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

TL;DR

This paper establishes sharp estimates and a Schwarz lemma for the symmetrized polydisc and extended symmetrized polydisc, providing new necessary and sufficient conditions especially for dimensions two and three.

Contribution

It introduces sharp estimates for the extended symmetrized polydisc and derives a Schwarz lemma for the symmetrized polydisc, extending known results and providing new conditions for low dimensions.

Findings

Sharp estimates for the extended symmetrized polydisc.

A Schwarz lemma for the symmetrized polydisc.

New necessary and sufficient conditions for low dimensions.

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.