  Carleson showed that the Bergman space for a domain on the plane is trivial
if and only if its complement is polar. Here we give a quantitative version of
this result. One is the Suita conjecture, established by the first-named author
in 2012, the other is an upper bound for the Bergman kernel in terms of
logarithmic capacity. We give some other estimates for those quantities as
well. We also show that the volume of sublevel sets for the Green function is
not convex for all regular non simply connected domains, generalizing a recent
example of Forn\ae ss.
