I'm trying to prove the following: let $u\in C^2(\mathbb{R}^n)$ be an harmonic function, then the function $w:=|\nabla(u)|^2$ is subharmonic, that is, $\Delta(w)\geq 0$.
As a matter of fact this exercise has as a previous task to prove that $v(x):=|u|^p$, $p\geq 1$ is subharmonic w.r.t. the mean, that is
\begin{equation} v(x)\leq \frac{1}{B_R}\int_{B_R}v(y) \ dy \end{equation} I have already proven that with the property of $u$ on the mean value of harmonic functions and Holder inequality. I don't really know if this part is supposed to be related with the other. It would obviously enforce my faith in test makers if it was.