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.


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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