Inequalities Involving Kernels

Abstract

Let (a _ij ) be an m × n matrix of nonnegative terms. Then

$$ mn{\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {{a_{ij}}\sum\limits_{r = 1}^m {{a_{rj}}} \sum\limits_{s = 1}^n {{a_{is}}} \geqslant \left( {\sum\limits_{i = 1}^m {\sum\limits_{j = 1}^n {{a_{ij}}} } } \right)} } ^3}, $$

(1.1)

with equality if and only if all the row sums are equal, or all the column sums are equal, or both.