Fixed points of analytic self-mappings of Riemann surfaces (Q1257129)

Let \(X\) be a hyperbolic Riemann surface and let \(\varphi\) be an analytic self-mapping of \(X\). The existence of a fixed point of \(\varphi\) combined with some other conditions implies that \(\varphi\) is a conformal automorphism of finite order or the identity mapping. The author obtains such conditions and estimates the order of a conformal automorphism.