  In this paper we consider the generalized surface quasi-geostrophic
$\alpha$-SQG equations, in the "sublinear regime" $\alpha \in (0, 1)$ and we
study the stability of vortex patches close to vortex discs. We shall prove
that for regular, Sobolev initial vortex patches $\varepsilon$-close to a
vortex disc, the solutions stay $\varepsilon$-close to a vortex disc for a time
interval of order $O(\varepsilon^{- 2})$. The proof is based on a
paradifferential Birkhoff normal form reduction, implemented in the case where
the dispersion relation is sublinear.
