  A major result concerning perturbations of integrable Hamiltonian systems is
given by Nekhoroshev estimates, which ensures exponential stability of all
solutions provided the system is analytic and the integrable Hamiltonian not
too degenerate. In the particular but important case where the latter is
quasi-convex, these exponential estimates have been generalized by Marco and
Sauzin if the Hamiltonian is Gevrey regular, using a method introduced by
Lochak in the analytic case. In this paper, using the same approach we will
investigate the situation where the Hamiltonian is assumed to be only finitely
differentiable, it is known that exponential stability does not hold but
nevertheless we will prove estimates of polynomial stability.
