Weak log-majorization and inequalities of power means

As non-commutative versions of the quasi-arithmetic mean, we consider the Lim-P\'{a}lfia's power mean, R\'{e}nyi right mean and R\'{e}nyi power means. We prove that the Lim-P\'{a}lfia's power mean of order $t \in [-1,0)$ is weakly log-majorized by the log-Euclidean mean and fulfills the Ando-Hiai inequality. We establish the log-majorization relationship between the R\'{e}nyi relative entropy and the product of square roots of given variables. Furthermore, we show the norm inequalities among power means and provide the boundedness of R\'{e}nyi power mean in terms of the quasi-arithmetic mean.