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 $K_0$,$$|\mathrm{\Delta }f(x)|\le K{r}^{-1}sup|\mathrm{\nabla }f|+K_0{r}^{-2}sup|f|,$$where the supremum is taken over $B_r(x)=\{z:|z-x|<r\}$. If in addition $K_0=0$, then $|\mathrm{\nabla }f{|}^p$ has a similar property.