A Schwarz lemma for the symmetrized polydisc via estimates on another family of domains

Abstract

We make some sharp estimates to obtain a Schwarz lemma for the \textit{symmetrized polydisc} ${\mathbb{G}}_n$, a family of domains naturally associated with the spectral interpolation, defined by \[ \mathbb G_n :=\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j \dots, \prod_{i=1}^n z_i \right): \,|z_i|<1, i=1,\dots,n \right \}. \] We first make a few estimates for the \textit{the extended symmetrized polydisc} ${\tilde{\mathbb{G}}}_n$, a family of domains introduced in \cite{pal-roy 4} and defined in the following way: \begin{align*} \widetilde{\mathbb G}_n := \Bigg\{ (y_1,\dots,y_{n-1}, q)\in \C^n :\; q \in \mathbb D, \; y_j = \be_j + \bar \be_{n-j} q, \; β_j \in \mathbb C &\text{ and }\\ |β_j|+ |β_{n-j}| < {n \choose j} &\text{ for } j=1,\dots, n-1 \Bigg\}. \end{align*} We then show that these estimates are sharp and provide a Schwarz lemma for $\Gn$. It is easy to verify that ${\mathbb{G}}_n={\tilde{\mathbb{G}}}_n$ for $n=1,2$ and that ${\mathbb{G}}_n⊊{\tilde{\mathbb{G}}}_n$ for $n\ge 3$. As a consequence of the estimates for $\widetilde{{\mathbb{G}}_n}$, we have analogous estimates for ${\mathbb{G}}_n$. Since for a point $(s_1,\dots ,s_{n-1},p)\in {\mathbb{G}}_n$, $\left(\genfrac{}{}{0ex}{}{n}{i}\right)$ is the least upper bound for $|s_i|$, which is same for $|y_i|$ for any $(y_1,\dots ,y_{n-1},q)\in \widetilde{{\mathbb{G}}_n}$, $1\le i\le n-1$, the estimates become sharp for ${\mathbb{G}}_n$ too. We show that these conditions are necessary and sufficient for $\widetilde{{\mathbb{G}}_n}$ when $n=1,2,3$. In particular for $n=2$, our results add a few new necessary and sufficient conditions to the existing Schwarz lemma for the symmetrized bidisc.

31 Pages. arXiv admin note: substantial text overlap with arXiv:1904.03745