The author proves the following Liouville-type theorem for a complete Riemannian manifold \(M\) whose Ricci curvature is bounded from below: If \(f\geq 0\) is a \(C^2\) function with \(\triangle f\geq c_0 f^n\) for some \(c_0>0\) and \(n>1\), then \(f\) vanishes. This generalizes a theorem of S. Nishikawa (\(c_0=2\) and \(n=2\)) [Nagoya Math. J. 95, 117-124 (1984; Zbl 0544.53050)].