Boundary limits of subharmonic functions in the disc

Abstract:

In this note we prove the following: let $f\not \equiv - \infty$ be subharmonic in $\left | z \right | < 1$ satisfying $\lim \limits _{r \to 1} \int _0^{2\pi } {f(r{e^{i\theta }})d\theta = 0}$ with $f(z) \leqslant 0$; then \[ \lim \limits _{r \to 1} \sup (1 - r)\inf \limits _{\left | z \right | = r} f(z) = 0\].