Some matrix inequalities for weighted power mean

Abstract

In this paper, we prove that, for any positive definite matrices $A,B$, and real numbers $\nu ,\mu ,p$ with $-1\le p<1$ and $0<\nu \le \mu <1$, we have

$$\frac{\nu }{\mu }(A{\nabla }_{\mu }B-A{♯}_{p,\mu }B)\le A{\nabla }_{\nu }B-A{♯}_{p,\nu }B\le \frac{1-\nu }{1-\mu }(A{\nabla }_{\mu }B-A{♯}_{p,\mu }B),$$ where ${\nabla }_{\nu }$ and ${♯}_{p,\nu }$ stand for weighted arithmetic and power mean, respectively. In the special cases when $p=0,1$, this inequality can be considered as a generalization of harmonic-arithmetic and geometric-arithmetic means inequalities and their reverses.

Applying this inequality, some inequalities for the Heinz mean and determinant inequalities related to weighted power means are obtained.