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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)


On a class of analytic functions

Author: Nobuyuki Suita
Journal: Proc. Amer. Math. Soc. 43 (1974), 249-250
MSC: Primary 30A40
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?)

  • [1] Lars Ahlfors and Arne Beurling, Conformal invariants and function-theoretic null-sets, Acta Math. 83 (1950), 101–129. MR 0036841 (12,171c)
  • [2] A. Denjoy, Sur fonctions analytiques uniformes qui restant continues sur un ensemble parfait discontinu de singularités, C.R. Acad. Sci. Paris 149 (1909), 1154-1156.
  • [3] S. Ja. Havinson, The analytic capacity of sets related to the non-triviality of various classes of analytic functions, and on Schwarz’s lemma in arbitrary domains, Mat. Sb. (N.S.) 54 (96) (1961), 3–50 (Russian). MR 0136720 (25 #182)

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PII: S 0002-9939(1974)0335785-0
Keywords: Extremal problem, compact family, bounded functions
Article copyright: © Copyright 1974 American Mathematical Society