On a Subclass of Bi-univalent Functions Affiliated with Bell and Gegenbauer Polynomials

Abstract

This research paper explores the development of a novel class of analytic bi-univalent functions, leveraging the Bell polynomials along with the Gegenbauer polynomials as a fundamental component for establishing the new subclass. Analytical techniques are employed to determine and evaluate the Maclaurin coefficients $|a_{2}|$ and $|a_{3}|$ and the Fekete-Szeg\"{o} functional problem for functions belonging to the constructed class. We demonstrate that several new results can be derived by specializing the parameters in our main findings. The conclusions drawn from this research enrich the theoretical foundation of this field and open new avenues for mathematical inquiry and application.