Generalized Hellinger metric and Audenaert's in-betweenness

Let σ and τ be Kubo-Ando means [1]. In this article we consider the in-betweenness property [2] for τ with respect to the generalized Hellinger metric induced by σ. That is, we show that when dσ represents the generalized Hellinger metric corresponding to σ,dσ(A,AτB)≤dσ(A,B).