On the matrix Heron means and Renyi divergences

Bhatia, Lim, and Yamazaki studied the norm minimality of several Kubo-Ando means of positive semidefinite matrices. Recently, Hiai proved a norm minimality result involving the the weighted geometric mean , which is a matrix function in the definition of the quantum alpha-z-Renyi divergence. In connection to these results, for positive semidefinite matrices, we show that the inequality p = 1, 2, , and , among other related inequalities.