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

 

A generalization of the $ {\rm cos}\ \pi \,\rho $ theorem


Author: Albert Baernstein
Journal: Trans. Amer. Math. Soc. 193 (1974), 181-197
MSC: Primary 30A66
MathSciNet review: 0344468
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Abstract: Let f be an entire function, and let $ \beta $ and $ \lambda $ be positive numbers with $ \beta \leq \pi $ and $ \beta \lambda < \pi $. Let $ E(r) = \{ \theta :\log \vert f(r{e^{i\theta }})\vert > \cos \beta \lambda \log M(r)\} $. It is proved that either there exist arbitrarily large values of r for which $ E(r)$ contains an interval of length at least $ 2\beta $, or else $ {\lim _{r \to \infty }}{r^{ - \lambda }}\log M(r,f)$ exists and is positive or infinite. For $ \beta = \pi $ this is Kjellberg's refinement of the cos $ \pi \rho $ theorem.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0344468-7
PII: S 0002-9947(1974)0344468-7
Article copyright: © Copyright 1974 American Mathematical Society