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

 

On the radial limits of analytic and meromorphic functions


Author: J. S. Hwang
Journal: Trans. Amer. Math. Soc. 270 (1982), 341-348
MSC: Primary 30D40
MathSciNet review: 642346
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Abstract: Early in the fifties, A. J. Lohwater proved that if $ f(z)$ is analytic in $ \vert z\vert < 1$ and has the radial limit 0 almost everywhere on $ \vert z\vert = 1$, then every complex number $ \zeta $ is an asymptotic value of $ f(z)$ provided the $ \zeta $-points satisfy the following Blaschke condition: $ \sum _{k = 1}^\infty (1 - \vert{z_k}\vert) < \infty $, where $ f({z_k}) = \zeta $, $ k = 1\,,2, \ldots $. We may, therefore, ask under the hypothesis on $ f(z)$ how many complex numbers $ \zeta $ are there whose $ \zeta $-points can satisfy the Blaschke condition. We show that there is at most one such number and this one number phenomenon can actually occur if the number is zero.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1982-0642346-1
PII: S 0002-9947(1982)0642346-1
Keywords: Analytic function, radial limit, Blaschke condition, Lusin-Privaloff's class, boundary behaviour
Article copyright: © Copyright 1982 American Mathematical Society