Bloch Constants for Meromorphic Functions Near an Isolated Singularity

Abstract

Suppose $f$ is meromorphic in a punctured neighborhood of the origin and has an essential singularity at the origin. Given any $\varepsilon > 0$ we show that the Riemann surface of $f$ contains an unramified disk of spherical radius $\pi /3 - \varepsilon$. The number $\pi /3$ can be replaced by $\pi /2$ if $f$ is locally schlicht and this value is best possible. If $f$ is actually holomorphic, then the Riemann surface of $f$ contains arbitrarily large unramified euclidean disks. These results generalize theorems of Valiron and Ahlfors dealing with holomorphic and meromorphic functions, respectively, on the complex plane which have an essential singularity at infinity.