  In this paper we prove Hessian and Laplacian comparison theorems for the
Lorentzian distance function in a spacetime with sectional (or Ricci) curvature
bounded by a certain function by means of a comparison criterion for Riccati
equations. Using these results, under suitable conditions, we are able to
obtain some estimates on the higher order mean curvatures of spacelike
hypersurfaces satisfying a Omori-Yau maximum principle for certain elliptic
operators.
