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

Al-Hawary, Tariq; Aldawish, Ibtisam; Frasin, Basem Aref

doi:10.3390/sym17122099

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

by

Tariq Al-Hawary

^1,†

,

Ibtisam Aldawish

^2,*,†

and

Basem Aref Frasin

^3,†

¹

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

²

Mathematics and Statistics Department, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 13318 , Saudi Arabia

³

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

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

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

Submission received: 8 November 2025 / Revised: 2 December 2025 / Accepted: 4 December 2025 / Published: 7 December 2025

(This article belongs to the Section Mathematics)

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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 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.

Keywords: univalent; bi-univalent; holomorphic; Liouville–Caputo; Maclaurin series; Fekete-Szegö

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MDPI and ACS Style

Al-Hawary, T.; Aldawish, I.; Frasin, B.A. A Utilization of Liouville–Caputo Fractional Derivatives for Families of Bi-Univalent Functions Associated with Specific Holomorphic Symmetric Function. Symmetry 2025, 17, 2099. https://doi.org/10.3390/sym17122099

AMA Style

Al-Hawary T, Aldawish I, Frasin BA. A Utilization of Liouville–Caputo Fractional Derivatives for Families of Bi-Univalent Functions Associated with Specific Holomorphic Symmetric Function. Symmetry. 2025; 17(12):2099. https://doi.org/10.3390/sym17122099

Chicago/Turabian Style

Al-Hawary, Tariq, Ibtisam Aldawish, and Basem Aref Frasin. 2025. "A Utilization of Liouville–Caputo Fractional Derivatives for Families of Bi-Univalent Functions Associated with Specific Holomorphic Symmetric Function" Symmetry 17, no. 12: 2099. https://doi.org/10.3390/sym17122099

APA Style

Al-Hawary, T., Aldawish, I., & Frasin, B. A. (2025). A Utilization of Liouville–Caputo Fractional Derivatives for Families of Bi-Univalent Functions Associated with Specific Holomorphic Symmetric Function. Symmetry, 17(12), 2099. https://doi.org/10.3390/sym17122099

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

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