Abstract
We obtain eigenvalue inequalities for matrix geometric means of positive definite matrices. This implies matrix norm inequalities for unitarily invariant norms, which are considered as complementary to a series of norm inequalities among geometric means. We give complements of the Ando–Hiai type inequality for the Karcher mean by means of the generalized Kantorovich constant. Finally, we consider the monotonicity of the eigenvalue function for the Karcher mean.
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Acknowledgements
The authors would like to express their cordial thanks to the referee for his/her valuable suggestions.
Notes
No potential conflict of interest was reported by the authors.