Automorphic Carath\'eodory-Julia Theorem

Let $w(\zeta)$ be a function analytic on $\mathbb D$, $|w(\zeta)|\le 1$. Let $|t_0|=1$. Assume that $w$ and $w'$ have nontangential boundary values $w_0$ and $w'_0$, respectively, at $t_0$, $|w_0|=1$. Then (Carath\'eodory - Julia) $t_0\dfrac{w'_0}{w_0}\ge 0$. 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)