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 U . These functions have inverses with a bi-univalent analytic continuation to U and are associated with orthogonal polynomials; namely, Gegenbauer polynomials that satisfy subordination conditions on U. Finally, we introduce new essential results derived by specializing the parameter τ employed in our foundational finding. © 2025 The Author(s).