A Schwarz-Jack lemma, circularly symmetric domains and numerical ranges
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913019117699072 |
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| author | Mashreghi, Javad Moucha, Annika O'Loughlin, Ryan Ransford, Thomas Roth, Oliver |
| author_facet | Mashreghi, Javad Moucha, Annika O'Loughlin, Ryan Ransford, Thomas Roth, Oliver |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23831 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Schwarz-Jack lemma, circularly symmetric domains and numerical ranges Mashreghi, Javad Moucha, Annika O'Loughlin, Ryan Ransford, Thomas Roth, Oliver Complex Variables Functional Analysis 30C20, 30C80, 47A12 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. |
| title | A Schwarz-Jack lemma, circularly symmetric domains and numerical ranges |
| topic | Complex Variables Functional Analysis 30C20, 30C80, 47A12 |
| url | https://arxiv.org/abs/2506.23831 |