*(English)*Zbl 1189.15030

Let $A,B\in {\u2102}^{n}$ be positive definite. For $t\in [0,1]$, the generalized geometric mean $A{\#}_{t}B$, of $A$ and $B$ is defined as $A{\#}_{t}B={A}^{\frac{1}{2}}{\left({A}^{-\frac{1}{2}}B{A}^{-\frac{1}{2}}\right)}^{t}{A}^{\frac{1}{2}}$. In particular, $A{\#}_{\frac{1}{2}}B$ henceforth denoted by $A\#B$ is called the geometric mean of $A$ and $B$. 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 $A,B,C,D$ that $A\#B=C\#D$ implies $(A\#C)\#(B\#D)=A\#B$.

The authors of the paper under review obtain the following generalization (Theorem 3.1) of the above result: In what follows, $X\le Y$ means that $Y-X$ is positive semi-definite and $0<X$ means that $X$ is positive definite. Suppose that $A,B,C,D$ are positive definite invertible matrices such that $0<mI\le A,B,C,D\le MI$ for some positive numbers $m$ and $M$. Suppose also that $A{\#}_{\alpha}B=C{\#}_{\alpha}D=G$ for some $\alpha \in (0,1)$. Set ${\alpha}_{0}=min\{\alpha ,1-\alpha \}$ and $h=\frac{M}{m}$. Then for each $\beta \in [0,1]$, the following two matrix inequalities hold:

The authors also obtain certain examples and counterexamples related to the main result.