Annals of Functional Analysis

Some matrix inequalities for weighted power mean

Maryam Khosravi

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Abstract

In this paper, we prove that, for any positive definite matrices A,B, and real numbers ν,μ,p with 1p<1 and 0<νμ<1, we have

νμ(AμBAp,μB)AνBAp,νB1ν1μ(AμBAp,μB), where ν and p,ν 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


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