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," R 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.