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Geometry and inequalities of geometric mean

Geometry and inequalities of geometric mean

Source title:

Czechoslovak Mathematical Journal, 66(3): 777-792,
2016
(ISI)

Academic year of acceptance:

2016-2017

Abstract:

We study some geometric properties associated with the *t*-geometric means *A* ♯ * _{t} B:= A ^{1/2}(A ^{−1/2} BA ^{−1/2}) ^{t} A ^{1/2}* of two

*n × n*positive definite matrices

*A*and

*B*. Some geodesical convexity results with respect to the Riemannian structure of the n × n positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding m pairs of positive definite matrices is posted.