Let 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 whose double is hyperelliptic. Besides we consider the problem when a linear transformation of an Ahlfors function on is again an Ahlfors function. We give a necessary and sufficient condition for this problem when the degree of the Ahlfors function is maximal.
Akira YAMADA. "Ahlfors functions on compact bordered Riemann surfaces." J. Math. Soc. Japan 53 (2) 261 - 283, April, 2001. https://doi.org/10.2969/jmsj/05320261