Automorphic Carathéodory-Julia Theorem

Abstract

Let $w(\zeta )$ be a function analytic on $\mathbb{D}$, $|w(\zeta )|\le 1$. Let $|t_0|=1$. Assume that $w$ and $w^{\prime }$ have nontangential boundary values $w_0$ and $w_0^{\prime }$, respectively, at $t_0$, $|w_0|=1$. Then (Carathéodory - Julia) $t_0\frac{w_0^{\prime }}{w_0}\ge 0$. The goal of this paper is to obtain a lower bound on this ratio if $w$ is character-automorphic with respect to a Fuchsian group (Theorem 6.1)