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

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 ${\mathbb{G}}_2$.