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Kyungpook Mathematical Journal 2024; 64(3): 487-497

Published online September 30, 2024 https://doi.org/10.5666/KMJ.2024.64.3.487

Copyright © Kyungpook Mathematical Journal.

Gegenbauer Polynomials For a New Subclass of Bi-univalent Functions

Gunasekar Saravanan, Sudharsanan Baskaran, Balasubramaniam Vanithakumari, Abbas Kareem Wanas*

Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Vengal, Chennai-601103, Tamil Nadu, India
ORCiD: https://orcid.org/0000-0002-5706-4174
e-mail : gsaran825@yahoo.com and g_saravanan@ch.amrita.edu

Department of Mathematics, Agurchand Manmull Jain College, Meenambakkam, Chennai-600061, Tamil Nadu, India
ORCiD: https://orcid.org/0000-0001-8980-9671
e-mail : sbas9991@gmail.com

Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Vengal, Chennai-601103, Tamil Nadu, India
Present address: Department of Mathematics, Agurchand Manmull Jain College, Meenambakkam, Chennai-600061, Tamil Nadu, India
ORCiD: https://orcid.org/0000-0001-8812-0725
e-mail : vanithagft@gmail.com

Department of Mathematics, College of Science, University of Al-Qadisiyah, Al- Qadisiyah, Al Diwaniyah 58001, Iraq
ORCiD: https://orcid.org/0000-0001-5838-7365
e-mail : abbas.kareem.w@qu.edu.iq

Received: December 2, 2023; Revised: June 4, 2024; Accepted: June 12, 2024

Abstract

In this study, we introduce and investigate a novel subclass of analytic biunivalent functions, which we define using Gegenbauer polynomials. We derive the initial coefficient bounds for |a2|, |a3|, and |a4|, and establish Fekete-Szegö inequalities for this class. In addition, we confirm that Brannan and Clunie’s conjecture, |a2| ≤ 2, is valid for this subclass. To facilitate better understanding, we provide visualizations of the functions, using appropriately chosen parameters.

Keywords: Analytic functions, Bi-univalent functions, Gegenbauer polynomials