For any real-valued smooth function $u$, we have the Kato inequality
$|\nabla|\nabla u||^2\leq(\operatorname{trace}(\operatorname{Hess}(u)))^2$,
which holds when $|\nabla u|\neq0$.
If moreover $u$ is harmonic (in an open set in $\mathbb{R}^n$), how would the Kato inequality be improved to
$|\nabla|\nabla u||^2\leq\frac{n−1}{n}(\operatorname{trace}(\operatorname{Hess}(u)))^2$ ?