  We show that a compact almost-K\"ahler four manifold $(M, g, \omega)$ with
harmonic self-dual Weyl curvature and constant scalar curvature is K\"ahler if
$c_{1}\cdot\omega\geq 0$. We also prove an integral curvature inequality for
compact almost-K\"ahler four manifolds with harmonic self-dual Weyl curvature.
