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

Wanas, Abbas Kareem; Shakir, Qasim Ali; Catas, Adriana

doi:10.3390/sym17091532

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

by

Abbas Kareem Wanas

^1,*

,

Qasim Ali Shakir

² and

Adriana Catas

^3,*

¹

Department of Mathematics, College of Education for Women, University of Al-Qadisiyah, Al Diwaniyah 58001, Iraq

²

Department of Computer Science, College of Computer Science and Information Technology, University of Al-Qadisiyah, Al Diwaniyah 58001, Iraq

³

Department of Mathematics and Computer Science, University of Oradea, 1 University Street, 410087 Oradea, Romania

^*

Authors to whom correspondence should be addressed.

Symmetry 2025, 17(9), 1532; https://doi.org/10.3390/sym17091532 (registering DOI)

Submission received: 30 July 2025 / Revised: 5 September 2025 / Accepted: 8 September 2025 / Published: 13 September 2025

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

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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 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.

Keywords: holomorphic functions; bi-univalent functions; Bazilevič functions; bi-univalent functions; λ-pseudo functions; q-Bernoulli polynomial; coefficient estimates; Fekete-Szegö problem

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MDPI and ACS Style

Wanas, A.K.; Shakir, Q.A.; Catas, A. Coefficient Estimates and Symmetry Analysis for Certain Families of Bi-Univalent Functions Defined by the 𝒒-Bernoulli Polynomial. Symmetry 2025, 17, 1532. https://doi.org/10.3390/sym17091532

AMA Style

Wanas AK, Shakir QA, Catas A. Coefficient Estimates and Symmetry Analysis for Certain Families of Bi-Univalent Functions Defined by the 𝒒-Bernoulli Polynomial. Symmetry. 2025; 17(9):1532. https://doi.org/10.3390/sym17091532

Chicago/Turabian Style

Wanas, Abbas Kareem, Qasim Ali Shakir, and Adriana Catas. 2025. "Coefficient Estimates and Symmetry Analysis for Certain Families of Bi-Univalent Functions Defined by the 𝒒-Bernoulli Polynomial" Symmetry 17, no. 9: 1532. https://doi.org/10.3390/sym17091532

APA Style

Wanas, A. K., Shakir, Q. A., & Catas, A. (2025). Coefficient Estimates and Symmetry Analysis for Certain Families of Bi-Univalent Functions Defined by the 𝒒-Bernoulli Polynomial. Symmetry, 17(9), 1532. https://doi.org/10.3390/sym17091532

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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

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