  We first prove a weighted inequality of Moser-Trudinger type depending on a
parameter, in the two-dimensional Euclidean space. The inequality holds for
radial functions if the parameter is larger than -1. Without symmetry
assumption, it holds if and only if the parameter is in the interval (-1,0].
The inequality gives us some insight on the symmetry breaking phenomenon for
the extremal functions of the Hardy-Sobolev inequality, as established by
Caffarelli-Kohn-Nirenberg, in two space dimensions. In fact, for suitable sets
of parameters (asymptotically sharp) we prove symmetry or symmetry breaking by
means of a blow-up method. In this way, the weighted Moser-Trudinger inequality
appears as a limit case of the Hardy-Sobolev inequality.
