  We extend Ando-Hiai's log-majorization for the weighted geometric mean of
positive definite matrices into that for the Cartan barycenter in the general
setting of probability measures on the Riemannian manifold of positive definite
matrices equipped with trace metric. The main key is the settlement of the
monotonicity problem of the Cartan barycenteric map on the space of probability
measures with finite first moment for the stochastic order induced by the cone.
We also derive a version of Lie-Trotter formula and related unitarily invariant
norm inequalities for the Cartan barycenter as the main application of
log-majorization.
