Near-order relation of power means

On the setting of positive definite operators we study the near-order properties of power means such as the quasi-arithmetic mean (Hölder mean) and Rényi 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 Rényi power mean to the log-Euclidean mean with respect to the near-order have been established.