Generalized Ahlfors functions

Abstract:

Let $\Sigma$ be a bordered Riemann surface with genus $g$ and $m$ boundary components. Let $\lbrace \gamma _{z}\rbrace _{z\in \partial \Sigma }$ be a smooth family of smooth Jordan curves in $\mathbb {C}$ which all contain the point $0$ in their interior. Let $p\in \Sigma$ and let ${\mathcal F}$ be the family of all bounded holomorphic functions $f$ on $\Sigma$ such that $f(p)\ge 0$ and $f(z)\in \widehat {\gamma _z}$ for almost every $z\in \partial \Sigma$. Then there exists a smooth up to the boundary holomorphic function $f_0\in {\mathcal F}$ with at most $2g+m-1$ zeros on $\Sigma$ so that $f_0(z)\in \gamma _z$ for every $z\in \partial \Sigma$ and such that $f_0(p)\ge f(p)$ for every $f\in {\mathcal F}$. If, in addition, all the curves $\lbrace \gamma _z\rbrace _{z\in \partial \Sigma }$ are strictly convex, then $f_0$ is unique among all the functions from the family ${\mathcal F}$.