Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

  • Lars Ahlfors and Arne Beurling, Conformal invariants and function-theoretic null-sets, Acta Math. 83 (1950), 101–129. MR 36841, DOI
  • 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

Keywords: Extremal problem, compact family, bounded functions
Article copyright: © Copyright 1974 American Mathematical Society