Radial images by holomorphic mappings,

Fernández, José; Pestana, Domingo

doi:10.1090/S0002-9939-96-02971-1

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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