Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles 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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


On a class of analytic functions
HTML articles powered by AMS MathViewer

by Nobuyuki Suita PDF
Proc. Amer. Math. Soc. 43 (1974), 249-250 Request permission


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.
  • 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
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30A40
  • Retrieve articles in all journals with MSC: 30A40
Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 43 (1974), 249-250
  • MSC: Primary 30A40
  • DOI:
  • MathSciNet review: 0335785