Poincaré metric on the Riemann sphere minus more than two points

Question

If we omit more than two points from the Riemann sphere, we will obtain a hyperbolic Riemann surface endowed with a canonical metric descending from its universal cover which is the Poincaré disk. Let us denote the hyperbolic metric on this surface by $d_h$, and the usual spherical metric on the Riemann sphere by $d$. Here is my question: Can we find a constant $C>0$ such that for any two points $x$ and $y$ in this punctured sphere we have: $$d(x,y)<C d_h(x,y).$$

Answer 1

Yes. The density of the Poincare metric with respect to the spherical metric is a positive continuous function which tends to infinity at the punctures. Thus it is bounded from below by some positive constant. The constant depends only on the configuration of the punctures. For some special configurations of punctures, the exact constant has been explicitly found:

MR1428102 Bonk, Mario; Cherry, William, Bounds on spherical derivatives for maps into regions with symmetries. J. Anal. Math. 69 (1996), 249–274.

The authors of this paper say that for general punctures, the explicit determination of the optimal constant is hopeless, and I agree with them.