On the matrix Heinz means and Rényi divergences

Bhatia, Lim and Yamazaki obtained results on the norm minimality of the Kubo-Ando Heinz means of positive semidefinite matrices. Namely, they proved that
|||A♯αB+A♯1−αB|||≤|||AαB1−α+A1−αBα|||,for α∈(0,1), in the case of the Schatten 1-norm or 2-norm. In this article, we refine these results via the matrix function in the definition of the Rényi divergence. We show that four different types of inequalities hold, under certain conditions, for Schatten 1- and 2-norms.