arXiv:2610.33031v1 Announce Type: new Abstract: In this paper, we introduce and investigate a parameterized Hankel-type functional $$H_{2,2}^{\mu}(f)=\mu a_2a_4-a_3^2,\qquad \mu\in[0,1], $$ 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.
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