Bounded holomorphic functions with given maximum modulus on all circles

Kot, Piotr

doi:10.1090/S0002-9939-08-09468-9

Bounded holomorphic functions with given maximum modulus on all circles
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Proc. Amer. Math. Soc. 137 (2009), 179-187 Request permission

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 $z\in \partial \Omega$.

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