Toeplitz determinants for starlike mappings of order α in several complex variables

Let $F(z)=zf(z)$, where $f$ is a normalized scalar holomorphic function on the unit ball $\mathbb{B}$ of a complex Banach space $X$ or on the unit polydisk ${\mathbb{U}}^n$ of ${\mathbb{C}}^n$. We establish sharp bounds for two Toeplitz determinants involving the second, third and fourth Fréchet derivatives of $F$ when $F$ is starlike of order $\alpha$, $0\le \alpha <1$. The first bound is sharp for the full range $0\le \alpha <1$, while the second is sharp for $0\le \alpha \le 2\mathrm{/}3$. The extremal mappings are explicitly identified, confirming the sharpness of the estimates.