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 μ_p(A, B, t) = (tA^p + (1 − t)B^p)^1/p, 1 $<em>\le </em>\mathrm{<em>p</em> }\le 2$ , satisfy Audenaert's “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.