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arXiv · 2506.23831

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

Javad Mashreghi·Annika Moucha·Ryan O'Loughlin·Thomas Ransford·Oliver Roth
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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.

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Javad Mashreghi, Annika Moucha, Ryan O'Loughlin, Thomas Ransford, Oliver Roth. 2025-06-30. A Schwarz-Jack lemma, circularly symmetric domains and numerical ranges. https://arxiv.org/abs/2506.23831

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