  We generalize A. Borb\'ely's condition for the conclusion of the Omori-Yau
maximum principle for the Laplace operator on a complete Riemannian manifold to
a second-order linear semi-elliptic operator $L$ with bounded coefficients and
no zeroth order term. Also, we consider a new sufficient condition for the
existence of a tamed exhaustion function. As a corollary, we show that the
existence of a tamed exhaustion function is more general than the hypotheses in
the version of the Omori-Yau maximum principle that was given by A. Ratto, M.
Rigoli and A.G. Setti.
