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} $\widetilde{\mathbb G}n$, 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, \; \beta_j \in \mathbb C &\text{ and }\ |\beta_j|+ |\beta_{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 $\mathbb G_n=\widetilde{\mathbb G}n$ for $n=1,2$ and that ${\mathbb G}_n \subsetneq \widetilde{\mathbb G}_n$ for $n\geq 3$. As a consequence of the estimates for $\widetilde{\mathbb G_n}$, we have analogous estimates for $\mathbb G_n$. Since for a point $(s_1,\dots, s{n-1},p)\in \mathbb G_n$, ${n \choose i}$ is the least upper bound for $|s_i|$, which is same for $|y_i|$ for any $(y_1,\dots ,y_{n-1},q) \in \widetilde{\mathbb G_n}$, $1\leq i \leq n-1$, 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.