Assume that $\rho$ and $\rho'$ are conformal metrics on the unit disk which is a geodesic disk of radius $1$ w.r.t. both metrics $\rho$ and $\rho'$, and assume that $\rho'$ has a constant Gauss curvature $K'$ which is greater than the curvature of $\rho$. Let $d$ and $d'$ be the corresponding distance functions defined on the unit disk. Which is the best reference for the inequality $d(0,a)\le d'(0,a)$.



Comparision theorem for distance function
See [Bob Osserman's 1999 Notices articl](http://www.ams.org/notices/199908/fea-osserman.pdf)e (which is beautifully written, and copiously referenced).


