  In this paper, we genelize the Heintze-Karcher type inequalities for
fractional Q-curvature $Q_{2\gamma}$ on conformally compact Einstein manifolds.
Such inequality holds for all $\gamma\in (0,1]$. In particular, for
$\gamma=\frac{1}{2}$ and $\gamma=1$, we obtain some rigidity theorems by
characterising the equalities. Moreover, we also verify the Heintze-Karcher
inequality on Riemannian manifolds with sectional curvature bounded below.
