  On a complete Riemannian manifold M with Ricci curvature satisfying
$$\textrm{Ric}(\nabla r,\nabla r) \geq -Ar^2(\log r)^2(\log(\log
r))^2...(\log^{k}r)^2$$ for $r\gg 1$, where A>0 is a constant, and r is the
distance from an arbitrarily fixed point in M. we prove some Liouville-type
theorems for a C^2 function $f:M\rightarrow \Bbb R$ satisfying $\Delta f\geq
F(f)$ for a function $F:\Bbb R\rightarrow \Bbb R$.
