Rényi means, Hölder inequality and trace characterizations

Abstract We prove that Hölder-type trace inequalities formulated for Rényi means and, as the limiting case, for the Log-Euclidean mean of positive definite matrices actually characterize the canonical trace functional among all positive linear functionals. Similar results involving the Fiedler–Pták spectral geometric mean are also presented. Our findings are connected to the recent paper [4], they generalize, extend several results in that work. Some further trace characterizations are also presented.