I have rational map $R$ of the Riemann sphere $\overline{\mathbb{C}}$ onto itself. Consider the spherical metric $\sigma$ on $\overline{\mathbb{C}}$, I want to show that it $R$ satisfies the Lipschitz condition on $\overline{\mathbb{C}}$. That is there exists positive constant $M$ such that \begin{align} \sigma(R(z), R(w)) < M \sigma(z,w) \end{align} for all $z$ and $w$ in the Riemann sphere.
I know one way of proving it is that we can work with chordal metric. Since chordal metric and spherical metric are equivalent then $R$ also satisfies the Lipschitz condition.
But I am stuck because I am starting doing this itself from the spherical metric. May be one of the reason is that I am not comfortable with spherical metric.