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 $((A^p+B^p)\mathrm{/}2{)}^{1\mathrm{/}p}\le ((A^q+B^q)\mathrm{/}2{)}^{1\mathrm{/}q}$ 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 $\mathrm{\Phi }(A^p{)}^{1\mathrm{/}p}\le \mathrm{\Phi }(A^q{)}^{1\mathrm{/}q}$ holds for every unital positive linear map $\mathrm{\Phi }$ and $A>0$ are also clarified.