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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
DOI: https://doi.org/10.1090/S0002-9947-1977-0438133-8
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 ${\text {Cl}}(f,\zeta )\backslash f(\Sha _\zeta )$ 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 $\zeta$, $\Sha _\zeta$ is the Shilov boundary of the fiber algebra, and $\lambda$ is the harmonic measure on $\mathfrak {M}(D)$.


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Article copyright: © Copyright 1977 American Mathematical Society