Operator means and the reduced relative quantum entropy

We study operator means more general than the Kubo-Ando means. They are given as minima of geodesically convex functions and allow uniquely defined extensions to any number of variables. Multivariate hyper-means is a class of means of this type bounded from below by the arithmetic mean. We extend the definition of a hyper-man in the bivariate case and discover bivariate means that are not restrictions of the multivariate means studied earlier. We introduce the notion of reduced relative quantum entropy and prove that it is convex. The result is used to give a simplified proof of a theorem of Lieb and Seiringer.