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(2003 / vol.6 / no.2)
Order preserving inequalities induced by some operator functions and its applications
Mariko Giga, Masahiro Yanagida
Pages. 149-153     



This paper is a resume based on our talk at KOTAC 2003,
and also this is an early announcement of cite{8}.

As an application of cite[Theorem 1]{10},
we show a simple proof of the following result:

{itshape%
If $A ge B ge C ge 0$ with $B>0$,
then for each $t in [0,1]$ and $p ge t$,
the following extup{(i)} and extup{(ii)} hold
for a fixed real number $q$,
and they are mutually equivalent:
egin{enumerate}
item
If $q ge 0$, then
[
G_{p,q,t}(A,B,C,r,s)=
A^{frac{-r}2}{A^{frac{r}2}(B^{frac{-t}2}C^p B^{frac{-t}2})^s
A^{frac{r}2}}^{frac{q-t+r}{(p-t)s+r}}A^{frac{-r}2}
]
is a decreasing function for $r ge t$ and $s ge 1$
such that $(p-t)s ge q-t$.
item
If $p ge q$, then
[
G_{p,q,t}(A,B,C,r,s)=
A^{frac{-r}2}{A^{frac{r}2}(B^{frac{-t}2}C^p B^{frac{-t}2})^s
A^{frac{r}2}}^{frac{q-t+r}{(p-t)s+r}}A^{frac{-r}2}
]
is a decreasing function for $s ge 1$ and $r ge max{t, t-q}$.
end{enumerate}
}

This result is an extension of cite[Theorem 2]{10}.
On the other hand,
M.Uchiyama cite{16} shows the following interesting result:

{itshape%
egin{enumerate}
addtocounter{enumi}{2}
item
If $A ge B ge C ge 0$ with $B > 0$,
then for each $t in [0,1]$ and $p ge 1$,
[
A^{1-t+r} ge
{A^{frac{r}2}(B^{frac{-t}2}C^p B^{frac{-t}2})^s
A^{frac{r}2}}^{frac{1-t+r}{(p-t)s+r}}
]
holds for $r ge t$ and $s ge 1$.
end{enumerate}
}
We show that (i) is equivalent to (iii),
that is, they follow from each other.
And also, as an application of cite[Theorem 1]{10},
we give a simple proof of M.Uchiyama's result cite[Theorem 3.4]{16}.



1. Introduction
2. Operator functions implying Theorem U
3. Equivalence relation associated with Theorem
4. Satellite inequalities
5. M.Uchiyama's result via Theorem