  We prove that the hypotheses in the version of the Omori-Yau maximum
principle that was given by Pigola-Rigoli-Setti are logically equivalent to the
assumption that the manifold carries a $C^2$ proper function whose gradient and
Hessian (Laplacian) are bounded. In particular, this result extends the scope
of the original Omori-Yau principle, formulated in terms of lower bounds for
curvature.
