Matrix Inequalities of Cubic Type

Let \(A = (a_{ij})\) be an \(m \times n\) nonnegative matrix, with row-sums \(r_i\) and column-sums \(c_j\). We show that \[
mn \sum\limits_{i,j} a_{ij} f(r_i) f(c_j) \geq \sum\limits_{i,j} a_{ij}\sum\limits_{i} f(r_i)\sum\limits_{j} f(c_j)
\] providing the function \(f\) meets certain conditions. When \(f\) is the identity function, this inequality is one proven by Atkinson, Watterson, and Moran in 1960. We also prove another inequality, of similar type, that refines a result of Ajtai, Komlós, and Szemerédi (1981).