  There exist lots of distinct geometric means on the cone of positive definite
Hermitian matrices such as the metric geometric mean, spectral geometric mean,
log-Euclidean mean and Wasserstein mean. In this paper, we prove the
log-majorization relation on the singular values of the product of given two
positive definite matrices and their (metric and spectral) geometric means. We
also establish the weak log-majorization between the spectra of two-variable
Wasserstein mean and spectral geometric mean. In particular, we verify with
certain condition on variables that two-variable Wasserstein mean converges
decreasingly to the log-Euclidean mean with respect to the weak
log-majorization.
