On matrix inequalities between the power means: counterexamples

TL;DR

This paper establishes the optimal conditions on parameters for matrix power mean inequalities to hold universally, using counterexamples to demonstrate the bounds' sharpness.

Contribution

It proves the best possible parameter conditions for matrix power mean inequalities and clarifies conditions for positive linear maps, using explicit counterexamples.

Findings

Identifies sharp bounds for matrix power mean inequalities.

Constructs explicit 2x2 counterexamples to demonstrate optimality.

Clarifies conditions for inequalities involving positive linear maps.

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 $2\times2$ 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.