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Convex maps on n and positive definite matrices
Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 645-649.

We obtain several convexity statements involving positive definite matrices. In particular, if A,B,X,Y are invertible matrices and A,B are positive, we show that the map

(s,t)TrlogX * A s X+Y * B t Y

is jointly convex on 2 . This is related to some exotic matrix Hölder inequalities such as

sinh i=1 m A i B i sinh i=1 m A i p 1/p sinh i=1 m B i q 1/q

for all positive matrices A i ,B i , such that A i B i =B i A i , conjugate exponents p,q and unitarily invariant norms ·. Our approach to obtain these results consists in studying the behaviour of some functionals along the geodesics of the Riemanian manifold of positive definite matrices.

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DOI : 10.5802/crmath.25
Classification : 47A30, 15A60

Jean-Christophe Bourin 1 ; Jingjing Shao 2

1 Laboratoire de mathématiques, Université de Franche-Comté, 25000 Besançon, France
2 College of Mathematics and Statistic Sciences, Ludong University, Yantai 264001, China
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     title = {Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices},
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Jean-Christophe Bourin; Jingjing Shao. Convex maps on $\protect \mathbb{R}^n$ and positive definite matrices. Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 645-649. doi : 10.5802/crmath.25. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.25/

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[3] Rajendra Bhatia Positive Definite Matrices, Princeton Series in Applied Mathematics, Princeton University Press, 2007 | Zbl

[4] Jean-Christophe Bourin; Eun-Young Lee Matrix inequalities from a two variables functional, Int. J. Math., Volume 27 (2016) no. 9, 16500771, 19 pages | MR | Zbl

[5] Constantin P. Niculescu Convexity according to the geometric mean, Math. Inequal. Appl., Volume 3 (2000) no. 2, pp. 155-167 | MR | Zbl

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