On matrix inequalities between the power means: counterexamples

Abstract

We prove that the known sufficient conditions on the real parameters $(p,q)$ for which the matrix power mean inequality $((A^p+B^p)/2{)}^{1/p}\le ((A^q+B^q)/2{)}^{1/q}$ holds for every pair of matrices $A,B>0$ are indeed best possible. The proof proceeds by constructing $2\times 2$ counterexamples. The best possible conditions on $(p,q)$ for which $\Phi (A^p{)}^{1/p}\le \Phi (A^q{)}^{1/q}$ holds for every unital positive linear map $\Phi$ and $A>0$ are also clarified.

18 pages, Theorems 2.4 and 2.5 added