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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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Radial images by holomorphic mappings,
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by José L. Fernández and Domingo Pestana PDF
Proc. Amer. Math. Soc. 124 (1996), 429-435 Request permission

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
  • Email: pando@ccuam3.sdi.uam.es
  • 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
  • Email: dompes@arwen.uc3m.es
  • 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
  • © Copyright 1996 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 124 (1996), 429-435
  • MSC (1991): Primary 30E25, 30F45
  • DOI: https://doi.org/10.1090/S0002-9939-96-02971-1
  • MathSciNet review: 1283549