Ahlfors functions on compact bordered Riemann surfaces

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