Given $u$ a harmonic function on a ball of radius $r$. i.e.
$$ -\Delta u=0 \qquad{\text{in $B_r(0)$}} $$
Then show that $$ |\nabla u(0)|\leq C \frac{1}{r}\def\avint{\mathop{\,\rlap{-}\!\!\int}\nolimits} \avint_{\partial B_r(0)}|u| $$
It would be a great help if I could set some insight on how this inequality changes in case of a non-homogeneous equation and when there is a $p$-Laplacian.