Radial images by holomorphic mappings,

Abstract

Let R \mathcal {R} be a nonexceptional Riemann surface, other than the punctured disk. We prove that if f f is a holomorphic mapping from the unit disk Δ \Delta of the complex plane into R \mathcal {R} , then the set of radial images that remain bounded in the Poincaré metric of R \mathcal {R} has Hausdorff dimension at least δ ( R ) \delta (\mathcal {R}) , the exponent of convergence of R \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.