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18 pages, 544 KB

Open AccessArticle

A Utilization of Liouville–Caputo Fractional Derivatives for Families of Bi-Univalent Functions Associated with Specific Holomorphic Symmetric Function

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

Symmetry 2025, 17(12), 2099; https://doi.org/10.3390/sym17122099 - 7 Dec 2025

Viewed by 203

Abstract

In this investigation, two new subfamilies of bi-univalent functions defined on the open unit disk are presented using Liouville–Caputo fractional derivatives. We determine bounds on the initial Maclaurin coefficients

| a_{2} |

and

| a_{3} |,

as well as Fekete–Szegö [...] Read more.

In this investigation, two new subfamilies of bi-univalent functions defined on the open unit disk are presented using Liouville–Caputo fractional derivatives. We determine bounds on the initial Maclaurin coefficients

| a_{2} |

and

| a_{3} |,

as well as Fekete–Szegö inequality results based on the bonds of

a_{2}

and

a_{3}

for functions belonging to certain bi-univalent function subfamilies. Additionally, some novel subfamilies are inferred that have not yet been examined within the context of Liouville–Caputo fractional derivatives. Full article

(This article belongs to the Section Mathematics)

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17 pages, 340 KB

Open AccessArticle

Certain Subclasses of Te-Univalent Functions Subordinate to q-Bernoulli Polynomials

by Sondekola Rudra Swamy, A. Alameer, Basem Aref Frasin and Savithri Shashidhar

Mathematics 2025, 13(23), 3841; https://doi.org/10.3390/math13233841 - 30 Nov 2025

Viewed by 181

Abstract

The present work centers on the significance of q-calculus in geometric function theory and its expanding applications within the domain of Te-univalent functions, especially those associated with special polynomials like the q-Bernoulli polynomials. Motivated by recent interest in these polynomials, our [...] Read more.

The present work centers on the significance of q-calculus in geometric function theory and its expanding applications within the domain of Te-univalent functions, especially those associated with special polynomials like the q-Bernoulli polynomials. Motivated by recent interest in these polynomials, our study introduces and analyzes a generalized subclass of Te-univalent functions that intimately relate to q-Bernoulli polynomials. For this new family, we establish explicit bounds for

| d_{2} |

and

| d_{3} |

, and provide estimates for the Fekete–Szegö functional

| d_{3} - ξ d_{2}^{2} |, ξ \in R

. Our findings contribute new results and demonstrate meaningful connections to prior work involving Te-univalent and subordinate functions, thereby broadening and integrating various strands of the existing literature. Full article

(This article belongs to the Special Issue Current Topics in Geometric Function Theory, 2nd Edition)

15 pages, 369 KB

Open AccessArticle

Certain Subclasses of Bi-Univalent Functions Involving Caputo Fractional Derivatives with Bounded Boundary Rotation

by Abbas Kareem Wanas, Mohammad El-Ityan, Adel Salim Tayyah and Adriana Catas

Mathematics 2025, 13(21), 3563; https://doi.org/10.3390/math13213563 - 6 Nov 2025

Viewed by 316

Abstract

In this paper, we introduce and investigate new subclasses of analytic bi-univalent functions defined via Caputo fractional derivatives with boundary rotation constraints. Utilizing the generalized operator

C_{ȷ}^{ϱ}

, which encompasses and extends classical operators such as the Salagean differential operator and [...] Read more.

In this paper, we introduce and investigate new subclasses of analytic bi-univalent functions defined via Caputo fractional derivatives with boundary rotation constraints. Utilizing the generalized operator

C_{ȷ}^{ϱ}

, which encompasses and extends classical operators such as the Salagean differential operator and the Libera–Bernardi integral operator, we establish sharp coefficient estimates for the initial Taylor Maclaurin coefficients of functions within these subclasses. Furthermore, we derive Fekete–Szegö-type inequalities that provide bounds on the second and third coefficients and their linear combinations involving a real parameter. Our approach leverages subordination principles through analytic functions associated with the classes

T_{ς} (ξ)

and

R_{Ω}^{ȷ, ϱ} (ϑ, ς, ξ)

, allowing a unified treatment of fractional differential operators in geometric function theory. The results generalize several known cases and open avenues for further exploration in fractional calculus applied to analytic function theory. Full article

(This article belongs to the Special Issue Advances in Nonlinear Differential Equations with Applications)

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13 pages, 319 KB

Open AccessArticle

Inclusive Subfamilies of Complex Order Generated by Liouville–Caputo-Type Fractional Derivatives and Horadam Polynomials

by Feras Yousef, Tariq Al-Hawary, Basem Frasin and Amerah Alameer

Fractal Fract. 2025, 9(11), 698; https://doi.org/10.3390/fractalfract9110698 - 30 Oct 2025

Cited by 1 | Viewed by 466

Abstract

In this paper, we introduce the inclusive subfamilies of complex order

E (δ_{1}, δ_{2}, δ_{3}, δ_{4}, a, b)

and [...] Read more.

In this paper, we introduce the inclusive subfamilies of complex order

E (δ_{1}, δ_{2}, δ_{3}, δ_{4}, a, b)

and

C (δ_{1}, δ_{2}, δ_{3}, δ_{4}, a, b)

, defined by means of the Liouville–Caputo-type derivative operator and subordination to the Horadam polynomials. For these subfamilies, we derive estimates for the initial coefficients

| q_{2} |

and

| q_{3} |

, as well as results concerning the Fekete–Szegö functional

| q_{3} - ϱ q_{2}^{2} |

. In addition, several related results are established as corollaries, accompanied by a concluding remark. Full article

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

16 pages, 315 KB

Open AccessArticle

Applications of Bernoulli Polynomials and q²-Srivastava–Attiya Operator in the Study of Bi-Univalent Function Classes

by Basem Aref Frasin, Sondekola Rudra Swamy, Ibtisam Aldawish and Paduvalapattana Kempegowda Mamatha

Mathematics 2025, 13(21), 3384; https://doi.org/10.3390/math13213384 - 24 Oct 2025

Viewed by 440

Abstract

The central focus of this study is the development and investigation of a generalized subclass of bi-univalent functions, defined using the

q^{2}

-Srivastava–Attiya operator in conjunction with Bernoulli polynomials. We derive initial coefficient estimates for functions in the newly proposed class and [...] Read more.

The central focus of this study is the development and investigation of a generalized subclass of bi-univalent functions, defined using the

q^{2}

-Srivastava–Attiya operator in conjunction with Bernoulli polynomials. We derive initial coefficient estimates for functions in the newly proposed class and also provide bounds for the Fekete–Szegö functional. In addition to presenting several new findings, we also explore meaningful connections with previously established results in the theory of bi-univalent and subordinate functions, thereby extending and unifying the existing literature in a novel direction. Full article

(This article belongs to the Special Issue New Trends in Polynomials and Mathematical Analysis)

10 pages, 1976 KB

Open AccessArticle

On Bi-Univalent Function Classes Defined via Gregory Polynomials

by Ibtisam Aldawish, Mallikarjun G. Shrigan, Sheza El-Deeb and Hari M. Srivastava

Mathematics 2025, 13(19), 3121; https://doi.org/10.3390/math13193121 - 29 Sep 2025

Viewed by 446

Abstract

In this paper, we introduce and study a new subclass of bi-univalent functions related to Mittag–Leffler functions associated with Gregory polynomials and satisfy certain subordination conditions defined in the open unit disk. We derive coefficient bounds for the Taylor–Maclaurin coefficients [...] Read more.

In this paper, we introduce and study a new subclass of bi-univalent functions related to Mittag–Leffler functions associated with Gregory polynomials and satisfy certain subordination conditions defined in the open unit disk. We derive coefficient bounds for the Taylor–Maclaurin coefficients

| γ_{2} |

and

| γ_{3} |

, and also explore the Fekete–Szegö functional. Full article

(This article belongs to the Section C4: Complex Analysis)

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13 pages, 290 KB

Open AccessArticle

Bi-Univalent Function Classes Defined by Imaginary Error Function and Bernoulli Polynomials

by Ibtisam Aldawish, Sondekola Rudra Swamy, Basem Aref Frasin and Supriya Chandrashekharaiah

Axioms 2025, 14(10), 731; https://doi.org/10.3390/axioms14100731 - 27 Sep 2025

Viewed by 376

Abstract

In recent years, special functions have played a significant role in the investigation of different subclasses within the class of bi-univalent functions. In this work, we present and investigate two new subclasses of bi-univalent functions defined in

U =

[...] Read more.

In recent years, special functions have played a significant role in the investigation of different subclasses within the class of bi-univalent functions. In this work, we present and investigate two new subclasses of bi-univalent functions defined in

U =

{ς \in C : | ς | < 1}

, characterized by Bernoulli polynomials associated with imaginary error functions. For functions belonging to these subclasses, we establish bounds for their initial coefficients. For these classes, we also tackle the Fekete–Szegö problem. Several new results are also obtained as special cases by specifying certain parameter values in the general findings. Full article

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

14 pages, 301 KB

Open AccessArticle

Coefficient Estimates, the Fekete–Szegö Inequality, and Hankel Determinants for Universally Prestarlike Functions Defined by Fractional Derivative in a Shell-Shaped Region

by Dina Nabil, Georgia Irina Oros, Awatef Shahin and Hanan Darwish

Axioms 2025, 14(9), 711; https://doi.org/10.3390/axioms14090711 - 21 Sep 2025

Viewed by 603

Abstract

In this paper, we introduce and investigate a new subclass

{(R_{ς}^{u})}^{g} (ϕ)

of universally prestarlike generalized functions of order

ς

, where

ς \leq 1

, associated with a shell-shaped region defined by [...] Read more.

In this paper, we introduce and investigate a new subclass

{(R_{ς}^{u})}^{g} (ϕ)

of universally prestarlike generalized functions of order

ς

, where

ς \leq 1

, associated with a shell-shaped region defined by

Λ = C ∖ [1, \infty)

for the present investigation, by utilizing the Srivastava–Owa fractional derivative of order

δ

. Coefficient inequalities for

| a_{2} |

and

| a_{3} |

for functions belonging to the newly introduced class are obtained. Additionally, the Fekete–Szegö inequality is investigated for this class of functions. In order to enhance the coefficient studies for this class, the second Hankel determinant is also evaluated. Full article

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

18 pages, 724 KB

Open AccessArticle

Coefficient Estimates and Symmetry Analysis for Certain Families of Bi-Univalent Functions Defined by the q-Bernoulli Polynomial

by Abbas Kareem Wanas, Qasim Ali Shakir and Adriana Catas

Symmetry 2025, 17(9), 1532; https://doi.org/10.3390/sym17091532 - 13 Sep 2025

Viewed by 614

Abstract

In the present work, we define certain families,

M_{Σ} (μ, Υ, ℷ, q; x)

and

N_{Σ} (μ, Υ, ℷ, q; x)

, of normalized holomorphic and bi-univalent functions associated with Bazilevič [...] Read more.

In the present work, we define certain families,

M_{Σ} (μ, Υ, ℷ, q; x)

and

N_{Σ} (μ, Υ, ℷ, q; x)

, of normalized holomorphic and bi-univalent functions associated with Bazilevič functions and

ℷ

-pseudo functions involving the

q

-Bernoulli polynomial, which is defined by the symmetric nature of quantum calculus in the open unit disk

U

. We determine the upper bounds for the initial symmetry Taylor–Maclaurin coefficients and the Fekete–Szegö-type inequalities of functions in the families we have introduced here. In addition, we indicate certain special cases and consequences for our results. Full article

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

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15 pages, 851 KB

Open AccessArticle

Third-Order Hankel Determinant for a Class of Bi-Univalent Functions Associated with Sine Function

by Mohammad El-Ityan, Mustafa A. Sabri, Suha Hammad, Basem Frasin, Tariq Al-Hawary and Feras Yousef

Mathematics 2025, 13(17), 2887; https://doi.org/10.3390/math13172887 - 6 Sep 2025

Cited by 5 | Viewed by 730

Abstract

This paper investigates a new subclass of bi-univalent analytic functions defined on the open unit disk in the complex plane, associated with the subordination to

1 + s i n z

. Coefficient bounds are obtained for the initial Taylor–Maclaurin coefficients, with a [...] Read more.

This paper investigates a new subclass of bi-univalent analytic functions defined on the open unit disk in the complex plane, associated with the subordination to

1 + s i n z

. Coefficient bounds are obtained for the initial Taylor–Maclaurin coefficients, with a particular focus on the second- and third-order Hankel determinants. To illustrate the non-emptiness of the proposed class, we consider the function

\sqrt{1 + tanh z}

, which maps the unit disk onto a bean-shaped domain. This function satisfies the required subordination condition and hence serves as an explicit member of the class. A graphical depiction of the image domain is provided to highlight its geometric characteristics. The results obtained in this work confirm that the class under study is non-trivial and possesses rich geometric structure, making it suitable for further development in the theory of geometric function classes and coefficient estimation problems. Full article

(This article belongs to the Special Issue New Trends in Polynomials and Mathematical Analysis)

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19 pages, 2685 KB

Open AccessArticle

Sharp Bounds and Electromagnetic Field Applications for a Class of Meromorphic Functions Introduced by a New Operator

by Abdelrahman M. Yehia, Atef F. Hashem, Samar M. Madian and Mohammed M. Tharwat

Axioms 2025, 14(9), 684; https://doi.org/10.3390/axioms14090684 - 5 Sep 2025

Cited by 1 | Viewed by 543

Abstract

In this paper, we present a new integral operator that acts on a class of meromorphic functions on the punctured unit disc

U^{*}

. This operator enables the definition of a new subclass of meromorphic univalent functions. We obtain sharp bounds for [...] Read more.

In this paper, we present a new integral operator that acts on a class of meromorphic functions on the punctured unit disc

U^{*}

. This operator enables the definition of a new subclass of meromorphic univalent functions. We obtain sharp bounds for the Fekete–Szegö inequality and the second Hankel determinant for this class. The theoretical approach is based on differential subordination. Furthermore, we link these theoretical insights to applications in 2D electromagnetic field theory by outlining a physical framework in which the operator functions as a field transformation kernel. We show that the operator’s parameters correspond to physical analogs of field regularization and spectral redistribution, and we use subordination theory to simulate the design of vortex-free fields. The findings provide new insights into the interaction between geometric function theory and physical field modeling. Full article

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

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13 pages, 345 KB

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

Cited by 1 | Viewed by 678

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 KB

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 771

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 KB

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

Cited by 7 | Viewed by 859

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 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 1 | Viewed by 692

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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