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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

$|\nabla|\nabla u||^2\leq\frac{n−1}{n}(\operatorname{trace}(\operatorname{Hess}(u)))^2$ ?

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It should be worth mentioning that this was also asked at MO link –  Ben A. Jan 14 '13 at 22:21

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