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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Author: Albert Baernstein
Journal: Trans. Amer. Math. Soc. 193 (1974), 181-197
MSC: Primary 30A66
DOI: https://doi.org/10.1090/S0002-9947-1974-0344468-7
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 $E(r) = \{ \theta :\log |f(r{e^{i\theta }})| > \cos \beta \lambda \log M(r)\}$. 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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Article copyright: © Copyright 1974 American Mathematical Society