The arithmetic mean — the geometric mean and related matrix inequalities

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.