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.