Bloch functions on fundamental regions (Q1919028)

Let \(\Gamma\) be a Fuchsian group consisting of Möbius maps of the (open) unit disc (in the finite complex plane) to itself. Let \(A(\Gamma)\) denote the collection of functions analytic on \(D\) and automorphic with respect to \(\Gamma\). For \(f\in A(\Gamma)\), \(f\) is Bloch (little Bloch) on \(\Omega=\{z\in D:|z|<|\gamma z|, id\neq\gamma\in\Gamma\}\) if \[ \text{Sup}_{z\in\overline\Omega}(1-|z|^2)|f'(z)|\leq \infty\quad (\textstyle{\lim_{|z|\to1,z\in\overline\Omega}}(1-|z|^2)f'(z)=0). \] A number of equivalent conditions characterizing Bloch (little Bloch) functions on \(\Omega\) are given.