In this note we prove the following: Let \(f\not\equiv -\infty\) be subharmonic in \(| z| <1\) satisfying \(\lim_{r\to 1}\int^{2\pi}_{0}f(re^{i\theta})d\theta =0\) with f(z)\(\leq 0\); then \(\limsup_{r\to 1}(1-r)\inf_{| z| =r}f(z)=0\).