Trace inequalities for positive definite matrix power products (Q913899)

Let A and B be Hermitian positive semidefinite \(N\times N\) matrices and let the eigenvalues of A and B be \(a_ 1,...,a_ N\) and \(b_ 1,...,b_ N\), respectively, with \(0\leq a_ 1\leq a_ 2\leq...\leq a_ N\) and \(0\leq b_ 1\leq b_ 2\leq...\leq b_ N\). The authors give an elementary proof for the inequalities: \[ \sum^{N}_{i=1}a^ n_ i b^ n_{N-i}\leq tr(AB)^ n\leq \sum^{N}_{i=1}a^ n_ i b^ n_ i, \] where n is a natural number.