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