Fekete-Szegö Functional Problem for Analytic and Bi-Univalent Functions Subordinate to Gegenbauer Polynomials

Abstract

This article aims to introduce a new qualitative subclass of bi-univalent and analytic functions that are intricately linked to Gegenbauer polynomials. These polynomials, known for their significant role in various areas of mathematics, provide a robust framework for exploring the properties of analytic functions. In our exploration, we will address the Fekete-Szego problem, which is pivotal in the field of complex analysis. By doing so, we will derive the coefficient bounds |h₂| and |h₃| for functions within this newly defined subclass, thereby enhancing our understanding of their behavior. Furthermore, by concentrating on the specific parameters that were utilized to achieve our primary results, we expect to generate a variety of additional outcomes. These results will not only deepen our insight into the characteristics of these functions but also contribute to the broader discourse on analytic function theory. We anticipate that the findings presented in this article will pave the way for future research and applications, particularly in the realms of mathematical analysis and applied mathematics.