On the hyperbolic distance of $n$-times punctured spheres

The length of the shortest closed geodesic in a hyperbolic surface $X$ is called the systole of $X\mathrm{.}$ When $X$ is an $n$-times punctured sphere $\hat{ $\mathbb{C}$} \setminus A$where$A \subset \hat{$\mathbb{C}$}$ is a finite set of cardinality $n\ge 4,$ we define a quantity $Q(A)$ in terms of cross ratios of quadruples in $A$ so that $Q(A)$ is quantitatively comparable with the systole of $X\mathrm{.}$ We next propose a method to construct a distance function $d_X$ on a punctured sphere $X$ which is Lipschitz equivalent to the hyperbolic distance $h_X$ on $X\mathrm{.}$ In particular, when the construction is based on a modified quasihyperbolic metric, $d_X$ is Lipschitz equivalent to $h_X$ with Lipschitz constant depending only on $Q(A)\mathrm{.}$