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17 pages, 313 KB

Open AccessArticle

Faber Polynomial Coefficient Estimates of m-Fold Symmetric Bi-Univalent Functions with Bounded Boundary Rotation

by Anandan Murugan, Srikandan Sivasubramanian, Prathviraj Sharma and Gangadharan Murugusundaramoorthy

Mathematics 2024, 12(24), 3963; https://doi.org/10.3390/math12243963 - 17 Dec 2024

Viewed by 1050

In the current article, we introduce several new subclasses of m-fold symmetric analytic and bi-univalent functions associated with bounded boundary and bounded radius rotation within the open unit disk

D .

Utilizing the Faber polynomial expansion, we derive upper bounds for the [...] Read more.

In the current article, we introduce several new subclasses of m-fold symmetric analytic and bi-univalent functions associated with bounded boundary and bounded radius rotation within the open unit disk

D .

Utilizing the Faber polynomial expansion, we derive upper bounds for the coefficients

| b_{m k + 1} |

and establish initial coefficient bounds for

| b_{m + 1} |

and

| b_{2 m + 1} | .

Additionally, we explore the Fekete–Szegö inequalities applicable to the functions that fall within these newly defined subclasses. Full article

(This article belongs to the Section B: Geometry and Topology)

14 pages, 280 KB

Open AccessArticle

Coefficient Estimates for New Subclasses of Bi-Univalent Functions with Bounded Boundary Rotation by Using Faber Polynomial Technique

by Huo Tang, Prathviraj Sharma and Srikandan Sivasubramanian

Axioms 2024, 13(8), 509; https://doi.org/10.3390/axioms13080509 - 28 Jul 2024

Cited by 2 | Viewed by 1315

Abstract

In this article, the authors use the Faber polynomial expansions to find the general coefficient estimates for a few new subclasses of bi-univalent functions with bounded boundary rotation and bounded radius rotation. Some of the results improve the existing coefficient bounds in the [...] Read more.

In this article, the authors use the Faber polynomial expansions to find the general coefficient estimates for a few new subclasses of bi-univalent functions with bounded boundary rotation and bounded radius rotation. Some of the results improve the existing coefficient bounds in the literature. Full article

(This article belongs to the Special Issue Advances in Geometric Function Theory and Related Topics)

16 pages, 329 KB

Open AccessArticle

Faber Polynomial Coefficient Inequalities for a Subclass of Bi-Close-To-Convex Functions Associated with Fractional Differential Operator

by Ferdous M. O. Tawfiq, Fairouz Tchier and Luminita-Ioana Cotîrlă

Fractal Fract. 2023, 7(12), 883; https://doi.org/10.3390/fractalfract7120883 - 14 Dec 2023

Cited by 2 | Viewed by 1610

Abstract

In this study, we begin by examining the

τ

-fractional differintegral operator and proceed to establish a novel subclass in the open unit disk E. The determination of the nth coefficient bound for functions within this recently established class is accomplished [...] Read more.

In this study, we begin by examining the

τ

-fractional differintegral operator and proceed to establish a novel subclass in the open unit disk E. The determination of the nth coefficient bound for functions within this recently established class is accomplished by the use of the Faber polynomial expansion approach. Additionally, we examine the behavior of the initial coefficients of bi-close-to-convex functions defined by the

τ

-fractional differintegral operator, which may exhibit unexpected reactions. We established connections between our current research and prior studies in order to validate our significant findings. Full article

(This article belongs to the Section General Mathematics, Analysis)

18 pages, 336 KB

Open AccessArticle

Certain New Applications of Faber Polynomial Expansion for a New Class of bi-Univalent Functions Associated with Symmetric q-Calculus

by Chetan Swarup

Symmetry 2023, 15(7), 1407; https://doi.org/10.3390/sym15071407 - 13 Jul 2023

Viewed by 1348

Abstract

In this study, we applied the ideas of subordination and the symmetric q-difference operator and then defined the novel class of bi-univalent functions of complex order

γ

. We used the Faber polynomial expansion method to determine the upper bounds for the [...] Read more.

In this study, we applied the ideas of subordination and the symmetric q-difference operator and then defined the novel class of bi-univalent functions of complex order

γ

. We used the Faber polynomial expansion method to determine the upper bounds for the functions belonging to the newly defined class of complex order

γ

. For the functions in the newly specified class, we further obtained coefficient bounds

|ρ_{2}|

and the Fekete–Szegő problem

|ρ_{3} - ρ_{2}^{2}|

, both of which have been restricted by gap series. We demonstrate many applications of the symmetric Sălăgean q-differential operator using the Faber polynomial expansion technique. The findings in this paper generalize those from previous studies. Full article

18 pages, 339 KB

Open AccessArticle

New Applications of Faber Polynomials and q-Fractional Calculus for a New Subclass of m-Fold Symmetric bi-Close-to-Convex Functions

by Mohammad Faisal Khan, Suha B. Al-Shaikh, Ahmad A. Abubaker and Khaled Matarneh

Axioms 2023, 12(6), 600; https://doi.org/10.3390/axioms12060600 - 16 Jun 2023

Cited by 2 | Viewed by 1583

Abstract

Using the concepts of q-fractional calculus operator theory, we first define a

(λ, q)

-differintegral operator, and we then use m-fold symmetric functions to discover a new family of bi-close-to-convex functions. First, we estimate the general Taylor–Maclaurin coefficient [...] Read more.

Using the concepts of q-fractional calculus operator theory, we first define a

(λ, q)

-differintegral operator, and we then use m-fold symmetric functions to discover a new family of bi-close-to-convex functions. First, we estimate the general Taylor–Maclaurin coefficient bounds for a newly established class using the Faber polynomial expansion method. In addition, the Faber polynomial method is used to examine the Fekete–Szegö problem and the unpredictable behavior of the initial coefficient bounds of the functions that belong to the newly established class of m-fold symmetric bi-close-to-convex functions. Our key results are both novel and consistent with prior research, so we highlight a few of their important corollaries for a comparison. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus)

19 pages, 352 KB

Open AccessArticle

Faber Polynomial Coefficient Estimates for Bi-Close-to-Convex Functions Defined by the q-Fractional Derivative

by Hari Mohan Srivastava, Isra Al-Shbeil, Qin Xin, Fairouz Tchier, Shahid Khan and Sarfraz Nawaz Malik

Axioms 2023, 12(6), 585; https://doi.org/10.3390/axioms12060585 - 13 Jun 2023

Cited by 12 | Viewed by 2164

Abstract

By utilizing the concept of the q-fractional derivative operator and bi-close-to-convex functions, we define a new subclass of

A

, where the class

A

contains normalized analytic functions in the open unit disk

E

and is invariant or symmetric under rotation. First, [...] Read more.

By utilizing the concept of the q-fractional derivative operator and bi-close-to-convex functions, we define a new subclass of

A

, where the class

A

contains normalized analytic functions in the open unit disk

E

and is invariant or symmetric under rotation. First, using the Faber polynomial expansion (FPE) technique, we determine the lth coefficient bound for the functions contained within this class. We provide a further explanation for the first few coefficients of bi-close-to-convex functions defined by the q-fractional derivative. We also emphasize upon a few well-known outcomes of the major findings in this article. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

12 pages, 325 KB

Open AccessArticle

Coefficient Estimation Utilizing the Faber Polynomial for a Subfamily of Bi-Univalent Functions

by Abdullah Alsoboh, Ala Amourah, Fethiye Müge Sakar, Osama Ogilat, Gharib Mousa Gharib and Nasser Zomot

Axioms 2023, 12(6), 512; https://doi.org/10.3390/axioms12060512 - 24 May 2023

Cited by 24 | Viewed by 2257

Abstract

The paper introduces a new family of analytic bi-univalent functions that are injective and possess analytic inverses, by employing a q-analogue of the derivative operator. Moreover, the article establishes the upper bounds of the Taylor–Maclaurin coefficients of these functions, which can aid [...] Read more.

The paper introduces a new family of analytic bi-univalent functions that are injective and possess analytic inverses, by employing a q-analogue of the derivative operator. Moreover, the article establishes the upper bounds of the Taylor–Maclaurin coefficients of these functions, which can aid in approximating the accuracy of approximations using a finite number of terms. The upper bounds are obtained by approximating analytic functions using Faber polynomial expansions. These bounds apply to both the initial few coefficients and all coefficients in the series, making them general and early, respectively. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory II)

18 pages, 348 KB

Open AccessArticle

Some New Applications of the Faber Polynomial Expansion Method for Generalized Bi-Subordinate Functions of Complex Order γ Defined by q-Calculus

by Mohammad Faisal Khan and Mohammed AbaOud

Fractal Fract. 2023, 7(3), 270; https://doi.org/10.3390/fractalfract7030270 - 18 Mar 2023

Viewed by 1846

Abstract

This work examines a new subclass of generalized bi-subordinate functions of complex order

γ

connected to the q-difference operator. We obtain the upper bounds

ρ_{m}

for generalized bi-subordinate functions of complex order

γ

using the Faber polynomial expansion technique. Additionally, we [...] Read more.

This work examines a new subclass of generalized bi-subordinate functions of complex order

γ

connected to the q-difference operator. We obtain the upper bounds

ρ_{m}

for generalized bi-subordinate functions of complex order

γ

using the Faber polynomial expansion technique. Additionally, we find coefficient bounds

|ρ_{2}|

and Feke–Sezgo problems

|ρ_{3} - ρ_{2}^{2}|

for the functions in the newly defined class, subject to gap series conditions. Using the Faber polynomial expansion method, we show some results that illustrate diverse uses of the Ruschewey q differential operator. The findings in this paper generalize those from previous efforts by a number of prior researchers. Full article

(This article belongs to the Special Issue Fractional Calculus and Hypergeometric Functions in Complex Analysis)

15 pages, 315 KB

Open AccessArticle

New Applications of Faber Polynomial Expansion for Analytical Bi-Close-to-Convex Functions Defined by Using q-Calculus

by Ridong Wang, Manoj Singh, Shahid Khan, Huo Tang, Mohammad Faisal Khan and Mustafa Kamal

Mathematics 2023, 11(5), 1217; https://doi.org/10.3390/math11051217 - 1 Mar 2023

Cited by 3 | Viewed by 2026

Abstract

In this investigation, the q-difference operator and the Sălăgean q-differential operator are utilized to establish novel subclasses of analytical bi-close-to-convex functions. We determine the general Taylor–Maclaurin coefficient of the functions in this class using the Faber polynomial method. We demonstrate the [...] Read more.

In this investigation, the q-difference operator and the Sălăgean q-differential operator are utilized to establish novel subclasses of analytical bi-close-to-convex functions. We determine the general Taylor–Maclaurin coefficient of the functions in this class using the Faber polynomial method. We demonstrate the unpredictable behaviour of initial coefficients

|a_{2}|

,

|a_{3}|

and investigate the Fekete–Szegő problem

|a_{3} - a_{2}^{2}|

for the subclasses of bi-close-to-convex functions. To highlight the connections between existing knowledge and new research, certain known and unknown corollaries are also highlighted. Full article

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

13 pages, 299 KB

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 2295

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)

9 pages, 292 KB

Open AccessArticle

A Family of Analytic and Bi-Univalent Functions Associated with Srivastava-Attiya Operator

by Adel A. Attiya and Mansour F. Yassen

Symmetry 2022, 14(10), 2006; https://doi.org/10.3390/sym14102006 - 25 Sep 2022

Cited by 2 | Viewed by 1750

Abstract

In this paper, we investigate a new family of normalized analytic functions and bi-univalent functions associated with the Srivastava–Attiya operator. We use the Faber polynomial expansion to estimate the bounds for the general coefficients

| a_{n} |

of this family. The bounds [...] Read more.

In this paper, we investigate a new family of normalized analytic functions and bi-univalent functions associated with the Srivastava–Attiya operator. We use the Faber polynomial expansion to estimate the bounds for the general coefficients

| a_{n} |

of this family. The bounds values for the initial Taylor–Maclaurin coefficients of the functions in this family are also established. Full article

(This article belongs to the Special Issue Symmetry in Geometric Function Theory)

11 pages, 274 KB

Open AccessArticle

Faber Polynomial Coefficient Estimates of Bi-Close-to-Convex Functions Associated with Generalized Hypergeometric Functions

by Jie Zhai, Rekha Srivastava and Jin-Lin Liu

Mathematics 2022, 10(17), 3073; https://doi.org/10.3390/math10173073 - 25 Aug 2022

Cited by 3 | Viewed by 1738

Abstract

A new subclass of bi-close-to-convex functions associated with the generalized hypergeometric functions defined in ∆

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

is introduced. The estimates for the general Taylor–Maclaurin coefficients of the functions in the introduced subclass [...] Read more.

A new subclass of bi-close-to-convex functions associated with the generalized hypergeometric functions defined in ∆

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

is introduced. The estimates for the general Taylor–Maclaurin coefficients of the functions in the introduced subclass are obtained by making use of Faber polynomial expansions. In particular, several previous results are generalized. Full article

13 pages, 297 KB

Open AccessFeature PaperArticle

Coefficients of a Comprehensive Subclass of Meromorphic Bi-Univalent Functions Associated with the Faber Polynomial Expansion

by Hari Mohan Srivastava, Ahmad Motamednezhad and Safa Salehian

Axioms 2021, 10(1), 27; https://doi.org/10.3390/axioms10010027 - 27 Feb 2021

Cited by 12 | Viewed by 3205

Abstract

In this paper, we introduce a new comprehensive subclass

Σ_{B} (λ, μ, β)

of meromorphic bi-univalent functions in the open unit disk

U

. We also find the upper bounds for the initial Taylor-Maclaurin coefficients [...] Read more.

In this paper, we introduce a new comprehensive subclass

Σ_{B} (λ, μ, β)

of meromorphic bi-univalent functions in the open unit disk

U

. We also find the upper bounds for the initial Taylor-Maclaurin coefficients

| b_{0} |

,

| b_{1} |

and

| b_{2} |

for functions in this comprehensive subclass. Moreover, we obtain estimates for the general coefficients

| b_{n} | (n ≧ 1)

for functions in the subclass

Σ_{B} (λ, μ, β)

by making use of the Faber polynomial expansion method. The results presented in this paper would generalize and improve several recent works on the subject. Full article

(This article belongs to the Collection Mathematical Analysis and Applications)

23 pages, 382 KB

Open AccessArticle

Coefficient Bounds for Certain Classes of Analytic Functions Associated with Faber Polynomial

by Adel A. Attiya, Abdel Moneim Lashin, Ekram E. Ali and Praveen Agarwal

Symmetry 2021, 13(2), 302; https://doi.org/10.3390/sym13020302 - 10 Feb 2021

Cited by 15 | Viewed by 3378

Abstract

In this paper, we intorduce a family of analytic functions in the open unit disk which is bi-univalent. By the virtue of the Faber polynomial expansions, the estimation of

n - t h

(n \geq 3)

Taylor–Maclaurin coefficients

|a_{n}|

is [...] Read more.

In this paper, we intorduce a family of analytic functions in the open unit disk which is bi-univalent. By the virtue of the Faber polynomial expansions, the estimation of

n - t h

(n \geq 3)

Taylor–Maclaurin coefficients

|a_{n}|

is obtained. Furthermore, the bounds value of the first two coefficients of such functions is established. Full article

(This article belongs to the Special Issue Advanced Calculus in Problems with Symmetry)

12 pages, 264 KB

Open AccessArticle

Faber Polynomial Coefficient Estimates for Bi-Univalent Functions Defined by Using Differential Subordination and a Certain Fractional Derivative Operator

by Hari M. Srivastava, Ahmad Motamednezhad and Ebrahim Analouei Adegani

Mathematics 2020, 8(2), 172; https://doi.org/10.3390/math8020172 - 1 Feb 2020

Cited by 70 | Viewed by 3803

Abstract

In this article, we introduce a general family of analytic and bi-univalent functions in the open unit disk, which is defined by applying the principle of differential subordination between analytic functions and the Tremblay fractional derivative operator. The upper bounds for the general [...] Read more.

In this article, we introduce a general family of analytic and bi-univalent functions in the open unit disk, which is defined by applying the principle of differential subordination between analytic functions and the Tremblay fractional derivative operator. The upper bounds for the general coefficients

| a_{n} |

of functions in this subclass are found by using the Faber polynomial expansion. We have thereby generalized and improved some of the previously published results. Full article

(This article belongs to the Special Issue Complex Analysis and Its Applications)

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