Matrix Inequalities of Cubic Type

Abstract

Let $A=(a_{ij})$ be an $m\times n$ nonnegative matrix, with row-sums $r_i$ and column-sums $c_j$. We show that $$mn\sum _{i,j}{a}_{ij}f(r_i)f(c_j)\ge \sum _{i,j}{a}_{ij}\sum _{i}f(r_i)\sum _{j}f(c_j)$$ 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).