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On a class of analytic functions


Author: Nobuyuki Suita
Journal: Proc. Amer. Math. Soc. 43 (1974), 249-250
MSC: Primary 30A40
DOI: https://doi.org/10.1090/S0002-9939-1974-0335785-0
MathSciNet review: 0335785
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Abstract: We show that the class $ {\mathfrak{E}_0}$ of analytic functions $ f$ in a plane region $ \Omega \notin {O_{AB}}$ vanishing at $ {z_0} \in \Omega $ and such that $ 1/f$ omits a set of values of area $ \geqq \pi $ is not compact. Here $ {O_{AB}}$ denotes the class of Riemann surfaces which have no nonconstant bounded analytic functions. We remark that the extremal functions maximizing $ \vert f'({z_0})\vert$ in $ {\mathfrak{E}_0}$ coincide with linear transformations $ w/(1 - cw)$ of those for the same problem in the class $ {\mathfrak{B}_0}$ consisting of functions $ f$ such that $ f({z_0}) = 0$ and $ \vert f(z)\vert \leqq 1$, i.e. so-called Ahlfors functions. Here $ 1/c$ is an omitted value of the Ahlfors function.


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

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DOI: https://doi.org/10.1090/S0002-9939-1974-0335785-0
Keywords: Extremal problem, compact family, bounded functions
Article copyright: © Copyright 1974 American Mathematical Society

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