Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 15 (2020), 315 -- 324

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This work is licensed under a Creative Commons Attribution 4.0 International License.

MATRIX POWER MEANS AND PÓLYA--SZEGÖ TYPE INEQUALITIES

Mohsen Kian and Fatemeh Rashid

Abstract. It has been shown that if μ is a compactly supported probability measure on Mn+, then for every unit vector η∈ℂn, there exists a compactly supported probability measure (denoted by <μ η,η> on ℝ+ so that the inequality

<Pt(μ)η,η>≤ Pt(<μ η,η>)     (t∈(0,1])
holds. In particular, we consider a reverse of the above inequality and present some Pólya--Szegö type inequalities for power means of probability measures on positive matrices.

2020 Mathematics Subject Classification: Primary 47A64; Secondary 47A63.
Keywords: Power means; Pólya--Szegö inequality; probability measure; positive matrix

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Mohsen Kian
Department of Mathematics University of Bojnord,
P. O. Box 1339, Bojnord 94531, Iran.
e-mail: kian@ub.ac.ir
https://ub.ac.ir/~kian

Fatemeh Rashid
Department of Mathematics University of Bojnord,
P. O. Box 1339, Bojnord 94531, Iran.
e-mail: f-rashid@yahoo.com

http://www.utgjiu.ro/math/sma