Generalized Ahlfors functions

Abstract

Let Σ be a bordered Riemann surface with genus g and m boundary components. Let {γ z } z∈∂Σ be a smooth family of smooth Jordan curves in C which all contain the point 0 in their interior. Let p ∈ Σ and let F be the family of all bounded holomorphic functions f on S such that f(p) > 0 and f(z) ∈ γz for almost every z ∈ ∂Σ. Then there exists a smooth up to the boundary holomorphic function f 0 ∈ F with at most 2g+m-1 zeros on S so that f 0 (z) ∈ γ z for every z ∈ ∂Σ and such that f 0 (p) ≥ f(p) for every f ∈ F. If, in addition, all the curves {γ z } z∈∂Σ are strictly convex, then f 0 is unique among all the functions from the family F.