A Schwarz lemma for the symmetrized tridisc and description of interpolating functions

TL;DR

This paper establishes a Schwarz lemma for the symmetrized tridisc, characterizes all related interpolating functions, and explores the complex geometry of this domain, providing new insights and characterizations.

Contribution

It introduces a Schwarz lemma for 3, describes all interpolating functions related to it, and analyzes the geometry of the symmetrized tridisc with new characterizations.

Findings

Interpolating function related to the Schwarz lemma is not unique.

Explicit description of all such interpolating functions is provided.

New characterizations for the open and closed symmetrized tridisc are presented.

Abstract

We produce a Schwarz lemma for the symmetrized tridisc \[ \mathbb G_3 =\{ (z_1+z_2+z_3,z_1z_2+z_2z_3+z_3z_1,z_1z_2z_3): \,|z_i|< 1, i=1,2,3 \}. \] We show that an interpolating function related to the Schward lemma for $\mathbb G_3$ is not unique and present an explicit description of all such interpolating functions. We also study the complex geometry of $\mathbb G_3$ and present a variety of new characterizations for the open and closed symmetrized tridisc.