On holomorphic functions satisfying $f(z)(1-z^{2})\leq 1$ in the unit disc
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- by Karl-Joachim Wirths PDF
- Proc. Amer. Math. Soc. 85 (1982), 19-23 Request permission
Abstract:
Let $f$ be holomorphic in $D = \{ \left . z \right |\left | z \right | < 1\}$, $\left | {f(z)} \right |(1 - {\left | z \right |^2}) \leqslant 1$ in $D$, ${\overline {\lim } _{\left | z \right | \to 1}}\left | {f(z)} \right |(1 - {\left | z \right |^2}) < 1$ and $L(f): = \{ \left . z \right |\left | {f(z)} \right |(1 - {\left | z \right |^2}) = 1\}$. 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
- Heinrich Behnke and Friedrich Sommer, Theorie der analytischen Funktionen einer komplexen VerÀnderlichen. , Die Grundlehren der mathematischen Wissenschaften, Band 77, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1962 (German). Zweite verÀnderte Auflage. MR 0147622, DOI 10.1007/978-3-662-01316-8
- Joseph A. Cima and Warren R. Wogen, Extreme points of the unit ball of the Bloch space ${\cal B}_{0}$, Michigan Math. J. 25 (1978), no. 2, 213â222. MR 486558
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 85 (1982), 19-23
- MSC: Primary 30D50; Secondary 30B40, 30H05
- DOI: https://doi.org/10.1090/S0002-9939-1982-0647889-8
- MathSciNet review: 647889