  Let $X$ be a convex co-compact hyperbolic surface and let $\delta$ be the
Hausdorff dimension of the limit set of the underlying discrete group. We show
that the density of the resonances of the Laplacian in strips ${\sigma\leq
\re(s) \leq \delta}$ with $|\im(s)| \leq T$ is less than
$O(T^{1+\delta-\epsilon(\sigma)})$ with $\epsilon>0$ as long as
$\sigma>\delta/2$. This improves the fractal Weyl upper bounds of Zworski and
supports numerical results obtained for various models of quantum chaotic
scattering.
