  On the space of positive definite matrices, several operator means are
popular and have been studied extensively. In this paper, we investigate the
near order and the L\"owner order relations on the curves defined by the
Wasserstein mean and the spectral geometric mean. We show that the near order
$\preceq $ is stronger than the eigenvalue entrywise order, and that
$A\natural_t B \preceq A\diamond_t B$ for $t\in [0,1]$. We prove the
monotonicity properties of the curves originated from the Wasserstein mean and
the spectral geometric mean in terms of the near order. The L\"owner order
properties of the Wasserstein mean and the spectral geometric mean are also
explored.
