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
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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