  We establish new functional versions of the Blaschke-Santal\'o inequality on
the volume product of a convex body which generalize to the non-symmetric
setting an inequality of K. Ball and we give a simple proof of the case of
equality. As a corollary, we get some inequalities for $\log$-concave functions
and Legendre transforms which extend the recent result of Artstein, Klartag and
Milman, with its equality case.
