Abstract
The difficulty of establishing a (noncommutative) matrix inequality involving the geometric mean was discussed in 1978 by K.V. Bhagwat and R. Subramanian [9] who pointed out that the problem of defining a geometric mean for non-commutative operators “makes it difficult to establish the validity or otherwise of the classical inequalities involving the geometric mean”. However, in a recent paper, Sagae and Tanabe [32] define a geometric mean and establish an AG-GM inequality for a finite number of positive definite matrices.
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Pečarić, J., Mond, B. (1997). The arithmetic mean — the geometric mean and related matrix inequalities. In: Bandle, C., Everitt, W.N., Losonczi, L., Walter, W. (eds) General Inequalities 7. ISNM International Series of Numerical Mathematics, vol 123. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8942-1_8
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