Bounded holomorphic functions in Siegel domains

Chen, Su Shing

doi:10.1090/S0002-9939-1973-0322211-X

Bounded holomorphic functions in Siegel domains
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Proc. Amer. Math. Soc. 40 (1973), 539-542 Request permission

Abstract:

A Siegel domain $D$ of the second kind (not necessarily affine homogeneous) is shown to be complete with respect to the Carathéodory distance. Thus $D$ is convex with respect to the bounded holomorphic functions, hence is a domain of holomorphy. A Phragmén-Lindelöf theorem for $D$ is also given. That is, if a holomorphic function $f$ in $D$ is continuous in $\bar D$, bounded on the distinguished boundary $S$ of $D$ and not of exponential growth, then $f$ is bounded in $\bar D$.

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Géométrie des domaines classiques et théorie des fonctions automorphes

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