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

 

Baire category principle and uniqueness theorem


Author: J. S. Hwang
Journal: Trans. Amer. Math. Soc. 266 (1981), 655-665
MSC: Primary 30D40; Secondary 30D50
MathSciNet review: 617558
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Abstract: Applying a theorem of Bagemihl and Seidel (1953), we prove that if $ {S_2}$ is a set of second category in $ (\alpha ,\beta )$, where $ 0 \leqslant \alpha < \beta \leqslant 2\pi $, and if $ f(z)$ is a function meromorphic in the sector $ \Delta (\alpha ,\beta ) = \{ z:0 < \left\vert z \right\vert < \infty ,\alpha < \arg z < \beta \} $ for which $ {\underline {{\operatorname{lim}}} _{r \to \infty }}\left\vert {f(r{e^{i\theta }})} \right\vert > 0$, for all $ \theta \in {S_2}$, then there exists a sector $ \Delta (\alpha ',\beta ') \subseteq \Delta (\alpha ,\beta )$ such that $ (\alpha ',\beta ') \subseteq {\bar S_2},{S_2}$ is second category in $ (\alpha ',\beta ')$, and $ f(z)$ has no zero in $ \Delta (\alpha ',\beta ')$. Based on this property, we prove several uniqueness theorems.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0617558-2
PII: S 0002-9947(1981)0617558-2
Keywords: Category principle, uniqueness theorem, meromorphic function, Blaschke product, harmonic measure, tangential approximation
Article copyright: © Copyright 1981 American Mathematical Society