Ahlfors functions on non-planar Riemann surfaces whose double are hyperelliptic

摘要

Let R be a finite bordered Riemann surface which is a regular subregion of a Riemann surface. The genus of R is finite and the boundary partial deriv R of R consists of a finite number of contours. Let J be the class of holomorphic functions h on R satisfying |h| < 1 on R. Let P be a point on R. If f ∈ J satisfies |(f o φ~(-1))′ (φ(P))| = sup {|(h o φ(-1))′ (φ(P))| : h ∈ J} for a fixed local parameter φ, then we call f the Ahlfors function at P on R. It is known, by Ahlfors, that the Ahlfors function exists at each point of R and is uniquely determined up to a constant multiple of absolute value 1. We note that the Ahlfors function does not depend on a choice of a local parameter φ.