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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Bounded holomorphic functions with given maximum modulus on all circles


Author: Piotr Kot
Journal: Proc. Amer. Math. Soc. 137 (2009), 179-187
MSC (2000): Primary 32A05, 32A35
Published electronically: July 31, 2008
MathSciNet review: 2439439
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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 $ \vert g(z)\vert<h(z)=h(\lambda z)$ for $ z\in\partial\Omega$ and $ \vert\lambda\vert=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

$\displaystyle \max_{\vert\lambda\vert=1}\vert(g+f_{1})(\lambda z)\vert=h(z)$

for $ z\in\partial\Omega$. Moreover we can choose $ f_{2}\in\mathbb{O}(\Omega)$ with $ \vert f_{2}^{*}(z)\vert=h(z)$ for almost all $ z\in\partial\Omega$ and $ \max_{\vert\lambda\vert<1}\vert f_{2}(\lambda z)\vert=h(z)$ for all $ z\in\partial\Omega$.


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Additional Information

Piotr Kot
Affiliation: Politechnika Krakowska, Instytut Matematyki, ul. Warszawska 24, 31-155 Kraków, Poland
Email: pkot@pk.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09468-9
PII: S 0002-9939(08)09468-9
Keywords: Homogeneous polynomials, maximum modulus set, inner function.
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.