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

Abstract

In this paper, we introduce and investigate a parameterized Hankel-type functional $$H_{2,2}^μ(f)=μa_2a_4-a_3^2,\qquad μ\in[0,1], $$ which reduces to the classical second-order Hankel determinant $H_{2,2}(f)$ when $μ=1$. We obtain estimates for $H_{2,2}^μ(f)$ over the class of starlike mappings of order $α$ 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 $H_{2,2}^μ(f)$ 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 $μ=1$, 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.