On the monotonicity of weighted power means for matrices

In this article, we provide an alternate proof of the fact that the weighted power means μp(A,B,t)=(tAp+(1−t)Bp)1/p, 1≤p≤2 satisfy Audenaert\u27s “in-betweenness” property for positive semidefinite matrices. We show that the “in-betweenness” property holds with respect to any unitarily invariant norm for p=1/2 and with respect to the Euclidean metric for p=1/4. We also show that the only Kubo–Ando symmetric mean that satisfies the “in-betweenness” property with respect to any metric induced by a unitarily invariant norm is the arithmetic mean. Finally, for p=6 we give a counterexample to a conjecture by Audenaert regarding the “in-betweenness” property