Question 46
Let $a_{ij}$, $i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$, be positive real numbers. Prove that
$$
\sum_{i=1}^{m} \left( \sum_{j=1}^{n} \frac{1}{a_{ij}} \right)^{-1} \le \left( \sum_{j=1}^{n} \left( \sum_{i=1}^{m} a_{ij} \right)^{-1} \right)^{-1}.
$$
When does the equality hold?
$$
\sum_{i=1}^{m} \left( \sum_{j=1}^{n} \frac{1}{a_{ij}} \right)^{-1} \le \left( \sum_{j=1}^{n} \left( \sum_{i=1}^{m} a_{ij} \right)^{-1} \right)^{-1}.
$$
When does the equality hold?