## Geometry and inequalities of geometric mean.(English)Zbl 1413.15039

Authors’ abstract: We study some geometric properties associated with the $$t$$-geometric means $$A\sharp_tB:=A^{1/2}(A^{-1/2}BA^{-1/2})^{t}A^{1/2}$$ of two $$n\times n$$ positive definite matrices $$A$$ and $$B$$. Some geodesical convexity results with respect to the Riemannian structure of the $$n\times n$$ positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of K. M. R. Audenaert [Electron. J. Linear Algebra 30, 80–84 (2015; Zbl 1326.15030)]. Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding $$m$$ pairs of positive definite matrices is posted.

### MSC:

 15A45 Miscellaneous inequalities involving matrices 15B48 Positive matrices and their generalizations; cones of matrices 15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory 47A64 Operator means involving linear operators, shorted linear operators, etc.

Zbl 1326.15030

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### References:

 [1] T. Ando: Concavity of certain maps on positive definite matrices and applications to Hadamard products. Linear Algebra Appl. 26 (1979), 203–241. · Zbl 0495.15018 [2] T. Ando, F. Hiai: Log majorization and complementary Golden-Thompson type inequalities. Linear Algebra Appl. 197/198 (1994), 113–131. · Zbl 0793.15011 [3] H. Araki: On an inequality of Lieb and Thirring. Lett. Math. Phys. 19 (1990), 167–170. · Zbl 0705.47020 [4] K. M. R. Audenaert: A norm inequality for pairs of commuting positive semidefinite matrices. Electron. J. Linear Algebra (electronic only) 30 (2015), 80–84. · Zbl 1326.15030 [5] R. Bhatia: The Riemannian mean of positive matrices. Matrix Information Geometry (F. Nielsen et al., eds.). Springer, Berlin, 2013, pp. 35–51. · Zbl 1271.15019 [6] R. Bhatia: Postitive Definite Matrices. Princeton Series in Applied Mathematics, Princeton University Press, Princeton, 2007. [7] R. Bhatia: Matrix Analysis. Graduate Texts in Mathematics 169, Springer, New York, 1997. · Zbl 0863.15001 [8] R. Bhatia, P. Grover: Norm inequalities related to the matrix geometric mean. Linear Algebra Appl. 437 (2012), 726–733. · Zbl 1252.15023 [9] J.-C. Bourin: Matrix subadditivity inequalities and block-matrices. Int. J. Math. 20 (2009), 679–691. · Zbl 1181.15030 [10] J.-C. Bourin, M. Uchiyama: A matrix subadditivity inequality for f(A + B) and f(A) + f(B). Linear Algebra Appl. 423 (2007), 512–518. · Zbl 1123.15013 [11] M. Fiedler, V. Pták: A new positive definite geometric mean of two positive definite matrices. Linear Algebra Appl. 251 (1997), 1–20. · Zbl 0872.15014 [12] S. Hayajneh, F. Kittaneh: Trace inequalities and a question of Bourin. Bull. Aust. Math. Soc. 88 (2013), 384–389. · Zbl 1305.15021 [13] M. Lin: Remarks on two recent results of Audenaert. Linear Algebra Appl. 489 (2016), 24–29. · Zbl 1326.15033 [14] M. Lin: Inequalities related to 2 {$$\times$$} 2 block PPT matrices. Oper. Matrices 9 (2015), 917–924. · Zbl 1345.15004 [15] A. W. Marshall, I. Olkin, B. C. Arnold: Inequalities: Theory of Majorization and Its Applications. Springer Series in Statistics, Springer, New York, 2011. · Zbl 1219.26003 [16] A. Papadopoulos: Metric Spaces, Convexity and Nonpositive Curvature. IRMA Lectures in Mathematics and Theoretical Physics 6, European Mathematical Society, Zürich, 2005. · Zbl 1115.53002 [17] W. Pusz, S. L. Woronowicz: Functional calculus for sesquilinear forms and the purification map. Rep. Math. Phys. 8 (1975), 159–170. · Zbl 0327.46032 [18] R. C. Thompson: Singular values, diagonal elements, and convexity. SIAM J. Appl. Math. 32 (1977), 39–63. · Zbl 0361.15009 [19] X. Zhan: Matrix Inequalities. Lecture Notes inMathematics 1790, Springer, Berlin, 2002.
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