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

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

doi:10.3390/sym17070982

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

¹

Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt 19117, Jordan

²

Department of Applied Science, Ajloun College, Al Balqa Applied University, Ajloun 26816, Jordan

³

Faculty of Science, Department of Mathematics, Al Al-Bayt University, Mafraq 25113, Jordan

⁴

Mathematics and Statistics Department, College of Science, IMSIU (Imam Mohammad Ibn Saud Islamic University), Riyadh 13327, Saudi Arabia

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Symmetry 2025, 17(7), 982; https://doi.org/10.3390/sym17070982 (registering DOI)

Submission received: 7 May 2025 / Revised: 18 June 2025 / Accepted: 19 June 2025 / Published: 21 June 2025

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

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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

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.

Keywords: (p,q)-derivative; Laguerre polynomials; Fekete–Szegö; subordination

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MDPI and ACS Style

El-Ityan, M.; Al-Hawary, T.; Frasin, B.A.; Aldawish, I. A New Subclass of Bi-Univalent Functions Defined by Subordination to Laguerre Polynomials and the (p,q)-Derivative Operator. Symmetry 2025, 17, 982. https://doi.org/10.3390/sym17070982

AMA Style

El-Ityan M, Al-Hawary T, Frasin BA, Aldawish I. A New Subclass of Bi-Univalent Functions Defined by Subordination to Laguerre Polynomials and the (p,q)-Derivative Operator. Symmetry. 2025; 17(7):982. https://doi.org/10.3390/sym17070982

Chicago/Turabian Style

El-Ityan, Mohammad, Tariq Al-Hawary, Basem Aref Frasin, and Ibtisam Aldawish. 2025. "A New Subclass of Bi-Univalent Functions Defined by Subordination to Laguerre Polynomials and the (p,q)-Derivative Operator" Symmetry 17, no. 7: 982. https://doi.org/10.3390/sym17070982

APA Style

El-Ityan, M., Al-Hawary, T., Frasin, B. A., & Aldawish, I. (2025). A New Subclass of Bi-Univalent Functions Defined by Subordination to Laguerre Polynomials and the (p,q)-Derivative Operator. Symmetry, 17(7), 982. https://doi.org/10.3390/sym17070982

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

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