 Rational map is Lipschitz in the chordal metric

A rational map is smooth as a map between compact round spheres, including at its poles in reciprocal coordinates. Its spherical differential norm therefore has a finite supremum $L$. Integrating along a shortest spherical geodesic gives $\rho (Rz,Rw)\le L\rho (z,w)$. Since $2\chi \le \rho \le \pi \chi$, we obtain $\chi (Rz,Rw)\le (\pi L/2)\chi (z,w)$. In finite coordinates the differential norm is $R^{\#}(z)=|R^{\prime }(z)|(1+|z{|}^2)/(1+|R(z){|}^2)$, extending continuously to the whole sphere. This uniform bound is useful for passing equicontinuity through compositions.