We derive lower bounds in rational approximation of given degree to functions in the Hardy space H of the unit disk. We apply these to asymptotic error rates in rational approximation to Blaschke products and to Cauchy integrals on geodesic arcs. We also explain how to compute such bounds, either using Adamjan-Arov-Krein theory or linearized errors, and we present a couple of numerical experiments. We dwell on a maximin principle developed in Baratchart and Seyfert (2002).