Extensions of inequalities for unitarily invariant norms via log majorization

Abstract

A capital letter means n×n matrix. T is said to be positive definite (denoted by T>0) if T is positive semidefinite and invertible. We shall show the following central results via log majorization obtained by an order preserving operator inequality.Theorem.IfA>0andB⩾0, then for0⩽α⩽1,t∈[0,1]andr⩾tA1-t2At♯αBA1-t2s≻(log)w2(Ar♯αBs)Aw2holds for(1-α)(r-t)1-αt+1⩾s⩾1,wherew=(1-α)(s-r)+α(1-t)s.Our result extends the following recent elegant inequality by Matharu and Aujlia.Let A,B be positive definite and α∈[0,1]. Then∏i=1kλj(A1-αBα)⩾∏i=1kλj(A♯αB)1⩽k⩽n.Also some results associated with log majorization are shown.