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.
Categories: math.CV