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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 $|z| < 1$ and has the radial limit $0$ almost everywhere on $|z| = 1$, then every complex number $\zeta$ is an asymptotic value of $f(z)$ provided the $\zeta$-points satisfy the following Blaschke condition: $\sum _{k = 1}^\infty (1 - |{z_k}|) < \infty$, 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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Keywords: Analytic function, radial limit, Blaschke condition, Lusin-Privaloff’s class, boundary behaviour
Article copyright: © Copyright 1982 American Mathematical Society