Third Hankel Determinant Estimates for a Subclass of Bi-Univalent Functions Associated with a Bean-Shaped Region

This paper investigates upper bounds for the third Hankel determinant \( \mathcal{H}_3(1) \) for certain subclasses of bi-univalent functions associated with a bean-shaped region. The class under study is defined via subordination to the mapping \( \mathcal{B}(z) = \sqrt{1 + \tanh z} \), which sends the open unit disk \( \mathcal{Q} \) conformally onto a convex, non-starlike domain in the complex plane. By employing coefficient estimates and sharp inequalities, we derive an explicit bound:\[| \mathcal{H}_3(1) | \leq |\varrho_3|\, |\varrho_2 \varrho_4 - \varrho_3^2| + |\varrho_4|\, |\varrho_4 - \varrho_2 \varrho_3| + |\varrho_5|\, |\varrho_3 - \varrho_2^2|.\]These results contribute to the ongoing development of the geometric theory of bi-univalent functions, particularly in the context of non-standard domains. The associated bean-shaped domain, being convex yet non-starlike, presents unique geometric features that enrich the analysis of such function classes. Our findings extend classical bounds and demonstrate the potential of using complex geometric regions to derive sharper analytic estimates. Future research may explore higher-order Hankel determinants and generalizations to other non-standard domains.