Inequalities for mixed Schur functions

Abstract

If A k =( a k ij ), k=1,2,…, n, are n× n positive semidefinite matrices and if α:S n → C , where S n is the symmetric group of degree n, an inequality is obtained for the “mixed Schur function,” ∑ σ,τ∈S n α(σ) α(τ) ∏ i=1 n a i σ(i)τ(i) When the matrices A k , k=1,2,…, n, are all equal, we get some known results due to Schur as consequences of the inequality. It is also deduced that the mixed discriminant of a set of positive semidefinite matrices exceeds or equals the geometric mean of their determinants.