Let $k$ and $l$ be two given positive integers and $a_{ij}(1 \leq i \leq k, 1 \leq j \leq l)$ be $kl$ positive integers. Show that if $q \geq p > 0$, then \[(\sum_{j=1}^{l}(\sum_{i=1}^{k}a_{ij}^{p})^{q/p})^{1/q}\leq (\sum_{i=1}^{k}(\sum_{j=1}^{l}a_{ij}^{q})^{p/q})^{1/p}.\]
Denote $r= \frac{q}{p}$ and $b_{ij}=a_{ij}^{p}$. Then $r \geq 1$ and the inequality is equivalent to:
$(\Sigma^{l}_{j=1}(\Sigma^{k}_{i=1}b_{ij})^{r})^{\frac{1}{r}}\leq \Sigma^{k}_{i=1}(\Sigma^{l}_{j=1}b_{ij}^{r})^{\frac{1}{r}}$.
We shall proceed by Mathematical Induction on $k$. For $k=1$, we have equality. For $k=2$ we have Minkowski's Inequality.
Suppose that for some $k \geq 3$ the inequality holds for $k-1$. Then by the induction hypothesis we have
$\Sigma^{k}_{i=1}(\Sigma^{l}_{j=1}b_{ij}^{r})^{\frac{1}{r}}\geq (\Sigma^{l}_{j=1}(\Sigma^{k-1}_{i=1}b_{ij})^{r})^{\frac{1}{r}}+(\Sigma^{l}_{j=1}b_{kj}^{r})^{\frac{1}{r}}$.
Using the induction assumption for $k=2$ with:
$B_{1}= \Sigma^{k-1}_{i=1}b_{ij}$, $B_{2}=b_{kj}$
we have
${(\Sigma^{l}_{j=1}(\Sigma^{k-1}_{i=1}b_{ij})^{r}})^{\frac{1}{r}}+(\Sigma^{l}_{j=1}b_{kj}^{r})^{\frac{1}{r}}\geq (\Sigma^{l}_{j=1}(\Sigma^{k}_{i=1}b_{ij})^{r})^{\frac{1}{r}}$.
Q.E.D.