Estimates for a Parameterized Hankel Functional of Starlike Mappings of Order $\alpha$ and Quasi-Convex Mappings of Type $B$ in Complex Banach Spaces

Abstract

In this paper, we introduce and investigate a parameterized Hankel-type functional $$H_{2,2}^{\mu }(f)=\mu a_2{a}_4-a_3^2,\mu \in [0,1],$$ which reduces to the classical second-order Hankel determinant $H_{2,2}(f)$ when $\mu =1$. We obtain estimates for $H_{2,2}^{\mu }(f)$ over the class of starlike mappings of order $\alpha$ on the unit ball of a complex Banach space and on the unit polydisk ${\mathbb{U}}^n$ in ${\mathbb{C}}^n$. We also establish estimates for $H_{2,2}^{\mu }(f)$ over the class of quasi-convex mappings of type $B$ on the unit ball of a complex Banach space and on the unit polydisk ${\mathbb{U}}^n$. For $\mu =1$, our results recover the corresponding estimates for the classical Hankel determinant and extend the known one-dimensional results of Cho et al. \cite{Cho-2018} and Thomas et al. \cite{Thomas-Tuneski} to higher-dimensional settings.