Automorphic Carathéodory–Julia Theorem

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