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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On a class of analytic functions
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by Nobuyuki Suita PDF
Proc. Amer. Math. Soc. 43 (1974), 249-250 Request permission

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 $|f’({z_0})|$ 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 $|f(z)| \leqq 1$, i.e. so-called Ahlfors functions. Here $1/c$ is an omitted value of the Ahlfors function.
References
  • Lars Ahlfors and Arne Beurling, Conformal invariants and function-theoretic null-sets, Acta Math. 83 (1950), 101–129. MR 36841, DOI 10.1007/BF02392634
  • 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.
  • 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
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • 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