  In this paper we study the behavior of the scalar curvature $S$ of a complete
hypersurface immersed with constant mean curvature into a Riemannian space form
of constant curvature, deriving a sharp estimate for the infimum of $S$. Our
results will be an application of a weak Omori-Yau maximum principle due to
Pigola, Rigoli and Setti \cite{PRS}.
