The Fekete-Szegö problem for starlike mappings of order $\alpha$ and quasi-convex mappings of type $B$ in the unit balls of complex Banach spaces

We establish sharp estimates for the generalized Zalcman functional corresponding to the index $n=2$ for normalized starlike mappings of order $\alpha$ and normalized quasi-convex mappings of type $B$ in the unit ball of a complex Banach space and on the unit polydisk in ${\mathbb{C}}^n$. We also derive sharp Fekete-Szegö inequalities for normalized starlike mappings of order $\alpha$ and normalized quasi-convex mappings of type $B$ in the unit ball of a complex Banach space. Our results extend classical coefficient estimates to higher-dimensional holomorphic mappings and provide a partial confirmation of the generalized Zalcman conjecture in higher dimensions.