A generalized Schwarz lemma for two domains related to $\mu$-synthesis

TL;DR

This paper establishes a generalized Schwarz lemma for the tetrablock and symmetrized bidisc domains, extending previous results and providing necessary and sufficient conditions for holomorphic mappings in these complex domains.

Contribution

It introduces a unified set of conditions that generalize existing Schwarz lemmas for the tetrablock and symmetrized bidisc domains.

Findings

Derived necessary and sufficient conditions for Schwarz lemmas in these domains.

Extended previous results to more general settings.

Unified framework for Schwarz lemmas in related complex domains.

Abstract

We present a set of necessary and sufficient conditions that provides a Schwarz lemma for the tetrablock $\mathbb E$. As an application of this result, we obtain a Schwarz lemma for the symmetrized bidisc $\mathbb G_2$. In either case, our results generalize all previous results in this direction for $\mathbb E$ and $\mathbb G_2$.