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 | References | Similar Articles | Additional Information
Abstract: We show that the class of analytic functions
in a plane region
vanishing at
and such that
omits a set of values of area
is not compact. Here
denotes the class of Riemann surfaces which have no nonconstant bounded analytic functions. We remark that the extremal functions maximizing
in
coincide with linear transformations
of those for the same problem in the class
consisting of functions
such that
and
, i.e. so-called Ahlfors functions. Here
is an omitted value of the Ahlfors function.
- [1] Lars Ahlfors and Arne Beurling, Conformal invariants and function-theoretic null-sets, Acta Math. 83 (1950), 101–129. MR 36841, https://doi.org/10.1007/BF02392634
- [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
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Additional Information
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