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 $((A^p+B^p)/2)^{1/p}\le((A^q+B^q)/2)^{1/q}$ 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.