Inequalities for mixed Schur functions

AbstractIf Ak =(akij), k=1,2,…,n, are n×n positive semidefinite matrices and if α:Sn→C, where Sn is the symmetric group of degree n, an inequality is obtained for the “mixed Schur function,” ∑σ,τ∈Sn α(σ)α(τ)∏i=1naiσ(i)τ(i) When the matrices Ak, 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