A Schwarz-Jack lemma, circularly symmetric domains and numerical ranges

TL;DR

This paper introduces a Schwarz-Jack lemma for specific holomorphic functions, explores properties of conformal maps of symmetric domains, and offers a novel proof of Crouzeix's theorem on numerical ranges without explicit conformal mappings.

Contribution

It establishes a new Schwarz-Jack lemma for symmetric holomorphic functions and provides a novel proof of Crouzeix's theorem independent of explicit conformal mappings.

Findings

Proved a Schwarz-Jack lemma for symmetric holomorphic functions.

Established monotonicity and convexity of conformal maps for symmetric domains.

Provided a new proof of Crouzeix's theorem for 2x2 matrices.

Abstract

We prove a Schwarz-Jack lemma for holomorphic functions on the unit disk with the property that their maximum modulus on each circle about the origin is attained at a point on the positive real axis. With the help of this result, we establish monotonicity and convexity properties of conformal maps of circularly symmetric and bi-circularly symmetric domains. As an application, we give a new proof of Crouzeix's theorem that the numerical range of any $2\times 2$ matrix is a $2$-spectral set for the matrix. Unlike other proofs, our approach does not depend on the explicit formula for the conformal mapping of an ellipse onto the unit disk.