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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cluster values of bounded analytic functions


Author: T. W. Gamelin
Journal: Trans. Amer. Math. Soc. 225 (1977), 295-306
MSC: Primary 46J15; Secondary 30A72
MathSciNet review: 0438133
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Abstract: Let D be a bounded domain in the complex plane, and let $ \zeta $ belong to the topological boundary $ \partial D$ of D. We prove two theorems concerning the cluster set $ {\text{Cl}}(f,\zeta )$ of a bounded analytic function f on D. The first theorem asserts that values in $ {\text{Cl}}(f,\zeta )\backslash f({\mathcyr{SH}_\zeta })$ are assumed infinitely often in every neighborhood of $ \zeta $, with the exception of those lying in a set of zero analytic capacity. The second asserts that all values in $ {\text{Cl}}(f,\zeta )\backslash f({\mathfrak{M}_\zeta } \cap {\text{supp}}\;\lambda )$ are assumed infinitely often in every neighborhood of $ \zeta $, with the exception of those lying in a set of zero logarithmic capacity. Here $ {\mathfrak{M}_\zeta }$ is the fiber of the maximal ideal space $ \mathfrak{M}(D)$ of $ {H^\infty }(D)$ lying over $ \zeta $, $ {\mathcyr{SH}_\zeta }$ is the Shilov boundary of the fiber algebra, and $ \lambda $ is the harmonic measure on $ \mathfrak{M}(D)$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0438133-8
PII: S 0002-9947(1977)0438133-8
Article copyright: © Copyright 1977 American Mathematical Society