Question 138
Let $m, n \in \mathbb{N}^*, m, n \gt 1, a_{ij}$ ($i = 1, 2, \dots, n$, $j = 1, 2, \dots, m$) be non-negative real numbers (not all zero). Find the maximum and minimum values of
$$
f = \frac{n \sum_{i=1}^{n} (\sum_{j=1}^{m} a_{ij})^2 + m \sum_{j=1}^{m} (\sum_{i=1}^{n} a_{ij})^2}{(\sum_{i=1}^{n} \sum_{j=1}^{m} a_{ij})^2 + mn \sum_{i=1}^{n} \sum_{j=1}^{m} a_{ij}^2}.
$$
$$
f = \frac{n \sum_{i=1}^{n} (\sum_{j=1}^{m} a_{ij})^2 + m \sum_{j=1}^{m} (\sum_{i=1}^{n} a_{ij})^2}{(\sum_{i=1}^{n} \sum_{j=1}^{m} a_{ij})^2 + mn \sum_{i=1}^{n} \sum_{j=1}^{m} a_{ij}^2}.
$$