Weak log-majorization and inequalities of power means

As non-commutative versions of the quasi-arithmetic mean, we consider the Lim-Pálfia’s power mean, Rényi right mean and Rényi power means. We prove that the Lim-Pálfia’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 Rényi 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 Rényi power mean in terms of the quasi-arithmetic mean.