Comparing invariant distances and conformal metrics on Riemann surfaces

We exhibit sharp inequalities comparing the hyperbolic distance land hy perbolic metric) to distances land associated metrics) defined via positive harmonic functions as well as bounded harmonic functions. In the simply connected case, all four inequalities are identities. For the non-simply connected case, we determine precisely when equality can hold: for a pair of points in a distance inequality, or a single point in a metric inequality.