Comptes Rendus
no. 6
Analyse fonctionnelle
Convex maps on ℝ n and positive definite matrices
Jean-Christophe Bourin1  ; Jingjing Shao2
1 Laboratoire de mathématiques, Université de Franche-Comté, 25000 Besançon, France
2 College of Mathematics and Statistic Sciences, Ludong University, Yantai 264001, China
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 ℝ 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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