Sharp Estimates for Hankel, Fekete-Szeg\"o and Zalcman Functionals for $\mathcal{S}_\mathbb{B}^{*}(\alpha)$ in Complex Banach Spaces

Abstract

Inspired by the sharp coefficient estimates established by Cho \emph{et al.}\cite{CKKLS2018} for starlike functions of order $\alpha$ in the unit disk, we investigate the corresponding problems for starlike mappings of order $\alpha$ defined on the unit ball of a complex Banach space. Employing Fr\'echet derivatives together with suitable auxiliary lemmas, we establish sharp upper bounds for the second-order Hankel determinant, the Fekete--Szeg\"o functional, and the Zalcman functional associated with this class of mappings. In each case, the obtained estimates are shown to be sharp by identifying the corresponding extremal mappings. Furthermore, our results reduce to the known one-dimensional sharp estimates when the underlying Banach space is the complex plane, thereby extending several classical results of Cho \emph{et al.} \cite{CKKLS2018} to the setting of complex Banach spaces.