## Annals of Functional Analysis

- Ann. Funct. Anal.
- Volume 7, Number 2 (2016), 348-357.

### 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

$$\frac{\nu}{\mu}(A{\nabla}_{\mu}B-A{\u266f}_{p,\mu}B)\le A{\nabla}_{\nu}B-A{\u266f}_{p,\nu}B\le \frac{1-\nu}{1-\mu}(A{\nabla}_{\mu}B-A{\u266f}_{p,\mu}B),$$ where ${\nabla}_{\nu}$ and ${\u266f}_{p,\nu}$ 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.

#### Article information

**Source**

Ann. Funct. Anal., Volume 7, Number 2 (2016), 348-357.

**Dates**

Received: 29 June 2015

Accepted: 18 October 2015

First available in Project Euclid: 8 April 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.afa/1460141560

**Digital Object Identifier**

doi:10.1215/20088752-3544480

**Mathematical Reviews number (MathSciNet)**

MR3484388

**Zentralblatt MATH identifier**

1337.15020

**Subjects**

Primary: 15A45: Miscellaneous inequalities involving matrices

Secondary: 47A64: Operator means, shorted operators, etc. 26E60: Means [See also 47A64]

**Keywords**

weighted arithmetic mean weighted power mean positive definite matrices matrix inequality

#### Citation

Khosravi, Maryam. Some matrix inequalities for weighted power mean. Ann. Funct. Anal. 7 (2016), no. 2, 348--357. doi:10.1215/20088752-3544480. https://projecteuclid.org/euclid.afa/1460141560