@article{1189.15030,
author="Ito, Masatoshi and Seo, Yuki and Yamazaki, Takeaki and Yanagida,
    Masahiro",
title="{On a geometric property of positive definite matrices cone.}",
language="English",
journal="Banach J. Math. Anal. ",
volume="3",
number="2",
pages="64-76",
year="2009",
abstract="{Let $A,B \in \mathbb{C}^n$ be positive definite. For $t \in [0,1]$,
    the generalized geometric mean $A {\#}_t B$, of $A$ and $B$ is defined as $A
    {\#}_t B=A^{\frac{1}{2}}(A^{-\frac{1}{2}}BA^{-\frac{1}{2}})^t
    A^{\frac{1}{2}}$. 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 {\it W. Pusz} and {\it S. L. Woronowicz} [Rep.
    Math. Phys. 8, 159--170 (1975; Zbl 0327.46032)]. Later, {\it 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.
    {\it 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, {\it T. Ando, C.-K. Li} and {\it 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, $X \leq
    Y$ 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 $0<mI \leq A,B,C,D \leq MI$ 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: 
 $$\left\{ \frac{(h^{2-\frac{1}{{\alpha}_0}}
    +1)^2}{4h^{2-\frac{1}{{\alpha}_0}}} \right\}^{-{\alpha}_0} ~G \leq (A
    {\#}_{\beta} C) {\#}_{\alpha}(B {\#}_{\beta} D) \leq \left\{
    \frac{(h^{2-\frac{1}{{\alpha}_0}} +1)^2}{4h^{2-\frac{1}{{\alpha}_0}}}
    \right\}^{-{\alpha}_0}G.$$
 The authors also obtain certain examples and
    counterexamples related to the main result.}",
reviewer="{K. C. Sivakumar (Chennai)}",
keywords="{positive definite matrix; geometric mean; Riemannian metric; matrix
    inequalities}",
classmath="{*15A45 (Miscellaneous inequalities involving matrices)
15B48 ( )
26E60 (Means)
}",
}