Huo Tang, Muhammad Arif, Mirajul Haq, Nazar Khan, Mustaqeem Khan, Khurshid Ahmad and Bilal Khan

Symmetry 2022, 14(4), 663; https://doi.org/10.3390/sym14040663 - 24 Mar 2022

Cited by 5 | Viewed by 2045

Abstract

In recent years, the Hankel determinant bounds for different subclasses of analytic, starlike and symmetric starlike functions have been discussed and studied by the many well-known authors. In this paper, we first consider a new subclass of analytic function and then we derive [...] Read more.

In recent years, the Hankel determinant bounds for different subclasses of analytic, starlike and symmetric starlike functions have been discussed and studied by the many well-known authors. In this paper, we first consider a new subclass of analytic function and then we derive the fourth Hankel determinant bound for this class. Full article

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

14 pages, 287 KiB

Open AccessArticle

Applications of the Atangana–Baleanu Fractional Integral Operator

by Alina Alb Lupaş and Adriana Cătaş

Symmetry 2022, 14(3), 630; https://doi.org/10.3390/sym14030630 - 21 Mar 2022

Cited by 7 | Viewed by 2460

Abstract

Applications of the Atangana–Baleanu fractional integral were considered in recent studies related to geometric function theory to obtain interesting differential subordinations. Additionally, the multiplier transformation was used in many studies, providing elegant results. In this paper, a new operator is defined by combining [...] Read more.

Applications of the Atangana–Baleanu fractional integral were considered in recent studies related to geometric function theory to obtain interesting differential subordinations. Additionally, the multiplier transformation was used in many studies, providing elegant results. In this paper, a new operator is defined by combining those two prolific functions. The newly defined operator is applied for introducing a new subclass of analytic functions, which is investigated concerning certain properties, such as coefficient estimates, distortion theorems, closure theorems, neighborhoods and radii of starlikeness, convexity and close-to-convexity. This class may have symmetric or asymmetric properties. The results could prove interesting due to the new applications of the Atangana–Baleanu fractional integral and of the multiplier transformation. Additionally, the univalence properties of the new subclass of functions could inspire researchers to conduct further investigations related to this newly defined class. Full article

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

16 pages, 298 KiB

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 3 | Viewed by 1731

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)

17 pages, 319 KiB

Open AccessArticle

On Third-Order Differential Subordination and Superordination Properties of Analytic Functions Defined by a Generalized Operator

by Waggas Galib Atshan, Rajaa Ali Hiress and Sahsene Altınkaya

Symmetry 2022, 14(2), 418; https://doi.org/10.3390/sym14020418 - 19 Feb 2022

Cited by 15 | Viewed by 2100

Abstract

In this current study, we aim to give some results for third-order differential subordination and superordination for analytic functions in

U = {z \in ℂ : | z | < 1}

involving the generalized operator [...] Read more.

In this current study, we aim to give some results for third-order differential subordination and superordination for analytic functions in

U = {z \in ℂ : | z | < 1}

involving the generalized operator

I_{α, β}^{j} f

. The results are derived by investigating relevant classes of admissible functions. Some new results on differential subordination and superordination with some sandwich theorems are obtained. Moreover, several particular cases are also noted. The properties and results of the differential subordination are symmetry to the properties of the differential superordination to form the sandwich theorems. Full article

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

13 pages, 274 KiB

Open AccessArticle

Applications of Borel-Type Distributions Series to a Class of Janowski-Type Analytic Functions

by Bakhtiar Ahmad, Muhammad Ghaffar Khan and Luminiţa-Ioana Cotîrlă

Symmetry 2022, 14(2), 322; https://doi.org/10.3390/sym14020322 - 4 Feb 2022

Cited by 2 | Viewed by 1603

Abstract

The main purpose of this article is to introduce the new subclass of analytic functions whose coefficients are Borel distributions in the Janowski domain. Further, we investigate some useful number of properties such as Fekete–Szegő inequality, necessary and sufficient condition, growth and distortion [...] Read more.

The main purpose of this article is to introduce the new subclass of analytic functions whose coefficients are Borel distributions in the Janowski domain. Further, we investigate some useful number of properties such as Fekete–Szegő inequality, necessary and sufficient condition, growth and distortion approximations, convex linear combination, arithmetic mean, radii of close-to-convexity and starlikeness and partial sums, followed by some extremal functions for this defined class. The symmetry properties and other properties of the subclass of functions introduced in this paper can be studied as future research directions. Full article

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

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