Convex geometric means

A class of multivariable weighted geometric means of positive definite matrices admitting Jensen-type inequalities for geodesically convex functions is considered. It is shown that there are infinitely many such geometric means including the weighted inductive, Bini–Meini–Poloni and Karcher means and each of these means provides a geometric mean majorization on the space of positive definite matrices. Some connections between our geometric mean majorizations and classical results of the standard majorization of real numbers are discussed. In particular, we establish the Hardy–Littlewood–Pólya majorization theorem and also Rado’s theorem and Schur’s convexity theorem for the weighted Karcher mean.