  Using an approach recently developed by Nourdin and Poly, we improve the rate
in an inequality for the total variation distance between two double
Wiener-It\^o integrals originally due to Davydov and Martynova. An application
to the rate of convergence of a functional of a correlated two-dimensional
fractional Brownian motion towards the Rosenblatt random variable is then
given, following a previous study by Maejima and Tudor.
