Subharmonic behaviour of smooth functions

Abstract

We prove that $|f|^p$, $p>0$, behaves like a subharmonic function if $f$ is a $C^2$-function such that, for some constants $K$ and $K_0$,$$|\Delta f(x)|\leq Kr^{-1}\sup|\nabla f|+K_0r^{-2}\sup|f|,$$where the supremum is taken over $B_r(x)=\{\,z\,:|z-x|<r\,\}$. If in addition $K_0=0$, then $|\nabla f|^p$ has a similar property.