  The paper gives an improvement of the Trudinger-Moser inequality, in which
the constraint set is defined not by the squared gradient norm, but with the
squared gradient norm minus a remainder term of the weighted L^p-type. This is
a two-dimensional counterpart of the Hardy-Sobolev-Mazya inequality in higher
dimensions, which is a similar refinement of the limiting Sobolev inequality.
In particular, we generalize two known cases of remainder terms of potential
type (i.e. weighted L^2-terms) found by Adimurthi and Druet and by Wang and Ye.
In addition, we prove the inequality with a L^p-remainder, p>2, as well as give
an analogous improvement for the Onofri-Beckner inequality.
