# A Schwarz lemma for two families of domains and complex geometry

## TL;DR

This paper establishes a Schwarz lemma for symmetrized polydiscs and their extensions, providing explicit interpolation methods and exploring geometric relationships within these complex domains.

## Contribution

It introduces sharp estimates and explicit interpolation functions for symmetrized polydiscs, advancing the understanding of their complex geometric properties.

## Key findings

- Derived sharp estimates for symmetrized polydiscs
- Constructed explicit interpolating functions under certain conditions
- Explored geometric relationships within families of these domains

## Abstract

We make sharp estimates to obtain a Schwarz type lemma for the symmetrized polydisc $\gn$ and for the extended symmetrized polydisc $\Gn$. We explicitly construct an interpolating function under certain condition. To do so, we followed the methods described in \cite{Young-LMS}. Also we find a few geometric interplay between the members of the family $\Gn$ and its closure $\widetilde{\Gamma}_n$.