Title: A note on Rational Functions with three branched points on the Riemann sphere

Abstract: Studying the existence of rational functions with given branched data is a classical problem in the field of complex analysis and algebraic geometry. This problem dates back to Hurwitz and remains open to this day. In this paper, we utilize complex analysis to establish a property of rational functions with three branched points on the Riemann sphere. As applications, we present some new types of exceptional branched data. These results cover some previous results mentioned in \cite{EKS84,PP06,Zhu19}. We also establish the existence of a certain type of rational functions on the Riemann sphere.