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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Bounded holomorphic functions with given maximum modulus on all circles
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by Piotr Kot
Proc. Amer. Math. Soc. 137 (2009), 179-187
DOI: https://doi.org/10.1090/S0002-9939-08-09468-9
Published electronically: July 31, 2008

Abstract:

We study $\Omega \subset \mathbb {C}^{d}$, a circular, bounded, strictly convex domain with $C^{2}$ boundary. Let $g$ and $h$ be continuous functions on $\partial \Omega$ with $|g(z)|<h(z)=h(\lambda z)$ for $z\in \partial \Omega$ and $|\lambda |=1$. First we prove that $h$ can be approximated by the maximum modulus values of $K$ homogeneous polynomials, where $K$ is independent from $h$. Next we construct $f_{1}\in A(\Omega )$ such that \[ \max _{|\lambda |=1}|(g+f_{1})(\lambda z)|=h(z)\] for $z\in \partial \Omega$. Moreover we can choose $f_{2}\in \mathbb {O}(\Omega )$ with $|f_{2}^{*}(z)|=h(z)$ for almost all $z\in \partial \Omega$ and $\max _{|\lambda |<1}|f_{2}(\lambda z)|=h(z)$ for all $z\in \partial \Omega$.
References
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Bibliographic Information
  • Piotr Kot
  • Affiliation: Politechnika Krakowska, Instytut Matematyki, ul. Warszawska 24, 31-155 Kraków, Poland
  • Email: pkot@pk.edu.pl
  • Received by editor(s): September 11, 2007
  • Received by editor(s) in revised form: December 12, 2007
  • Published electronically: July 31, 2008
  • Communicated by: Mei-Chi Shaw
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 179-187
  • MSC (2000): Primary 32A05, 32A35
  • DOI: https://doi.org/10.1090/S0002-9939-08-09468-9
  • MathSciNet review: 2439439