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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

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1435-5337
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Volume 30, Issue 3

# The quasi-arithmetic means and Cartan barycenters of compactly supported measures

Sejong Kim
Published Online: 2017-10-06 | DOI: https://doi.org/10.1515/forum-2017-0136

## Abstract

Since positive definite Hermitian matrices have become fundamental objects in many areas, a variety of theoretical and computational research topics have been arisen. Especially, the average of positive definite matrices is a very important notion to see the central tendency of objects. There are many different kinds of averages for a finite number of positive definite matrices such as quasi-arithmetic means, power means and Cartan barycenters. We generalize these averages to the setting of positive definite matrices equipped with probability measures of compact support, and show the monotonicity of quasi-arithmetic means for parameters $\ge 1$, and connections with inequalities between quasi-arithmetic means and power means, and between quasi-arithmetic means and Cartan barycenters.

MSC 2010: 28A25; 15B48

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Revised: 2017-09-12

Published Online: 2017-10-06

Published in Print: 2018-05-01

Funding Source: National Research Foundation of Korea

Award identifier / Grant number: 2015R1C1A1A02036407

This work was supported by a Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Science, ICT and Future Planning (2015R1C1A1A02036407).

Citation Information: Forum Mathematicum, Volume 30, Issue 3, Pages 753–765, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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