Construction of nearly hyperbolic distance on punctured spheres

We define a distance function on the bordered punctured disk 0 < | z| ≤ 1 / e in the complex plane, which is comparable with the hyperbolic distance of the punctured unit disk 0 < | z| < 1. As an application, we will construct a distance function on an n-times punctured sphere which is comparable with the hyperbolic distance. We also propose a comparable quantity which is not necessarily a distance function on the punctured sphere but easier to compute.