Some new rearrangement inequalities having application in stability analysis

Abstract

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.