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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 Phragmén-Lindelöf theorem conjectured by D. J. Newman


Author: W. H. J. Fuchs
Journal: Trans. Amer. Math. Soc. 267 (1981), 285-293
MSC: Primary 30C80
MathSciNet review: 621988
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ D$ be a region of the complex plane, $ \infty \in \partial D$. If $ f(z)$ is holomorphic in $ D$, write $ M(r) = {\sup _{\vert z\vert = r,\,z \in D}}\vert f(z)\vert$.

Theorem 1. If $ f(z)$ is holomorphic in $ D$ and $ \lim {\sup _{z \to \zeta ,\,z \in D}}\vert f(z)\vert \leqslant 1$ for $ \zeta \in \partial D$, $ \zeta \ne \infty $, then one of the following holds (a) $ \vert f(z)\vert < 1(z \in D)$, (b)$ f(z)$ has a pole at $ \infty $, (c) $ \log \,M(r)/\log r \to \infty $ as $ r \to \infty $. If $ M(r)/r \to 0(r \to \infty )$, then (a) must hold.


References [Enhancements On Off] (What's this?)

  • [1] Rolf Nevanlinna, Eindeutige analytische Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd XLVI, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). 2te Aufl. MR 0057330 (15,208c)
  • [2] M. Tsuji, Potential theory in modern function theory, Maruzen Co., Ltd., Tokyo, 1959. MR 0114894 (22 #5712)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1981-0621988-2
PII: S 0002-9947(1981)0621988-2
Article copyright: © Copyright 1981 American Mathematical Society