  We consider sharp inequalities for Bergman spaces of the unit disc,
establishing analogues of the inequality in Carleman's proof of the
isoperimetric inequality and of Weissler's inequality for dilations. By
contractivity and a standard tensorization procedure, the unit disc
inequalities yield corresponding inequalities for the Bergman spaces of
Dirichlet series. We use these results to study weighted multiplicative Hankel
forms associated with the Bergman spaces of Dirichlet series, reproducing most
of the known results on multiplicative Hankel forms associated with the Hardy
spaces of Dirichlet series. In addition, we find a direct relationship between
the two type of forms which does not exist in lower dimensions. Finally, we
produce some counter-examples concerning Carleson measures on the infinite
polydisc.
