Automorphic Carathéodory-Julia Theorem

Let $w(\zeta )$ be a function analytic on $\mathbb{D}$, $\mathrm{\mid }w(\zeta )\mathrm{\mid }\le 1$. Let $\mathrm{\mid }{t}_0\mathrm{\mid }=1$. Assume that $w$ and $w^{\mathrm{\prime }}$ have nontangential boundary values $w_0$ and $w_0^{\mathrm{\prime }}$, respectively, at $t_0$, $\mathrm{\mid }{w}_0\mathrm{\mid }=1$. Then (Carathéodory - Julia) $t_0{\displaystyle \frac{w_0^{\mathrm{\prime }}}{w_0}}\ge 0$. 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)