  In this paper we show that the eigenvalue map and the symplectic eigenvalue
map of positive definite matrices are Lipschitz for the Cartan-Hadamard
Riemannian metric, and establish log-majorizations for the (symplectic)
eigenvalues of the Cartan barycenter of integrable probability Borel measures.
This leads a version of Jensen's inequality for geometric integrals of
matrix-valued integrable random variables.
