Annals of Functional Analysis

Some matrix inequalities for weighted power mean

Maryam Khosravi

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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

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

Received: 29 June 2015
Accepted: 18 October 2015
First available in Project Euclid: 8 April 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A45: Miscellaneous inequalities involving matrices
Secondary: 47A64: Operator means, shorted operators, etc. 26E60: Means [See also 47A64]

weighted arithmetic mean weighted power mean positive definite matrices matrix inequality


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

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