  We derive some lower bounds in rational approximation of given degree to
functions in the Hardy space $H^2$ of the disk. We apply these to asymptotic
errors rates in 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 on several types of functions. We dwell on the
Adamjan-Arov-Krein theory and a maximin principle developed in the article "An
L^p analog of AAK theory for p \textgreater{}= 2", by L. Baratchart and F.
Seyfert, in the Journal of Functional Analysis, 191 (1), pp. 52-122, 2012.
