  In this paper, we prove a new Heintze-Karcher type inequality for shifted
mean convex hypersurfaces in hyperbolic space. As applications, we prove an
Alexandrov type theorem for closed embedded hypersurfaces with constant shifted
$k$th mean curvature in hyperbolic space. Furthermore, a uniqueness result for
$h$-convex hypersurfaces satisfying certain curvature equations is obtained.
