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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Analytic functions with large sets of Fatou points


Authors: J. S. Hwang and Peter Lappan
Journal: Proc. Amer. Math. Soc. 90 (1984), 293-298
MSC: Primary 30D40
MathSciNet review: 727253
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Abstract: For a function $ f$ analytic in the unit disc $ D$, and for each $ \lambda > 0$, let $ L\left( \lambda \right) = \left\{ {z \in D:\left\vert {f\left( z \right)} \right\vert = \lambda } \right\}$ denote a level set for $ f$. We introduce a class $ \mathcal{L}$, of functions characterized by geometric properties of a collection of sets $ \left\{ {L\left( {{\lambda _n}} \right)} \right\}$, where $ \left\{ {{\lambda _n}} \right\}$ is an unbounded sequence. We show that $ {\mathcal{L}_1}$, is a proper subclass of the class $ \mathcal{L}$ of G. R. MacLane. Let $ {A_\infty }$ denote the set of points $ {e^{i\theta }}$ at which the function $ f$ has $ \infty $ as an asymptotic value, and let $ F\left( f \right)$ denote the set of Fatou points of $ f$. We prove that for a function $ f$ in the class $ {\mathcal{L}_1}$, if $ \Gamma $ is an arc of the unit circle such that $ \Gamma \cap {A_\infty } = \emptyset $, then almost every point of $ \Gamma $ belongs to $ F\left( f \right)$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1984-0727253-5
PII: S 0002-9939(1984)0727253-5
Keywords: Level set, Fatou point
Article copyright: © Copyright 1984 American Mathematical Society