  We give an elementary proof of the recent hook inequality given in [MPP3]:
$\prod_{u\in \lambda} h(u) \, \le \, \prod_{u\in \lambda} h^\ast(u),$ where
$h(u)$ is the usual hook in Young diagram $\lambda$, and $h^\ast(i,j)=i+j-1$.
We then obtain a large variety of similar inequalities and their
high-dimensional generalizations.
