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13 pages, 345 KiB

Open AccessArticle

An Application of Liouville–Caputo-Type Fractional Derivatives on Certain Subclasses of Bi-Univalent Functions

by Ibtisam Aldawish, Hari M. Srivastava, Sheza M. El-Deeb, Gangadharan Murugusundaramoorthy and Kaliappan Vijaya

Fractal Fract. 2025, 9(8), 505; https://doi.org/10.3390/fractalfract9080505 - 31 Jul 2025

Viewed by 121

Abstract

In this study, we present two novel subclasses of bi-univalent functions defined in the open unit disk, utilizing Liouville–Caputo fractional derivatives. We find constraints on initial Taylor coefficients

| c_{2} |

,

| c_{3} |

for functions in these subclasses of [...] Read more.

In this study, we present two novel subclasses of bi-univalent functions defined in the open unit disk, utilizing Liouville–Caputo fractional derivatives. We find constraints on initial Taylor coefficients

| c_{2} |

,

| c_{3} |

for functions in these subclasses of bi-univalent functions. Additionally, by using the values of

a_{2}, a_{3}

we determine the Fekete–Szegö inequality results. Moreover, a few new subclasses are deduced that have not been studied in relation to Liouville–Caputo fractional derivatives so far. The implications of the results are also emphasized. Our results are concrete examples of several earlier discoveries that are not only improved but also expanded upon. Full article

(This article belongs to the Special Issue Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives)

16 pages, 291 KiB

Open AccessArticle

Initial Coefficient Bounds for Bi-Close-to-Convex and Bi-Quasi-Convex Functions with Bounded Boundary Rotation Associated with q-Sălăgean Operator

by Prathviraj Sharma, Srikandan Sivasubramanian, Adriana Catas and Sheza M. El-Deeb

Mathematics 2025, 13(14), 2252; https://doi.org/10.3390/math13142252 - 11 Jul 2025

Viewed by 252

Abstract

In this article, through the application of the q-Sălăgean operator associated with functions characterized by bounded boundary rotation, we propose a few new subclasses of bi-univalent functions that utilize the q-Sălăgean operator with bounded boundary rotation in the open unit disk [...] Read more.

In this article, through the application of the q-Sălăgean operator associated with functions characterized by bounded boundary rotation, we propose a few new subclasses of bi-univalent functions that utilize the q-Sălăgean operator with bounded boundary rotation in the open unit disk

E

. For these classes, we establish the initial bounds for the coefficients

| a_{2} |

and

| a_{3} |

. Additionally, we have derived the well-known Fekete–Szegö inequality for this newly defined subclasses. Full article

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

14 pages, 569 KiB

Open AccessArticle

A New Subclass of Bi-Univalent Functions Defined by Subordination to Laguerre Polynomials and the (p,q)-Derivative Operator

by Mohammad El-Ityan, Tariq Al-Hawary, Basem Aref Frasin and Ibtisam Aldawish

Symmetry 2025, 17(7), 982; https://doi.org/10.3390/sym17070982 - 21 Jun 2025

Viewed by 425

Abstract

In this work, we introduce a new subclass of bi-univalent functions using the

(p, q)

-derivative operator and the concept of subordination to generalized Laguerre polynomials

L_{t}^{ς} (k)

, which satisfy the differential equation [...] Read more.

In this work, we introduce a new subclass of bi-univalent functions using the

(p, q)

-derivative operator and the concept of subordination to generalized Laguerre polynomials

L_{t}^{ς} (k)

, which satisfy the differential equation

k y^{″} + (1 + ς - k) y^{'} + t y = 0,

with

1 + ς > 0

,

k \in R

, and

t \geq 0

. We focus on functions that blend the geometric features of starlike and convex mappings in a symmetric setting. The main goal is to estimate the initial coefficients of functions in this new class. Specifically, we obtain sharp upper bounds for

| a_{2} |

and

| a_{3} |

and for the Fekete–Szegö functional

| a_{3} - η a_{2}^{2} |

for some real number

η

. In the final section, we explore several special cases that arise from our general results. These results contribute to the ongoing development of bi-univalent function theory in the context of

(p, q)

-calculus. Full article

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

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22 pages, 810 KiB

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

Viewed by 354

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)

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24 pages, 360 KiB

Open AccessArticle

Sharp Coefficient Bounds for Analytic Functions Related to Bounded Turning Functions

by Sudhansu Palei, Madan Mohan Soren, Luminiţa-Ioana Cotîrlǎ and Daniel Breaz

Mathematics 2025, 13(11), 1845; https://doi.org/10.3390/math13111845 - 1 Jun 2025

Viewed by 393

Abstract

Let

B

denote the class of bounded turning functions

F

analytic in the open unit disk, where the image of

F^{'} (z)

is contained in the domain [...] Read more.

Let

B

denote the class of bounded turning functions

F

analytic in the open unit disk, where the image of

F^{'} (z)

is contained in the domain

Ω (z) = \cosh z + \frac{2 z}{2 - z^{2}}

. This article determines sharp coefficient bounds, a Fekete–Szegö-type inequality, and second- and third-order Hankel determinants for functions in the class

B

. Additionally, we obtain sharp Krushkal and Zalcman functional-type inequalities related to the logarithmic coefficient for functions belonging to

B

. Full article

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14 pages, 287 KiB

Open AccessArticle

On the Third Hankel Determinant of a Certain Subclass of Bi-Univalent Functions Defined by (p,q)-Derivative Operator

by Mohammad El-Ityan, Qasim Ali Shakir, Tariq Al-Hawary, Rafid Buti, Daniel Breaz and Luminita-Ioana Cotîrlă

Mathematics 2025, 13(8), 1269; https://doi.org/10.3390/math13081269 - 11 Apr 2025

Cited by 2 | Viewed by 506

Abstract

In this study, the generalized

(p, q)

-derivative operator is used to define a novel class of bi-univalent functions. For this class, we define constraints on the coefficients up to

| ℓ_{5} |

. The functions are analyzed using [...] Read more.

In this study, the generalized

(p, q)

-derivative operator is used to define a novel class of bi-univalent functions. For this class, we define constraints on the coefficients up to

| ℓ_{5} |

. The functions are analyzed using a suitable operational method, which enables us to derive new bounds for the Fekete–Szegö functional, as well as explicit estimates for important coefficients like

| ℓ_{2} |

and

| ℓ_{3} |

. In addition, we establish the upper bounds of the second and third Hankel determinants, providing insights into the geometrical and analytical properties of this class of functions. Full article

(This article belongs to the Special Issue Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions)

22 pages, 378 KiB

Open AccessArticle

A Novel Family of Starlike Functions Involving Quantum Calculus and a Special Function

by Baseer Gul, Daniele Ritelli, Reem K. Alhefthi and Muhammad Arif

Fractal Fract. 2025, 9(3), 179; https://doi.org/10.3390/fractalfract9030179 - 14 Mar 2025

Cited by 2 | Viewed by 607

Abstract

The intent of quantum calculus, briefly q-calculus, is to find q-analogues of mathematical entities so that the original object is achieved when a certain limit is taken. In the case of q-analogue of the ordinary derivative, the limit is [...] Read more.

The intent of quantum calculus, briefly q-calculus, is to find q-analogues of mathematical entities so that the original object is achieved when a certain limit is taken. In the case of q-analogue of the ordinary derivative, the limit is

q \to 1^{-}

. Also, the study of integral as well as differential operators has remained a significant field of inquiry from the early developments of function theory. In the present article, a subclass

S_{s c} (μ, q)

of functions being analytic in

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

is introduced. The definition of

S_{s c} (μ, q)

involves the concepts of subordination, that of q-derivative and q-Ruscheweyh operators. Since coefficient estimates and coefficient functionals provide insights into different geometric properties of analytic functions, for this newly defined subclass, we investigate coefficient estimates up to

a_{4}

, in which both bounds for

| a_{2} |

and

| a_{3} |

are sharp, while that of

| a_{4} |

is sharp in one case. We also discuss the sharp Fekete–Szegö functional for the said class. In addition, Toeplitz determinant bounds up to

T_{3} (2)

(sharp in some cases) and sufficient condition are obtained. Several consequences derived from our above-mentioned findings are also part of the discussion. Full article

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

25 pages, 401 KiB

Open AccessArticle

Coefficient Bounds for Alpha-Convex Functions Involving the Linear q-Derivative Operator Connected with the Cardioid Domain

by Sudhansu Palei, Madan Mohan Soren and Luminiţa-Ioana Cotîrlǎ

Fractal Fract. 2025, 9(3), 172; https://doi.org/10.3390/fractalfract9030172 - 12 Mar 2025

Cited by 2 | Viewed by 659

Abstract

Scholars from several disciplines have recently expressed interest in the field of fractional q-calculus based on fractional integrals and derivative operators. This article mathematically applies the fractional q-differential and q-integral operators in geometric function theory. The linear q-derivative operator [...] Read more.

Scholars from several disciplines have recently expressed interest in the field of fractional q-calculus based on fractional integrals and derivative operators. This article mathematically applies the fractional q-differential and q-integral operators in geometric function theory. The linear q-derivative operator

S_{μ, δ, q}^{n, m}

and subordination are used in this study to define and construct new classes of

α

-convex functions associated with the cardioid domain. Additionally, this paper explores acute inequality problems for newly defined classes

R_{q}^{α} (a, c, m, L, P),

of

α

-convex functions in the open unit disc

U_{s}

, such as initial coefficient bounds, coefficient inequalities, Fekete–Szegö problems, the second Hankel determinants, and logarithmic coefficients. The results presented in this paper are simple to comprehend and demonstrate how current research relates to earlier research. We found all of the estimates, and they are sharp. Full article

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

25 pages, 322 KiB

Open AccessArticle

On Coefficient Inequalities for Functions of Symmetric Starlike Related to a Petal-Shaped Domain

by Muhammad Abbas, Reem K. Alhefthi, Daniel Breaz and Muhammad Arif

Axioms 2025, 14(3), 165; https://doi.org/10.3390/axioms14030165 - 24 Feb 2025

Cited by 1 | Viewed by 526

Abstract

The research on coefficient inequalities in various classes of univalent holomorphic functions focuses on interpreting their coefficients through the coefficients associated with Carathéodory functions. Therefore, researchers can investigate the behavior of coefficient functionals by applying the known inequalities for Carathéodory functions. This study [...] Read more.

The research on coefficient inequalities in various classes of univalent holomorphic functions focuses on interpreting their coefficients through the coefficients associated with Carathéodory functions. Therefore, researchers can investigate the behavior of coefficient functionals by applying the known inequalities for Carathéodory functions. This study will explore various coefficient inequalities employing the techniques developed for the previously discussed family of functions. These coefficient inequalities include the Krushkal, Zalcman, and Fekete-Szegö inequalities, along with the second and third Hankel determinants. The class of symmetric starlike functions linked with a petal-shaped domain is the primary focus of our study. Full article

(This article belongs to the Special Issue Theory of Functions and Applications, 2nd Edition)

11 pages, 257 KiB

Open AccessArticle

Comprehensive Subfamilies of Bi-Univalent Functions Defined by Error Function Subordinate to Euler Polynomials

by Tariq Al-Hawary, Basem Frasin and Jamal Salah

Symmetry 2025, 17(2), 256; https://doi.org/10.3390/sym17020256 - 8 Feb 2025

Viewed by 605

Abstract

Recently, several researchers have estimated the Maclaurin coefficients, namely

|q_{2}|

and

|q_{3}|,

and the Fekete–Szegö functional problem of functions belonging to some special subfamilies of analytic functions related to certain polynomials, such as Lucas polynomials, Legendrae polynomials, Chebyshev polynomials, and [...] Read more.

Recently, several researchers have estimated the Maclaurin coefficients, namely

|q_{2}|

and

|q_{3}|,

and the Fekete–Szegö functional problem of functions belonging to some special subfamilies of analytic functions related to certain polynomials, such as Lucas polynomials, Legendrae polynomials, Chebyshev polynomials, and others. This study obtains the bounds of coefficients

|q_{2}|

and

|q_{3}|

, and the Fekete–Szegö functional problem for functions belonging to the comprehensive subfamilies

T (ζ, ϵ, δ)

and

J (φ, δ)

of analytic functions in a symmetric domain

U

, using the imaginary error function subordinate to Euler polynomials. After specializing the parameters used in our main results, a number of new special cases are also obtained. Full article

(This article belongs to the Special Issue Symmetry and Its Applications in Complex Analysis by the Means of Special Functions)

15 pages, 306 KiB

Open AccessArticle

Inclusive Subclasses of Bi-Univalent Functions Defined by Error Functions Subordinate to Horadam Polynomials

by Tariq Al-Hawary, Basem Frasin, Daniel Breaz and Luminita-Ioana Cotîrlă

Symmetry 2025, 17(2), 211; https://doi.org/10.3390/sym17020211 - 30 Jan 2025

Viewed by 648

Abstract

In this paper, by utilizing error functions subordinate to Horadam polynomials, we introduce the inclusive subclasses

A (a, ς, r, u, η, ρ, σ),

[...] Read more.

In this paper, by utilizing error functions subordinate to Horadam polynomials, we introduce the inclusive subclasses

A (a, ς, r, u, η, ρ, σ),

B (a, ς, r, u, τ, θ)

and

C (a, ς, r, u, τ, θ)

of bi-univalent functions in the symmetric unit disk U. For functions in these subclasses, we derive estimations for the Maclaurin coefficients

| k_{2} |

and

| k_{3} |,

as well as the Fekete–Szegö functional. Additionally, some related results are also obtained. Full article

(This article belongs to the Special Issue Geometric Function Theory and Special Functions II)

12 pages, 250 KiB

Open AccessArticle

On a New Class of Concave Bi-Univalent Functions Associated with Bounded Boundary Rotation

by Prathviraj Sharma, Srikandan Sivasubramanian, Gangadharan Murugusundaramoorthy and Nak Eun Cho

Mathematics 2025, 13(3), 370; https://doi.org/10.3390/math13030370 - 23 Jan 2025

Viewed by 821

Abstract

In this research article, we introduce a new subclass of concave bi-univalent functions associated with bounded boundary rotation defined on an open unit disk. For this new class, we make an attempt to find the first two initial coefficient bounds. In addition, we [...] Read more.

In this research article, we introduce a new subclass of concave bi-univalent functions associated with bounded boundary rotation defined on an open unit disk. For this new class, we make an attempt to find the first two initial coefficient bounds. In addition, we investigate the very famous Fekete–Szegö inequality for functions belonging to this new subclass of concave bi-univalent functions related to bounded boundary rotation. For some particular choices of parameters, we derive the earlier estimates on the coefficient bounds, which are stated at the end. Full article

(This article belongs to the Special Issue Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions)

9 pages, 280 KiB

Open AccessArticle

Applications of Lucas Balancing Polynomial to Subclasses of Bi-Starlike Functions

by Gangadharan Murugusundaramoorthy, Luminita-Ioana Cotîrlă, Daniel Breaz and Sheza M. El-Deeb

Axioms 2025, 14(1), 50; https://doi.org/10.3390/axioms14010050 - 10 Jan 2025

Viewed by 809

Abstract

The Lucas balancing polynomial is linked to a family of bi-starlike functions denoted as

S_{s c}^{c} (ϑ, Ξ (x))

, which we present and examine in this work. These functions are defined with respect to symmetric [...] Read more.

The Lucas balancing polynomial is linked to a family of bi-starlike functions denoted as

S_{s c}^{c} (ϑ, Ξ (x))

, which we present and examine in this work. These functions are defined with respect to symmetric conjugate points. Coefficient estimates are obtained for functions in this family. The classical Fekete–Szegö inequality of functions in this family is also obtained. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 3rd Edition)

17 pages, 313 KiB

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 704

Abstract

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)

17 pages, 364 KiB

Open AccessArticle

Two Families of Bi-Univalent Functions Associating the (p, q)-Derivative with Generalized Bivariate Fibonacci Polynomials

by Sondekola Rudra Swamy, Basem Aref Frasin, Daniel Breaz and Luminita-Ioana Cotîrlă

Mathematics 2024, 12(24), 3933; https://doi.org/10.3390/math12243933 - 13 Dec 2024

Cited by 3 | Viewed by 789

Abstract

Making use of generalized bivariate Fibonacci polynomials, we propose two families of regular functions of the type

ϕ (ζ) = ζ + \sum_{j = 2}^{\infty} d_{j} ζ^{j}

, which are bi-univalent in the disc [...] Read more.

Making use of generalized bivariate Fibonacci polynomials, we propose two families of regular functions of the type

ϕ (ζ) = ζ + \sum_{j = 2}^{\infty} d_{j} ζ^{j}

, which are bi-univalent in the disc

{ζ \in C : | ζ | < 1}

involving the (p, q)-derivative operator. We find estimates on the coefficients

| d_{2} |

,

| d_{3} |

and the of Fekete–Szegö functional for members of these families. Relevant connections to the existing results and new consequences of the main result are presented. Full article

(This article belongs to the Special Issue Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions)

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