Minkowski Inequality

Minkowski Inequality states:

Let $r>s$ be a nonzero real number, then for any positive numbers $a_{ij}$ , the following inequality holds:

$(\sum_{j=1}^{m}(\sum_{i=1}^{n}a_{ij}^r)^{s/r})^{1/s}\geq (\sum_{i=1}^{n}(\sum_{j=1}^{m}a_{ij}^s)^{r/s})^{1/r}$

Notics that if one of $r,s$ is zero, the inequality is equivelant to Holder's Inequality.

Page