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Radial images by holomorphic mappings

Authors: José L. Fernández and Domingo Pestana
Journal: Proc. Amer. Math. Soc. 124 (1996), 429-435
MSC (1991): Primary 30E25, 30F45
MathSciNet review: 1283549
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Abstract: Let $\mathcal{R}$ be a nonexceptional Riemann surface, other than the punctured disk. We prove that if $f$ is a holomorphic mapping from the unit disk $\Delta $ of the complex plane into $\mathcal{R}$, then the set of radial images that remain bounded in the Poincaré metric of $\mathcal{R}$ has Hausdorff dimension at least $\delta (\mathcal{R})$, the exponent of convergence of $\mathcal{R}$. The result is best possible. This is a hyperbolic analog of the result of N. G. Makarov that Bloch functions are bounded on a set of radii of dimension one.

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Additional Information

José L. Fernández
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

Domingo Pestana
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Address at time of publication: Departamento de Matemáticas, Universidad Carlos III de Madrid, 28911 Leganes, Spain

Received by editor(s): May 6, 1994
Received by editor(s) in revised form: June 16, 1994
Additional Notes: Research supported by a grant of CICYT, Ministerio de Educación y Ciencia, Spain.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1996 American Mathematical Society

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