Automorphic Carathéodory–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 (Carathéodory–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).