Some new rearrangement inequalities having application in stability analysis

The Sylvester test for establishing the positivity of quadratic forms is a basic tool For nonquadratic forms, however, necessary and sufficient conditions for positivity are generally not known. Given here are some simple necessary and suffcient conditions for forms of the type
\sum\min{k,l=1}\max{n} x_{k}m_kl}f(x_{1})
to be positive. These results are derived by combining a classic result of Hardy, Littlewood, and Polya with the Birkhoff characterization of doubly stochastic matrices. The results are applied to the difference equations governing a nonlinear feedback loop. In this setting they yield new and quite general conditions for stability.