  By improving the analysis developed in the study of $\s_k$-Yamabe problem, we
prove in this paper that the De Lellis-Topping inequality is true on
3-dimensional Riemannian manifolds of nonnegative scalar curvature. More
precisely, if $(M^3, g)$ is a 3-dimensional closed Riemannian manifold with
non-negative scalar curvature, then \[\int_M |Ric-\frac{\bar R} 3 g|^2 dv
(g)\le 9\int_M |Ric-\frac{R} 3 g|^2dv(g), \] where $\bar R=vol (g)^{-1} \int_M
R dv(g)$ is the average of the scalar curvature $R$ of $g$. Equality holds if
and only if $(M^3,g)$ is a space form. We in fact study the following new
conformal invariant \[\ds \widetilde Y([g_0]):=\sup_{g\in {\cal
C}_1([g_0])}\frac {\ds vol(g)\int_M \s_2(g) dv(g)} {\ds (\int_M \s_1(g)
dv(g))^2}, \] where ${\cal C}_1([g_0]):=\{g=e^{-2u}g_0\,|\, R>0\}$ and prove
that $\widetilde Y([g_0])\le 1/3$, which implies the above inequality.
