The fixed points of an analytic self-mapping

Let R be a hyperbolic Riemann surface embedded in a compact Riemann surface of genus g and let f be an analytic function mapping R into R, f not the identity function. Then f has as most 2g + 2 distinct fixed points in R; equality may hold. If f has 2 or more distinct fixed points, then f is a periodic conformai automorphism of R onto itself. This paper contains a proof of this theorem and several related results.