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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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closed as off-topic by Michael Albanese, Iuʇǝƃɹɐʇoɹ, Jean-Claude Arbaut, user7530, Claude Leibovici Dec 14 '14 at 3:42

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It should be worth mentioning that this was also asked at MO link – Ben Jan 14 '13 at 22:21
This question has been asked on MO and has answers there. – Michael Albanese Dec 14 '14 at 2:12