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Proceedings of the American Mathematical Society

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On holomorphic functions satisfying $ \,f(z)(1-z\sp{2})\leq 1$ in the unit disc


Author: Karl-Joachim Wirths
Journal: Proc. Amer. Math. Soc. 85 (1982), 19-23
MSC: Primary 30D50; Secondary 30B40, 30H05
MathSciNet review: 647889
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Abstract: Let $ f$ be holomorphic in $ D = \{ \left. z \right\vert\left\vert z \right\vert < 1\} $, $ \left\vert {f(z)} \right\vert(1 - {\left\vert z \right\vert^2}) \leqslant 1$ in $ D$, $ {\overline {\lim } _{\left\vert z \right\vert \to 1}}\left\vert {f(z)} \right\vert(1 - {\left\vert z \right\vert^2}) < 1$ and $ L(f): = \{ \left. z \right\vert\left\vert {f(z)} \right\vert(1 - {\left\vert z \right\vert^2}) = 1\} $. It is shown that the set $ L(f)$ consists of one simple closed curve $ \gamma $ and a finite number of points in the bounded component of $ {\mathbf{C}}\backslash \gamma $ if $ L(f)$ is an infinite set.


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

  • [1] Heinrich Behnke and Friedrich Sommer, Theorie der analytischen Funktionen einer komplexen Veränderlichen., Zweite veränderte Auflage. Die Grundlehren der mathematischen Wissenschaften, Bd. 77, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1962 (German). MR 0147622
  • [2] Joseph A. Cima and Warren R. Wogen, Extreme points of the unit ball of the Bloch space \cal𝐵₀, Michigan Math. J. 25 (1978), no. 2, 213–222. MR 0486558

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0647889-8
Keywords: Holomorphic function, algebraic function, identity principle
Article copyright: © Copyright 1982 American Mathematical Society