## 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.## References

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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