Let be positive definite. For , the generalized geometric mean , of and is defined as . In particular, henceforth denoted by is called the geometric mean of and . Apparently, this notion first appeared in W. Pusz and S. L. Woronowicz [Rep. Math. Phys. 8, 159–170 (1975; Zbl 0327.46032)]. Later, T. Ando [Linear Algebra Appl. 26, 203–241 (1979; Zbl 0495.15018)] developed many of the fundamental properties of the geometric mean in a systematic manner. G. E. Trapp [Linear Multilinear Algebra 16, 113–123 (1984; Zbl 0548.15013)] presented a more recent survey of matrix means. In a recent paper, T. Ando, C.-K. Li and R. Mathias [Linear Algebra Appl. 385, 305–334 (2004; Zbl 1063.47013)] have shown for positive definite matrices that implies .
The authors of the paper under review obtain the following generalization (Theorem 3.1) of the above result: In what follows, means that is positive semi-definite and means that is positive definite. Suppose that are positive definite invertible matrices such that for some positive numbers and . Suppose also that for some . Set and . Then for each , the following two matrix inequalities hold:
The authors also obtain certain examples and counterexamples related to the main result.