  Studying the existence of rational functions with given branching datum 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 3
branching points on the Riemann sphere. Given two compact Riemann surfaces $M$
and $N$, a pair $(d,\mathcal{D})$ of an integer $d\geq2$ and a collection
$\mathcal{D}$ of nontrivial partitions of $d$ is called a candidate branching
datum if it satisfies the Riemann-Hurwitz formula. And a candidate branching
datum is exceptional if there does not exist a rational function realization
it. As applications, we present some new types of exceptional branching datum.
These results cover some previous results mentioned in \cite{EKS84,PP06,Zhu19}.
We also deduce the realizability of a certain type of candidate branching datum
on the Riemann sphere.
