Second Hankel Determinant for a Bi-univalent Function Subclass Based Gegenbauer (Ultraspherical) Polynomials

In this paper, we aim to establish a new upper bound approximation for the second Hankel determinant utilizing a certain subclass of the class of normalized analytic and bi-univalent functions in the open unit disk \,$\mathbb{U}$\,. These functions have inverses with a bi-univalent analytic continuation to \,$\mathbb{U}$\, and are associated with orthogonal polynomials; namely, Gegenbauer polynomials that satisfy subordination conditions on \,$\mathbb{U}$. Finally, we introduce new essential results derived by specializing the parameter \,$\tau$\, employed in our foundational finding.