  On the setting of positive definite operators we study the near-order
properties of power means such as the quasi-arithmetic mean (H\"{o}lder mean)
and R\'{e}nyi power mean. We see the monotonicity of spectral geometric mean
and Wasserstein mean on parameters with respect to the near-order and the
near-order relationship between the spectral geometric mean and Wasserstein
mean. Furthermore, the monotonicity of quasi-arithmetic mean on parameters and
the convergence of R\'{e}nyi power mean to the log-Euclidean mean with respect
to the near-order have been established.
