A generalization of POWERS-STØRMER inequality

Abstract

Let $A,B$ be the positive semidefinite matrices. A matrix version of the famous Powers-Størmer's inequality $$2Tr(A^{\alpha }{B}^{1-\alpha })\ge Tr(A+B-|A-B|),0\le \alpha \le 1,$$ was proven by Audenaert et. al. We establish a comparison of eigenvalues for the matrices $A^{\alpha }{B}^{1-\alpha }$ and $A+B-|A-B|,0\le \alpha \le 1,$ subsuming the Powers-Størmer's inequality. We also prove several related norm inequalities.