MDPI - Publisher of Open Access Journals

23 pages, 512 KB

Open AccessArticle

Toeplitz and Hankel Determinants for Certain Subclasses Associated with Sakaguchi Type Functions

by Mohammed Ali Alamri, Adriana Catas, Bushra Kanwal, Arooj Iman, Fethiye Müge Sakar and Saqib Hussain

Math. Comput. Appl. 2026, 31(4), 141; https://doi.org/10.3390/mca31040141 - 20 Jul 2026

Viewed by 426

Abstract

In this work, we establish bounds for Hermitian Toeplitz determinants of orders two and three for distinct subclasses of symmetric starlike functions associated with balloon-, limaçon-, and bean-shaped domains. Using subordination theory, we derive explicit upper and lower bounds for Toeplitz determinants of [...] Read more.

In this work, we establish bounds for Hermitian Toeplitz determinants of orders two and three for distinct subclasses of symmetric starlike functions associated with balloon-, limaçon-, and bean-shaped domains. Using subordination theory, we derive explicit upper and lower bounds for Toeplitz determinants of order two and three for each class. Our results reveal a clear geometric hierarchy: the bean-shaped domain imposes the tightest restrictions, while the limaçon- domain permits the widest variation. The analysis is further extended to 2-fold and 3-fold symmetric functions, for which bounds for the third Hankel determinant are obtained. This work demonstrates how symmetry and domain geometry jointly govern coefficient estimates, offering new insights into the interplay between shape and analytic structure. Full article

► Show Figures

Figure 1

16 pages, 2232 KB

Open AccessArticle

Coefficient Inequalities and Geometric Behavior of a Sakaguchi-Type Bi-Univalent Class Associated with the Four-Leaf Domain

by Arzu Akgül, Gangadharan Murugusundaramoorthy and Yasemin Demirel

Axioms 2026, 15(6), 453; https://doi.org/10.3390/axioms15060453 - 17 Jun 2026

Viewed by 595

Abstract

The main objective of this paper is to introduce and investigate a new subclass of Sakaguchi-type analytic bi-univalent functions in the open unit disk, defined by a two-sided subordination condition to the four-leaf function [...] Read more.

The main objective of this paper is to introduce and investigate a new subclass of Sakaguchi-type analytic bi-univalent functions in the open unit disk, defined by a two-sided subordination condition to the four-leaf function

Λ_{4 L} (z) = 1 + \frac{5}{6} z + \frac{1}{6} z^{5}

, whose image is a four-leaf-shaped domain. For suitable choices of the complex parameter b with

| b | \leq 1

,

b \neq 1

, and the real parameters

α \in [0, 1]

and

ρ \geq 0

, we study the class

G_{Σ, b}^{α, ρ} (Λ_{4 L})

of bi-univalent functions f for which both f and its inverse

g = f^{- 1}

satisfy a Sakaguchi-type subordination condition involving the fundamental ratios

f (z) / z

and

f^{'} (z)

together with the difference quotient

{(f (z) - f (b z))}^{- 1}

. Coefficient inequalities for the initial Taylor–Maclaurin coefficients are obtained, and Fekete–Szegő-type estimates are derived for the associated coefficient functional. Furthermore, graphical representations of the image domains are provided, which illustrate the geometric behavior of the functions in

G_{Σ, b}^{α, ρ} (Λ_{4 L})

and confirm that the proposed class is non-trivial. The present investigation extends recent studies on subclasses of analytic and bi-univalent functions associated with geometrically structured mappings and contributes to the visual interpretation of their analytic properties. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 4th Edition)

► Show Figures

Figure 1

22 pages, 810 KB

Open AccessArticle

Gregory Polynomials Within Sakaguchi-Type Function Classes: Analytical Estimates and Geometric Behavior

by Arzu Akgül and Georgia Irina Oros

Symmetry 2025, 17(6), 884; https://doi.org/10.3390/sym17060884 - 5 Jun 2025

Cited by 3 | Viewed by 1120

Abstract

This work introduces a novel family of analytic and univalent functions formulated through the integration of Gregory coefficients and Sakaguchi-type functions. Employing subordination techniques, we obtain sharp bounds for the initial coefficients in their Taylor expansions. The influence of parameter variations is examined [...] Read more.

This work introduces a novel family of analytic and univalent functions formulated through the integration of Gregory coefficients and Sakaguchi-type functions. Employing subordination techniques, we obtain sharp bounds for the initial coefficients in their Taylor expansions. The influence of parameter variations is examined through comprehensive geometric visualizations, which confirm the non-emptiness of the class and provide insights into its structural properties. Furthermore, Fekete–Szegö inequalities are established, enriching the theory of bi-univalent functions. The combination of analytical methods and geometric representations offers a versatile framework for future research in geometric function theory. Full article

(This article belongs to the Special Issue Applications of Special Functions in Complex Analysis and Their Symmetries)

► Show Figures

Figure 1

14 pages, 322 KB

Open AccessArticle

Coefficient Functionals of Sakaguchi-Type Starlike Functions Involving Caputo-Type Fractional Derivatives Subordinated to the Three-Leaf Function

by Kholood M. Alsager, Sheza M. El-Deeb, Gangadharan Murugusundaramoorthy and Daniel Breaz

Mathematics 2024, 12(14), 2273; https://doi.org/10.3390/math12142273 - 20 Jul 2024

Viewed by 1841

Abstract

A challenging part of studying geometric function theory is figuring out the sharp boundaries for coefficient-related problems that crop up in the Taylor–Maclaurin series of univalent functions. Using Caputo-type fractional derivatives to define the families of Sakaguchi-type starlike functions with respect to symmetric [...] Read more.

A challenging part of studying geometric function theory is figuring out the sharp boundaries for coefficient-related problems that crop up in the Taylor–Maclaurin series of univalent functions. Using Caputo-type fractional derivatives to define the families of Sakaguchi-type starlike functions with respect to symmetric points, this article aims to investigate the first three initial coefficient estimates, the bounds for various problems such as Fekete–Szegő inequality, and the Zalcman inequalities, by subordinating to the function of the three leaves domain. Fekete–Szegő-type inequalities and initial coefficients for functions of the form

H^{- 1}

and

\frac{ζ}{H (ζ)}

and

\frac{1}{2} log (\frac{H (ζ)}{ζ})

connected to the three leaves functions are also discussed. Full article

(This article belongs to the Special Issue Geometric Function Theory and Applications―Festschrift for Grigore-Stefan Salagean's 75th Birthday)

11 pages, 293 KB

Open AccessArticle

Applications of Horadam Polynomials for Bazilevič and λ-Pseudo-Starlike Bi-Univalent Functions Associated with Sakaguchi Type Functions

by Isra Al-Shbeil, Abbas Kareem Wanas, Hala AlAqad, Adriana Cătaş and Hanan Alohali

Symmetry 2024, 16(2), 218; https://doi.org/10.3390/sym16020218 - 11 Feb 2024

Cited by 7 | Viewed by 1817

Abstract

In this study, we introduce a new class of normalized analytic and bi-univalent functions denoted by

D_{Σ} (δ, η, λ, t, r)

. These functions are connected to the Bazilevič functions and the

λ

-pseudo-starlike functions. [...] Read more.

In this study, we introduce a new class of normalized analytic and bi-univalent functions denoted by

D_{Σ} (δ, η, λ, t, r)

. These functions are connected to the Bazilevič functions and the

λ

-pseudo-starlike functions. We employ Sakaguchi Type Functions and Horadam polynomials in our survey. We establish the Fekete-Szegö inequality for the functions in

D_{Σ} (δ, η, λ, t, r)

and derive upper bounds for the initial Taylor–Maclaurin coefficients

| a_{2} |

and

| a_{3} |

. Additionally, we establish connections between our results and previous research papers on this topic. Full article

(This article belongs to the Special Issue Selected Papers from International Symposium on Geometric Function Theory and Applications (GFTA))

14 pages, 337 KB

Open AccessArticle

Sakaguchi Type Starlike Functions Related with Miller-Ross-Type Poisson Distribution in Janowski Domain

by Sheza M. El-Deeb, Asma Alharbi and Gangadharan Murugusundaramoorthy

Mathematics 2023, 11(13), 2918; https://doi.org/10.3390/math11132918 - 29 Jun 2023

Cited by 1 | Viewed by 1708

Abstract

In this research, using the Poisson-type Miller-Ross distribution, we introduce new subclasses Sakaguchi type of star functions with respect to symmetric and conjugate points and discusses their characteristic properties and coefficient estimates. Furthermore, we proved that the class is closed by an integral [...] Read more.

In this research, using the Poisson-type Miller-Ross distribution, we introduce new subclasses Sakaguchi type of star functions with respect to symmetric and conjugate points and discusses their characteristic properties and coefficient estimates. Furthermore, we proved that the class is closed by an integral transformation. In addition, we pointed out some new subclasses and listed their geometric properties according to specializing in parameters that are new and no longer studied in conjunction with a Miller-Ross Poisson distribution. Full article

21 pages, 399 KB

Open AccessArticle

Investigation of the Second-Order Hankel Determinant for Sakaguchi-Type Functions Involving the Symmetric Cardioid-Shaped Domain

by Khalil Ullah, Muhammad Arif, Ibtisam Mohammed Aldawish and Sheza M. El-Deeb

Fractal Fract. 2023, 7(5), 376; https://doi.org/10.3390/fractalfract7050376 - 30 Apr 2023

Viewed by 2228

Abstract

Determining the sharp bounds for coefficient-related problems that appear in the Taylor–Maclaurin series of univalent functions is one of the most difficult aspects of studying geometric function theory. The purpose of this article is to establish the sharp bounds for a variety of [...] Read more.

Determining the sharp bounds for coefficient-related problems that appear in the Taylor–Maclaurin series of univalent functions is one of the most difficult aspects of studying geometric function theory. The purpose of this article is to establish the sharp bounds for a variety of problems, such as the first three initial coefficient problems, the Zalcman inequalities, the Fekete–Szegö type results, and the second-order Hankel determinant for families of Sakaguchi-type functions related to the cardioid-shaped domain. Further, we study the logarithmic coefficients for both of these classes. Full article

(This article belongs to the Special Issue Fractional Operators and Their Applications)

8 pages, 294 KB

Open AccessArticle

Applications of Laguerre Polynomials for Bazilevič and θ-Pseudo-Starlike Bi-Univalent Functions Associated with Sakaguchi-Type Functions

by Luminiţa-Ioana Cotîrlǎ and Abbas Kareem Wanas

Symmetry 2023, 15(2), 406; https://doi.org/10.3390/sym15020406 - 3 Feb 2023

Cited by 8 | Viewed by 2270

Abstract

The aim of the present article is to introduce and investigate a new family

L_{Σ} (δ, η, θ, t; h)

of normalized holomorphic and bi-univalent functions that involve the Sakaguchi-type Bazilevič functions and Sakaguchi-type

θ

-pseudo-starlike [...] Read more.

The aim of the present article is to introduce and investigate a new family

L_{Σ} (δ, η, θ, t; h)

of normalized holomorphic and bi-univalent functions that involve the Sakaguchi-type Bazilevič functions and Sakaguchi-type

θ

-pseudo-starlike functions associated with Laguerre polynomials. We obtain estimates on the initial Taylor–Maclaurin coefficients and the Fekete–Szegö problem for functions in this family. Properties of symmetry can be studied for this newly family of functions. Full article

(This article belongs to the Special Issue Selected Papers from International Symposium on Geometric Function Theory and Applications (GFTA))

11 pages, 266 KB

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 2365

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)

Search Results (9)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (9)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI