On matrix inequalities between the power means: Counterexamples

We prove that the known sufficient conditions on the real parameters (p,q) for which the matrix power mean inequality ((Ap+Bp)/2)1/p⩽((Aq+Bq)/2)1/q holds for every pair of matrices A,B>0 are indeed best possible. The proof proceeds by constructing 2×2 counterexamples. The best possible conditions on (p,q) for which Φ(Ap)1/p⩽Φ(Aq)1/q holds for every unital positive linear map Φ and A>0 are also clarified.