Annals of Functional Analysis

Recent developments of matrix versions of the arithmetic–geometric mean inequality

Jun Ichi Fujii, Masatoshi Fujii, Yuki Seo, and Hongliang Zuo

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The main aim of this survey article is to present recent developments of matrix versions of the arithmetic–geometric mean inequality. Among others, we show improvements and generalizations of the arithmetic–geometric mean inequality for unitarily invariant norms via the Hadamard product, and for singular values via the operator monotone functions.

Article information

Ann. Funct. Anal. Volume 7, Number 1 (2016), 102-117.

Received: 25 March 2015
Accepted: 26 May 2015
First available in Project Euclid: 27 November 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A63: Operator inequalities
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors 15A42: Inequalities involving eigenvalues and eigenvectors 47A30: Norms (inequalities, more than one norm, etc.)

arithmetic–geometric mean inequality Heinz inequality unitarily invariant norm singular value Hadamard product


Fujii, Jun Ichi; Fujii, Masatoshi; Seo, Yuki; Zuo, Hongliang. Recent developments of matrix versions of the arithmetic–geometric mean inequality. Ann. Funct. Anal. 7 (2016), no. 1, 102--117. doi:10.1215/20088752-3429400.

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