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