Ahlfors functions on compact bordered Riemann surfaces

Abstract

Let $R$ be a compact bordered Riemann surface which is non-planar. We solve a conjecture posed by Gouma concerning the distribution of degrees of Ahlfors functions on $R$ whose double is hyperelliptic. Besides we consider the problem when a linear transformation of an Ahlfors function on $R$ is again an Ahlfors function. We give a necessary and sufficient condition for this problem when the degree of the Ahlfors function is maximal.