Axioms

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11 pages, 271 KiB

Open AccessArticle

Second Hankel Determinant Bound Application to Certain Family of Bi-Univalent Functions

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

Axioms 2024, 13(12), 819; https://doi.org/10.3390/axioms13120819 (registering DOI) - 24 Nov 2024

Viewed by 293

Abstract

A novel family of bi-univalent holomorphic functions is introduced by the use of the Lindelöf principle. The upper bound of the second Hankel determinant,

H_{2, 2} (χ)

, is evaluated. Furthermore, specific results are obtained as special cases of [...] Read more.

A novel family of bi-univalent holomorphic functions is introduced by the use of the Lindelöf principle. The upper bound of the second Hankel determinant,

H_{2, 2} (χ)

, is evaluated. Furthermore, specific results are obtained as special cases of the main conclusion. These cases coincide with certain recently obtained results and improve or enhance specific ones. Full article

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

12 pages, 300 KiB

Open AccessArticle

Initial Coefficient Bounds for Bi-Close-to-Convex Classes of n-Fold-Symmetric Bi-Univalent Functions

by P. Gurusamy, M. Çağlar, L. I. Cotirla and S. Sivasubramanian

Axioms 2024, 13(11), 735; https://doi.org/10.3390/axioms13110735 - 25 Oct 2024

Viewed by 466

Abstract

In this article, the strong class of bi-close-to-convex functions of order

α

and

β

in n-fold symmetric bi-univalent functions, which is the subclass of

σ

, is introduced. The upper bound value for

|a_{n + 1}|

, [...] Read more.

In this article, the strong class of bi-close-to-convex functions of order

α

and

β

in n-fold symmetric bi-univalent functions, which is the subclass of

σ

, is introduced. The upper bound value for

|a_{n + 1}|

,

|a_{2 n + 1}|

for functions in these classes are obtained. Moreover, the Fekete–Szegö relation for our classes of functions are established. Full article

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

12 pages, 299 KiB

Open AccessArticle

Sălăgean Differential Operator in Connection with Stirling Numbers

by Basem Aref Frasin and Luminiţa-Ioana Cotîrlă

Axioms 2024, 13(9), 620; https://doi.org/10.3390/axioms13090620 - 12 Sep 2024

Viewed by 449

Abstract

Sălăgean differential operator

D^{κ}

plays an important role in the geometric function theory, where many studies are using this operator to introduce new subclasses of analytic functions defined in the open unit disk. Studies of Sălăgean differential operator

D^{κ}

in connection [...] Read more.

Sălăgean differential operator

D^{κ}

plays an important role in the geometric function theory, where many studies are using this operator to introduce new subclasses of analytic functions defined in the open unit disk. Studies of Sălăgean differential operator

D^{κ}

in connection with Stirling numbers are relatively new. In this paper, the differential operator

D^{κ}

involving Stirling numbers is considered. A new sufficient condition involving Stirling numbers for the series

Υ_{θ}^{s} (ϰ)

written with the Pascal distribution are discussed for the subclass

T_{κ} (ϵ, ♭)

. Also, we provide a sufficient condition for the inclusion relation

I_{θ}^{s} (R^{ϖ} (E, D)) \subset T_{κ} (ϵ, ♭)

. Further, we consider the properties of an integral operator related to Pascal distribution series. New special cases as a consequences of the main results are also obtained. Full article

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

13 pages, 2069 KiB

Open AccessArticle

Analysis of a Normalized Structure of a Complex Fractal–Fractional Integral Transform Using Special Functions

by Rabha W. Ibrahim, Soheil Salahshour and Ágnes Orsolya Páll-Szabó

Axioms 2024, 13(8), 522; https://doi.org/10.3390/axioms13080522 - 2 Aug 2024

Viewed by 523

Abstract

By using the most generalized gamma function (parametric gamma function, or p-gamma function), we present the most generalized Rabotnov function, called the p-Rabotnov function. Consequently, new fractal–fractional differential and integral operators of a complex variable in an open unit disk are [...] Read more.

By using the most generalized gamma function (parametric gamma function, or p-gamma function), we present the most generalized Rabotnov function, called the p-Rabotnov function. Consequently, new fractal–fractional differential and integral operators of a complex variable in an open unit disk are defined and investigated analytically and geometrically. We address some inequalities involving the generalized fractal–fractional integral operator in some spaces of analytic functions. A novel complex fractal–fractional integral transform (CFFIT) is presented. A normalization of the proposed CFFIT is observed in the open unit disk. Examples are illustrated for power series of analytic functions. Full article

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

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8 pages, 242 KiB

Open AccessArticle

Toeplitz Matrices for a Class of Bazilevič Functions and the λ-Pseudo-Starlike Functions

by Abbas Kareem Wanas, Salam Abdulhussein Sehen and Ágnes Orsolya Páll-Szabó

Axioms 2024, 13(8), 521; https://doi.org/10.3390/axioms13080521 - 2 Aug 2024

Viewed by 503

Abstract

In the present paper, we define and study a new family of holomorphic functions which involve the Bazilevič functions and the

λ

-pseudo-starlike functions. We establish coefficient estimates for the first four determinants of the symmetric Toeplitz matrices

T_{2} (2)

[...] Read more.

In the present paper, we define and study a new family of holomorphic functions which involve the Bazilevič functions and the

λ

-pseudo-starlike functions. We establish coefficient estimates for the first four determinants of the symmetric Toeplitz matrices

T_{2} (2)

,

T_{2} (3)

,

T_{3} (1)

and

T_{3} (2)

for the functions in this family. Further, we investigate several special cases and consequences of our results. Full article

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

14 pages, 280 KiB

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

Viewed by 564

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)

19 pages, 297 KiB

Open AccessArticle

Certain New Applications of Symmetric q-Calculus for New Subclasses of Multivalent Functions Associated with the Cardioid Domain

by Hari M. Srivastava, Daniel Breaz, Shahid Khan and Fairouz Tchier

Axioms 2024, 13(6), 366; https://doi.org/10.3390/axioms13060366 - 29 May 2024

Viewed by 796

Abstract

In this work, we study some new applications of symmetric quantum calculus in the field of Geometric Function Theory. We use the cardioid domain and the symmetric quantum difference operator to generate new classes of multivalent q-starlike and q-convex functions. We [...] Read more.

In this work, we study some new applications of symmetric quantum calculus in the field of Geometric Function Theory. We use the cardioid domain and the symmetric quantum difference operator to generate new classes of multivalent q-starlike and q-convex functions. We examine a wide range of interesting properties for functions that can be classified into these newly defined classes, such as estimates for the bounds for the first two coefficients, Fekete–Szego-type functional and coefficient inequalities. All the results found in this research are sharp. A number of well-known corollaries are additionally taken into consideration to show how the findings of this research relate to those of earlier studies. Full article

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

19 pages, 521 KiB

Open AccessArticle

Subclasses of Analytic Functions Subordinated to the Four-Leaf Function

by Saravanan Gunasekar, Baskaran Sudharsanan, Musthafa Ibrahim and Teodor Bulboacă

Axioms 2024, 13(3), 155; https://doi.org/10.3390/axioms13030155 - 27 Feb 2024

Cited by 2 | Viewed by 1494

Abstract

The purpose of this research is to unify and extend the study of the well-known concept of coefficient estimates for some subclasses of analytic functions. We define the new subclass

A_{4}^{r, s}

of analytic functions related to the four-leaf domain, [...] Read more.

The purpose of this research is to unify and extend the study of the well-known concept of coefficient estimates for some subclasses of analytic functions. We define the new subclass

A_{4}^{r, s}

of analytic functions related to the four-leaf domain, to increase the adaptability of our investigation. The initial findings are the bound estimates for the coefficients

| a_{n} |

,

n = 2, 3, 4, 5

, among which the bound of

| a_{2} |

is sharp. Also, we include the sharp-function illustration. Additionally, we obtain the upper-bound estimate for the second Hankel determinant for this subclass as well as those for the Fekete–Szegő functional. Finally, for these subclasses, we provide an estimation of the Krushkal inequality for the function class

A_{4}^{r, s}

. Full article

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

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