Operator Functions on Chaotic Order Involving Order Preserving Operator Inequalities

数理解析研究所講究録第 1778 巻 2012 年 88-98 88

OPERATOR FUNCTIONS ON CHAOTIC ORDER INVOLVING ORDER PRESERVING OPERATOR INEQUALITIES

弘前大学 (名警教授) 古田考之 (Takayuki Furuta) Hirosaki University (Emeritus)

$T/|.r^{J}f\dot{o}r|,r\cdot tl’,$ $(\iota r\}.\uparrow\iota i_{1})er\cdot,\backslash ^{\backslash }\iota\cdot r\tau)$ of Prvf $\cdot.\cdot 7a\backslash \cdot$ .

$A_{11}$ operator $T$ is said to be positive (denoted by $T\geq 0$ ) if $(T.\iota\cdot.:r:)\geq 0$ for all vectol $:\backslash .1^{\cdot}$

$\backslash ^{\neg}di$ $|$ $T$ ill a $Hill)ert\backslash paec$. $i\lambda 11(1T$ is (1 to ) $t^{3}$ strictly positiv$e(rle$11ot $(^{Jd1)\backslash \cdot T}>0)$ if is positive

$1(1$ $arll/ﬁ,\prime J,\cdot)d\delta\geq()$ al invertible. Let $log.\prime 4\geq 1(^{-})gB(\iota r\iota(l?_{1}$ . $r\underline{\cdot)},$ $\ldots.",$ $\geq()$ and , and

$\geq\frac{\delta+\prime_{1+_{-}}?\cdot\prime+\ldots+r_{k-1}}{q[A\cdot-1]}$ $\ldots..p_{t}\geq\frac{\delta+r_{1}+r_{2+\cdots+\prime_{lt-1}}}{q[1\iota-1]}$ $l) \iota\geq\delta.)\geq\frac{\dot{\delta}+1}{l)_{1}+}r_{I}\perp$ ...... $p_{k}$. . .

Let $\mathfrak{F}_{r\iota}(l),,.Y_{\}}.)$ be deﬁned by

$\mathfrak{F}_{?t}.(p_{n}, r_{n})=A^{-.r-\prime}arrow 1-\mathbb{C}_{A.B}[n]^{\frac{\delta+,.+\prime\cdot+\ldots+|?|}{q|,\iota]}A\overline{2}^{g}}$ . Then the following inequalities (i). (ii) und (iii) hold:

(i) $A_{\theta}^{\frac{1)k\underline{.}-I}{}\text{へ_{}A\cdot-\downarrow()_{4}4^{\frac{J^{J}A-1}{2}}}}l^{J}A\cdot-1,$$’.k-1\geq S_{A}\cdot(r)$ for $\lambda:s\uparrow ich$ thaf. $1\leq k\leq?l$ .

$t\backslash +,\perp$ , $-^{-}$ (ii) $B^{\delta}\geq./\{^{\frac{-1}{\prime\sim)}\perp}(\Lambda_{\sim B^{f)_{\downarrow}}A^{\lrcorner}}^{\lrcorner^{r_{2}}}’\cdot, )$ $\overline{\rho_{1+r_{1}}}A2$

$\geq.\sim^{J}1^{\underline{-(r}_{\frac{+r}{\circ\sim},\underline{)}}}.\{\cdot 4^{\underline{r_{2}}}[3^{p\iota}A^{r\prime 0}\sim\iota\lrcorner,(PJ^{J\underline{\backslash }+r}$

$\geq A^{\infty’}\sim\{.--(r+.’|)|..44’-\underline,$ $(\cdot\cdot\cdot zJ_{-44^{\grave{\mathscr{C}}}]^{l)}.)}3/1^{\lrcorner}\}^{\frac{\delta+r1\vdash t_{-}-\succ 1}{\{ip_{1}+r_{1})\rho+r\underline{o}\};,s\dashv\prime}}4\sim’..\cdot.\underline{\cdot.,}\cdot.,,...’\underline{-}(’+.’,+r)$

. $\geq_{A}4\hat{2}.,.\mathbb{C}_{4.B}[7l]^{\underline{\delta}|r}\frac{+\cdot.)\vdash\ldots+\prime_{i}}{q|\prime\cdot|}A_{:}^{\underline{-}(\prime\underline{+}}\underline{-}(r\dashv,\dashv\underline{\ldots\dashv_{\tau\iota})}.\cdot,.\frac{\backslash .\cdot \text{ト}\ldots-\vdash_{l},)}{\underline{\backslash }}$

$|$ $\tilde{\delta}_{l},.(l)_{7},$ $.\iota$ $(th$ (iii) . $t,,)$ is ( decreasing function of ) . $\uparrow 7t\geq 0(r,r|,d\ell)_{\gamma\downarrow}\geq\frac{\delta+\gamma\cdot+r_{\Omega}.+\ldots+\prime}{q[n-||}$ . $wf’.er^{J}\mathbb{C}_{A.B}[n]$ and $q[n,]nre$ (leﬁ$7|,ed$ as follows:

$\mathbb{C}_{4,B}[r’]=A’\circ\sim\{A^{\frac{\prime\prime\prime-1}{\prime\sim)}}$ [.... $A_{\sim}^{;}\{A^{\frac{9}{\underline{9}}}(22\lrcorner’A^{\lrcorner’}A^{\lrcorner:}\ldots\cdot.\}^{p,\iota}A_{-}^{\underline{|}\lrcorner 1}$

$a71(l$

$q[’|.]=[\ldots..\{()+r_{1})_{l)+72}2^{\cdot}\}\uparrow_{3}+\ldots..r_{r\iota-1}]p_{n}+r_{\gamma\iota}$ .

We rernark tbat (ii) can be considered as a satellite inequality to chaotic $0\prime der^{:}$ .

This paper is early anouncement of the resul $ts$ in [19]. 89

$\{\backslash |1$ Introduction

$1$ $T$ $1$ An operator is said fo ) $e$ positi $l\cdot$}, (delloted )$vT\geq 0)$ if $(T\cdot r..r)\geq 0$ for all vectors ,:

$T$ $|$ in a Hilbert space. (llld is said fo ) $e$ strictly positive (denoted by $T>()$ ) if $T$ is positive and invertiblo.

$i_{llG(1^{11}}ali\dagger\}^{r}$ . This LH was originally proved ill [28] and then $i_{l1}[22]$ . $h,Ian\}$ nice proofs of

$\dot{c}11^{\cdot}$ $\iota,Ve\ln\epsilon^{\backslash }nti-[29]\dot{\zeta}\iota nd[3]$ $f_{\grave{c}}’$ $th_{e1.\uparrow J^{t}}\prime 1\geq/3\geq$ $()$ LH known. . Although LH $a.s9e$1 ) $e.r\},s\cdot u’\cdot\prime s$

$A”\geq B^{\Gamma t}$ for any ($\}\in[0,1],\cdot$ unfortunately $A^{\mathfrak{c}\iota}\geq B^{c\iota}$ doe.s not $al_{1lt)}.\backslash \cdot$ hold for $(t>1$ . The following result has been obt,ained from this point of view.

The $01^{\cdot}i\mathscr{D}11_{r1]}^{r}$ proof of $T1$ leorelil A is sliown in [10]. an elelnentary one-page proof is in . [11] $r$ and alternative ones are in [4]. [25]. It is shown in [30] tliat the conditiol$1sp$ . $q$ ancl in FIGURE 1 are $|)(^{\backslash }st$ possible.

Theorem $B(e.g,\cdot[12][6][25][26][20])$ . Let $A\geq \mathcal{B}\geq 0ui$ th $A>$ $()$ . $p\geq 1on(l$

$r\cdot\geq 0$ .

$(_{A.B(p.\cdot r)=A:(A^{\underline{\frac{r}{\backslash }}}B^{\mu}A^{\frac{l}{2}})^{\frac{1\vdash r}{\rho-\vdash r}A^{\frac{-r}{\sim\backslash }}}}^{\urcorner}7\underline{-t\cdot}.$.

$de_{\nearrow}crc:asin.g$ $G_{4_{\sim}4}\triangleright,,$ $i.s^{\neg}$ $\Gamma$ $\geq G_{A.B}[p.\uparrow\cdot]$ is a function of $p$ and $r_{:}$ and ] holds. that .

$\Lambda^{1+r}\geq(_{A}\angle 1^{\frac{r}{2}}B^{p}A^{\frac{1}{2}})^{\frac{1\dashv\cdot\tau}{p+r}}$ holrls for $p\geq 1$ and $r\geq 0$ . (1.1) 90

$\mathcal{B}>()$ $c1$ $\epsilon\dagger otic$ . NVe write $A\gg B$ if $\log^{2}41\geq 1_{0_{h}^{()}}\mathcal{B}$ for A. . wllich is catlled tlle 1 order.

Theorem C. For A. $T3>0_{:}$ the following (i) $tl’n,d$ (ii) hold:

$\geq(A_{-}^{-}B^{p}\Lambda’\overline{2}’,)^{\frac{r}{l)+r}}$ (i) $44\gg B$ holds if und $071./!l$ if il’‘ for $p$ . $’\cdot\geq 0$ .

$.(l$ (ii) $A\gg B$ holds if $a\uparrow$ } on.ly if $f\dot{o}r\cdot(\iota yfire(l\delta\geq 0$ ,

$F_{1,B}(\downarrow).r\cdot)=A^{-\prime}\overline{2}(A^{\frac{r}{\circ}}\cdot B^{\prime J}A\cdot\underline{\overline{\cdot}},)^{\frac{\dot{\delta}+r}{\rho\dashv-}\cdot A_{-}^{\underline{-r}}}$.

is a $de:C’;ea.s^{F}i,ngf\dot{?}\iota r\iota(^{\backslash }tio,\}$. of $\int J\geq(f$ and $r\geq 0$ .

(i) in Theoreni $C$ is sliowli $i\iota 1[12][6]$ and $aI1e\backslash cclle$ 1 $1t$ proof in [32] and a proof in the case . $\rho=?$ in [1]. \‘and (ii) in [12][6] and etc.

$c^{j}\uparrow nto\cdot’$ . $]’$ $in\cdot t)ertible$ Lemma $D[13]$ . Let $XbJn$ positive $in\cdot ner\cdot tibleol$) and be an

$nurril$)$cr\cdot\lambda$ $opernto7^{\cdot}$. For $a\gamma’.y$ real .

$(J’X1^{*})^{\lambda}=1\cdot’ X_{-}^{\underline{1}},$ $(X_{-}^{\underline{1}}, \}^{\nearrow*}\}’X^{\underline{\frac{I}{9}}})^{\lambda-1}X_{\sim}^{\underline{1}},\}^{r*}$ .

$\backslash \iota_{e}^{r}$ state the following result on the chaotic order which inspired us.

Theorem FKN-2 [9]. If $A\gg \mathcal{B}fo’\cdot.4-$ . $B>0$ . then,

$44^{t-\prime,}.:_{\frac{I\cdot\vdash r-l}{(\rho-,|s+r}(_{d}4_{Q}^{t1}}^{1}$ . $B^{p})\leq_{A}\angle 1^{t_{b_{\frac{1-\ell}{p-t}}}}.B^{p}\leq B$

holds for $p\geq 1$ . $\sigma\cdot\geq 1,$ $l^{\alpha}\geq 0$ an.d $t\leq 0$ .

$C$ lVe shall discuss further extensiol $1S$ of Theorem B. Theorem and Theorem FKN-2.

$Th,e$ purpose of th is paper is to emphasize that the $ch_{(}iotico\uparrow der_{4}4\gg B$ is sometimes more $com$ enient and more $n_{\iota};ef\tau/_{l}l$ than the usual ordcr $\mathcal{A}\geq B\geq 0$ for discussing some order pre serving opera,$tor$ inequalities.

Related results in this paper $\epsilon\iota\iota\cdot e$ discussed in [5].[7].[8].[14],[15],[16].[21].[31],[33] ancl etc.

\S 2 Deﬁnitions of $\mathbb{C}_{r^{-t,B}}[\gamma\iota:p_{1}, \iota)\underline{\cdot)},$ $\ldots/y_{n-1,l^{j_{?1}1\cdot r_{2}\ldots,r_{l-1\cdot 7_{7l}]}}}’\iota\cdot,\cdot$ . (denoted by $\mathbb{C}_{A.B}[n]$

or $\mathbb{C}_{[\}z]}br^{\vee}iefly$ sometime) and $q[,1_{?,-1},,|r_{1}.\gamma,\underline{)},$ $\ldots r_{\iota-1}.\cdot r_{t}]$ (de 71, oted by $q[n]$

$b\uparrow iefly.)$

$A,$ $\ldots p_{n}\geq 0(md?’|\cdot\Gamma_{2:}\ldots?,,$ $\geq 0$ Let $B\geq 0,$ $p_{1\cdot l_{\sim}})\cdot)$ . for a natural $nu7nbcrn$ .

$\mathbb{C}_{A.B}[n;p_{1},$ $p_{n}|r_{1},$ $r_{r\iota-1},$ $r_{r\iota}]$ Let $p_{2},$ $..,$ $p_{\tau 1.-1},$ $r_{2},$ $..,$ be deﬁned by 91

$\mathbb{C}_{A.B}[\cdot|?:,$ Denote $p_{1},$ $p_{2},$ $p_{r\iota-1},$ $p,,,|r_{1},$ $r_{2},$ $?_{r\iota}]$ $\mathbb{C}_{4,B}’[n.]$ $..,$ $..,$ $r_{t-1},$ by brieﬂv. For exal $\iota ipleb$ .

$\mathbb{C}_{A.B}[1]=A^{\frac{r_{1}}{\sim)}}B^{p_{1}}A^{\frac{l}{\underline{)}}}$ and $\mathbb{C}_{A.B}[2]=A^{\frac{\prime 0}{2}}(A^{\frac{r_{1}}{9\sim}}B^{11})A^{\frac{r_{1}}{2}})^{p_{2}}A^{\frac{1\underline{\cdot)}}{2}}$ and

$\mathbb{C}_{A,B}[4]=A_{\sim}^{\underline{\prime\ovalbox{\tt\small REJECT}}}[A_{\sim^{J}\{A^{\frac{\prime}{2}}(2}^{-1^{\underline{\prime}}4}\sim’.\cdot\cdot\underline,....\cdot\lrcorner$.

$Pa$ $tic\cdot nla$ l 1 $1y$ put $A=B$ in $\mathbb{C}_{A.B}[\cdot n]$ in (2.1). Then $\mathbb{C}_{4.A}[r_{1,2}(\gamma\cdot,$ $..,$ .

$=A’2arrow r\{A^{\frac{l,l-I}{2}}$ $[....A^{\frac{\prime\cdot;}{2}}\{A^{\frac{\prime\cdot\rangle}{2}}(A’\lrcorner 2_{d}4^{p_{1}}A_{\sim}^{\perp})^{I)_{\sim^{2}}}A^{\frac{r}{2}}’\cdot\underline,\}^{\int J}\cdot {}^{t}A^{\lrcorner}\ldots]l’ n-\downarrow A^{\frac{\prime\prime).-1}{2}}r_{2}.\}^{p,l}A’\overline{\underline{\cdot\supset}}$ (2.2)

$=A^{[\ldots..\{(p_{1}+r_{1})\cdot\uparrow 9}’\iota-|..\cdot$ (2.3)

$1et_{(}q[n:p_{1},$ $p_{n}|r_{1},$ $r_{r\iota}]$ $p_{2},$ $p_{t-1},$ $r_{2},$ Next $..,$ $..,$ $r_{1.-1},$ be deﬁned by

$q[n;p_{1},$ $p_{7l}|r_{1},$ $I$ $p_{2},$ $r_{\gamma\iota-1},$ $r_{n}]$ $vq[t^{3_{1}},$ $..,$ $r_{2},$ ) $p_{2},$ $p_{7?}.]$ $p,,$ $..,$ denoted $..,$ $p_{7l-1},$

$\uparrow$ $yq[r_{1},$ $7_{7t-1},7_{1\iota}]$ or ) $r_{2},$ $\backslash ;inll$)$licitv$ $q[n]|)riefl\gamma\cdot$ denoted $..,$ for or solnetimes denoted by . For exanlples. $q[1]=p_{1}+r_{1}$ and $q[2]=(p_{1}+7_{1})p_{2}+r_{2}$ and

$q[4]=[\{(p_{1}+r_{1})p_{2}+r_{2}\}p_{3}+r_{3}]p_{4}+7_{4}$ .

For the sake of convenience. $Wt_{\text{ノ}}^{\backslash }$ deﬁne

$\mathbb{C}_{A.B}[0]=B$ and $q[0]=1$ (2.5)

and these deﬁnitions in (2.5) ma,$V$ be reasonable by (2.1) and (2.4).

Lemma 2.1. For $A,$ $B\geq 0$ and any $7l.at\prime n7(\iota l_{71,un7}.ber\tau\}$ . the $follo\cdot n)i\uparrow\iota g(i)ar\iota d$ (ii) $l_{l,O}ld$ .

(i) $\mathbb{C}_{A_{:}B}[n]=A^{l}:_{\vee}arrow 7)\mathbb{C}_{A.B}[n-1]^{p,}\dagger\cdot A$争．

(ii) $q[n]=q[n-1]p,$ . $+r_{n}$ . 92

$\backslash ,V\epsilon!$ $\mathbb{C}_{1.B}=[\prime 1,]$ $q[\}\iota]$ state t,wo exaluples using $t1$ 1 $\epsilon^{\backslash t_{\backslash }’}\{^{\backslash }$ notations of alld for $reader’s$ convenience. . $\prime 1^{r}\geq(\mathcal{A}\underline{\overline{\cdot\backslash }}B\uparrow J\Lambda^{\underline{\frac{r}{\backslash }}})\overline{\})\vdash r}’.\Leftrightarrow.l4^{\uparrow}$ $\geq \mathbb{C}_{A.B}[1]\neg^{r_{1}}$ . . $A^{1+r}\geq(.:|_{\overline{2}}’B^{l^{J}}\Lambda_{\sim}\overline{|})^{\frac{1+r}{j)+\prime}}\cdot\Leftrightarrow.4^{1+\prime}$ $\geq \mathbb{C}_{1.B}[1]^{1}1\dot{T}^{|t}1|$ .

Remark 2.1. We remark that quite similar deﬁnitions t,o $\mathbb{C}_{A,B}[n.]$ and $q[n]$ are given in [18] alld related results are discussed in [18], [23]. [24]. [35] and etc.

\S 3 Basic results associated with $\mathbb{C}_{4,B}[n]$ and $q[n]$

$n.un|,ber\cdot n.$ Theorem 3.1. Let $A\gg B$ and $?_{1}.r_{2_{\dot{\prime}}}\ldots.r_{7t}\geq 0$ for $(l$, natural . The 71, the following inequality holds.

$A^{r_{1}+r_{-}\ldots+r_{\iota}\prime}=\mathbb{C}_{A.A}[\cdot tl]^{\frac{||+\prime 2\cdots+r_{\eta}}{q[,\iota]}}\geq \mathbb{C}_{A.B}[n]^{\frac{r_{1}+r_{2}..+r,|}{q[n]}}$ (3.1)

. $so,ti.gf.\iota/in\backslash c/$ for $l^{J_{1}}\cdot l^{J_{2}}\cdot\cdots\cdot/)_{1},$ .

$l^{\prime y}.’ \geq\frac{|\iota+\uparrow_{\sim^{1}}+\ldots+r_{j-1}}{q[)^{-1]}}$ for $j=1.2,$ $\ldots.\uparrow$ . $($ , $0=0$ and $q[0]=1)$ . (3.2)

th $(\iota t\cdot is$ .

$l)\iota\geq 0,$ $p:; \geq\frac{r_{1}+r_{-}}{(p\downarrow+r_{1})_{l^{22}}+r_{-}},\ldots..l^{J_{n}\geq\frac{r_{1}+r_{-+\ldots+l,?-1}}{q|r\iota-1]}}$ $l^{J}2 \geq\frac{7|}{p_{I}+\tau_{1}}$ . .

ivhe $re\mathbb{C}_{A.B}[n]i_{n}s$ deﬁned in (2.1) and $q[n]$ is $deﬁr,c.d$ in (2.4).

Corollary 3.2. Let $A\gg B$ and $\prime_{1}.\gamma_{2}.\gamma_{3}\geq 0$ . Then

(i) $A^{1+\}_{\sim},+\uparrow 3}\geq\{.’\lrcorner^{r_{\underline{Y}}}-[A^{\frac{r\cdot\}}{2}}(\mathcal{A}^{\lrcorner}B^{l1})A^{\lrcorner’}2)^{\rho_{2}}.2\cdot$ .

$lll_{Sfl2} \dot{o}^{l},.\geq\frac{1}{p_{I}}+|_{1}\mapsto(J,7t.dp\tau\geq\frac{r\cdot\iota+r\cdot\underline{.\underline{\supset}}}{(p_{1}+r_{1})p+r2}\cdot$

(ii) $A^{r_{1+2}}’\cdot\geq\{..A^{-.1^{r_{\underline{9}}}\prime}\cdot,,.....$

$p_{1}\geq 0$ holds for and $f’ 2 \geq\frac{1}{p_{I+1}\iota}$ .

Theorem 3.3. Let $A\geq B\geq 0$ and $’\iota_{:}^{r_{2},\ldots.r_{7l}}’\geq$ $()$ for a natuml number $n$ . $Tl\iota$ en the

$f\dot{o}llowi_{7}\tau gi_{71}.equalit_{l/}/|,olds$,

$A^{1+r_{1}+r_{2}\ldots+7}.,\}$ $=\mathbb{C}_{A.A}[n]^{\frac{1+\prime_{1\underline{\circ}\cdots 1l}+\cdot+\cdot\prime}{q[\cdot\iota]}}\geq \mathbb{C}_{A.B}[n.]^{\frac{1+\cdot+r\underline{v}\ldots+\prime}{q[n]}}$ (3.6)

$f_{07}\cdot\uparrow^{j}|\cdot p_{2}\ldots.,$ $\uparrow j,,$ $s$ atisfying

$l^{J_{j}} \geq\frac{\downarrow+\cdot+\prime}{q\lfloor\dot{\gamma}-1]}$ for $j=1.2_{:}\ldots,$ $\uparrow|$ . $(r_{0}=0$ and $q[0]=1)$ . (3.7) that $i,s$ .

$l)1 \geq 1_{l^{J_{2}\geq}},\frac{1+\prime\iota}{p_{1}+,\iota}$ . $\int’ 3\geq\frac{1+r_{1+,2}}{(\iota)1+r\downarrow)_{l}?_{2}+r_{\sim}}’\ldots..p_{r\iota}\geq\frac{1+r_{1}+r\underline{\cdot\backslash }+..+\prime_{r\iota-1}}{q|r\iota-J]}$. 93

’ Corollary 3.4 Let $A\geq \mathcal{B}\geq 0$ and , $’$ . $l.3\geq 0$ . Then

(i) $A^{1+\prime\iota+\cdot\cdot\underline{\backslash }+\prime_{3}}. \geq\{\lrcorner:),\cdot,A_{\sim}^{r_{1}}l3^{i,\downarrow}A^{\lrcorner})^{l_{A}^{J^{t}\supset}}\lrcorner^{r_{2}\underline{r}_{\wedge}}\wedge’\dashv\cdot\angle]^{\prime 3})A^{r_{\hat{2}}}.\}\dagger\frac{1\dashv I\dashv\cdot;\underline{o}.\dashv-i)}{|(lI+r_{1})1_{-2}^{r\cdot,\{\cdot,1\rho\ddot{.}+\prime\cdot:;}}$.

$p_{t}\geq 1$ $2 \geq\frac{\downarrow+l\downarrow}{p_{1}+r_{1}}$ holds for . $l$) and $p,$} $\geq\frac{1+11+r_{2}}{(\int,\downarrow+r_{1})_{l^{\underline{9}\urcorner}},,-r_{2}}$ .

(ii) $A^{1+r_{I+\prime_{2}}}.\geq\{A^{\frac{\prime}{\underline{9}}}(A^{r}\circ\lrcorner\sim B^{\gamma r}{}^{t}A^{r_{\underline{Y}}}\lrcorner^{1\dashv.\dashv})^{p_{2}}A^{\frac{r\underline{.}0}{}}\}^{(p_{1}-\vdash 1)p_{\wedge^{\vee\vdash 2}\sim}}\infty’)..$,

$f_{\dot{O}7}\cdot p_{1}\geq 1$ holds $and/$) $2\geq\frac{1+\prime}{)1+11},\cdot$

$\backslash 4^{\gamma}e$ Remark 3.2. reniark that Theorem 3.3 is a parallel result to Theorem 3.1 $al\cdot ld$ also

Corollary 3.4 is a parallel one to Corollary 3.2. $a1$1$d$ Theorem 3.1 is usually obtailled from

Theorem 3.3 by applying $Uchi\backslash ^{r}\epsilon 1\ln_{\dot{r}1S}$ nice technique [32] aft,er proving Theorem 3.3.

$Alth_{on(}hr\mathfrak{j}$ $en_{\rho}de’\cdot it$ ,($\iota\uparrow\iota y$ results on the $cl\downarrow,aot\cdot lcorde\uparrow\cdot(A\gg B)$ have be ) $e^{\lambda}df\uparrow 0771\prime tfi,eCO\uparrow’\prime t^{J}-$

$spor\iota li\uparrow\{,g\uparrow esul\dagger_{\text{ノ}}.90\prime\prime$ $th,e$ $0\uparrow d(,’,$ . . usual $(A\geq f3\geq 0)$ by applying $Uc:hi,yama8^{\int}nice$ method. $we$

$s/\iota 0?nC\cdot lln^{\nu}y5.4$ $e\cdot usual$ $f\cdot tl\cdot tl$ shall on th, orcler $(\Lambda\geq B\geq 0)$ , which is a 1 $e1^{\cdot}$ extension of

$by\uparrow\iota sing$ $c^{J}.\backslash ^{\backslash }?r,it$ $(;l_{l_{\text{ノ}}}ootic$ Theoren13.3, the $\Gamma’O’,7$ cspg Corolla $r^{v}$ ). on the order $(A\gg B)$ at the end of \S 5. \S 4 Monotonicity property on operator functions

$\tilde{s}_{A}\cdot(p_{k^{7}k})=.\triangleleft^{\frac{-}{2}L}\mathbb{C}_{A,Bd^{\dot{L}}}[k]^{\frac{\dot{\delta}+r_{1}+r\underline{\circ}+\ldots\dashv-\prime k}{||k1}}]^{\frac{-r}{2}A}$

’ Theorem 4.1. Let $A\gg B$ and $l_{1,\underline{)}}’$ . $\ldots.r_{l}\geq 0f_{07lr.atur(\iota lnu\tau\prime l},berr|..Fo7$ any

$\delta\geq 0$ ﬁkixed , let $p_{1\cdot l^{J}2,\ldots.p_{n}}$ be satisﬁed by

$p_{j} \geq\frac{\dot{\delta}+\prime_{1+1}\cdot-+..+|j-|}{q[j-1]}$ $f\dot{o}\uparrow j=1.2,$ $r|$ $\ldots,$ , (4.1) $fJ_{1,0},t$ is.

$p_{1} \geq|_{:}p_{2}\geq\frac{(J+r}{p_{1}+r_{1}}\ldots\ldots p_{k}\geq\frac{\tilde{\delta}\prime_{\sim}}{q[k-1]},\ldots.,$ $l’ n \geq\frac{\dot{\delta}+1\iota\cdot}{q[r\iota-1]}$ .

. $(\text{\’{u}}_{nu\prime\gamma}.be\gamma\cdot A\alpha$ $Thr:ope\cdot\iota t7^{\cdot}fur),ctionS_{k}(p, r\cdot.)fo’$ any natur such that $1\leq A\cdot\leq r\iota\dot{?}s$ $deﬁnc,d$ by $s_{k}(’\cdot’..’..--\prime A$ (4.2) Then the follo wing inequality holds:

$A^{\frac{k-1}{2}s_{k-1}(\rho_{k-l.k-1})A^{\frac{r_{k-1}}{2}}}\geq S_{k}(t)\Gamma)$ $(So(p_{0}, \cdot r_{0})=B^{\delta})$ (4.3)

for $e\prime ner^{\nu}ynat\uparrow\iota rulriuml_{J}er^{i}k$ such thut $1\leq k\leq n$ .

Remark 4.1. We $:’1_{1a}$ ] $1$ give an alternative proof of Theoreni 4.1 in Remark 6.1 via Theorem 6.1 at the $e11(1$ of ﬁ6. 94

$!^{\backslash }|5$ Order preserving operator inequalities via operator functions in \S 4

$\iota’\backslash :_{e}$ shall give order preserving operator inequalities as $c=u1$ application of $T1$leoreln 4.1.

Theorem 5.1. $LctA\gg Ba\uparrow\iota d,.\uparrow’\cdot,\geq 0$ for a natural number $n$ . $The\uparrow|$. $ﬂ’,e$ following $i?eq\prime 4itixhold$ for any ftxed $\delta\geq 0$ :

$B^{\delta}\geq A^{\frac{-i}{2}\perp}(.\angle 1^{r}\lrcorner-B^{l1}\mathcal{A}^{\lrcorner}\underline{\backslash })^{\frac{\delta\dashv\prime}{1’’1}}A^{\frac{-r}{2}}$

$\geq./-1^{-}-\frac{(l+7\backslash )}{-}\{A_{\sim}^{z_{(\sim}’}Q2^{\cdot}\backslash \lrcorner^{f\cdot\frac{\dot{\phi}-\vdash\prime_{1+r\underline{\cdot)}}}{(\rho+r)+r_{-}}\underline{-(r}_{\frac{+r\cdot 0)}{\sim\backslash \backslash }}}.\cdot$ , $\geq.L|\mapsto-:-\{2^{\cdot},\mathcal{A}^{\lrcorner^{1}}\prime 2B^{p1}\mathcal{A}^{\lrcorner})^{l)_{\sim}}.,A^{\frac{r}{2}}]^{p_{3}}A_{-\}.4^{\frac{-(\cdot++l)}{2}}}^{\lrcorner\frac{\dot{\delta}\vdash\prime_{1^{-\vdash\prime}2.::}+\prime}{\{)}};\ldots$

$\geq A^{\frac{-\dot{\iota}\prime\cdot\prime}{2}\mathbb{C}_{4.B}[\prime\iota]^{\frac{\delta+r\cdot|\dashv-z\dashv-\ldots+r_{?t}}{q|n|}A^{\frac{-(r\iota+r\cdot+\ldots+\prime_{t?)}}{n\sim}}}}".\underline,$ (5.1)

$p_{1}.\rho_{2},$ $p,,$ $9^{\backslash }ati.\backslash ^{\backslash }fi/ing$ for $\ldots,$

$l)j \geq\frac{\dot{\delta}+\prime_{1}+12+\cdot\cdot+r_{?}-\iota}{q\lfloor_{1}-1]}$ $f_{\dot{O}7\dot{|}}=1.2\ldots..r\iota$ , (4.1)

that $i,s$ .

$[)1 \geq\delta, p_{2}\geq\frac{\dot{\delta}+\prime\iota}{\iota)1+r_{1}} ...... \mu_{k}\geq\frac{\dot{\delta}+r_{1+2\prime_{\backslash }\cdot-1}r+\ldots+1}{q|k-1]}\ldots..’?)_{7t}\geq\frac{\dot{\delta}+r_{1+2+\cdots+|,|-1}\prime}{q[|\iota-1|}$ .

$snf\iota e\gamma e\mathbb{C}_{A.B}[n]$ is $(lc^{J}fi\cdot\}\}.ed$ in (2.1) and $q[n]$ is deﬁned in (2.4).

$\ldots,\cdot\geq 0$ Corollary 5.2. Let $A\gg B$ and $\gamma\cdot l,$ $r\underline{)}$ . for $n$ natrrral number ?. Then the $follou\prime i\gamma’,g(i)$ an $d$ (ii) $h)ld$ .

(i) $\mathcal{B}\geq \mathcal{A}_{-}^{-.r}-\lrcorner^{r_{0B^{p\iota}A^{\lrcorner}}}(A_{\sim}^{\lrcorner^{\prime\frac{1\dashv}{\rho_{1\dashv}-\prime|}-,\perp}}-).A^{-}-\cdot$

$\geq_{\supsetneq}4^{\underline{-(}_{\frac{\{r\cdot,)}{2}}}..\{.4_{-(_{4}\prime}^{2^{r_{2}}})\underline{r}_{0}\lrcorner^{\Gamma}\}rightarrow_{),)}^{\downarrow\vdash r\vdash\backslash }\underline{-(r}_{arrow\vdash\underline{\cdot\cdot)}}1p_{\iota+},.1\cdot 2+r_{2A-}’,’$

$\geq_{d}4^{\frac{-(r\vdash 1\circ+r\dagger)}{o\sim}\{A^{\lrcorner:\lrcorner’}}r_{2[A^{\frac{\prime}{2}}(2}.s_{\wedge}’\cdot\cdot,.’\cdot.1+.\underline{r\cdot+|.:\backslash )}Q^{\cdot},\underline,$

$\geq A--\mathbb{C}_{A.B}[n,]^{\frac{1\cdot\vdash,.+r_{-},-\vdash\ldots+\prime}{q|\prime\iota|}}d4^{\frac{-(r+ro++r_{\cap})}{\circ\sim}}$ (5.2)

h.ol $(ls^{\backslash }for\rho_{1\cdot l^{y_{2\prime}}\cdot\cdots\cdot P_{l}}$, satisfyi71. $q(3.7),$ $t$ hat is.

$\ell)_{1}\geq 1$ . $p_{2} \geq\frac{1+\prime\iota}{l)1+\prime_{1}}l_{\backslash }\}\geq\frac{1+\cdot\iota\prime}{(p_{l+l}\cdot\iota)\gamma_{J_{2}}+\tau_{-}},\ldots..l)_{n}\geq\frac{1+r_{1}+r_{\sim}o+\ldots+\prime_{\iota-1}}{q[1l-1]}$:

(ii) (5.2) holds for $I’ 1\cdot l^{J_{2}}\cdot\cdots\cdot l)_{n}\geq 1$ . where $\mathbb{C}_{4.B}[n]$ is deﬁ $rn^{J}.d$ in (2.1) and $q[n]$ is deﬁned in (2.4). 95

” $\ldots.\uparrow$ $,|,at_{\text{ノ}}uralnurr\iota.ber\cdot,l$ $Tl’,er\iota$ Corollary 5.3. Let $A\gg/3(xr\iota(lr_{1},$ $\Gamma_{\sim}l,,$ $t\geq 0$ for a .

$I\geq A^{\frac{-.\tau}{\sim?}\perp}(A^{1}\underline{\circ}\mathcal{B}_{4}^{l1_{\angle 1_{4}^{\lrcorner})^{\frac{r_{1}}{p_{1}+I}}\angle 1_{\sim}^{-.\prime}}}\lrcorner^{r_{L}},,.\overline{\supset}$

$\geq 4^{-(1}\{4^{-1^{\frac{)}{f}}(.4’-f3^{l^{J}1}.4_{\sim}^{\lrcorner})^{f2}A^{\frac{r\underline{\cdot)}}{2}}\}^{\frac{r_{1}.\dashv\cdot,2}{(+)\rho_{\sim},-\vdash 2}}A2^{+\underline{r\cdot\supset)}}}-,,,\cdot.,.$.

$\geq A^{\frac{-(|\vdash|1\vdash:)}{\underline{\rangle}}}\{_{d}4\lrcorner’\underline{\supset}[A^{\frac{)}{Q}}(A_{\sim}^{\lrcorner})B^{p_{1}}A^{?}\lrcorner\underline{)})^{p_{2}}.\cdot\not\in-\cdot\overline{2}’]^{p_{3}}$ $A$ I2” $\}^{\frac{\prime\prime_{1*\vdash\cdot?}\underline{\rangle}}{\{(-\vdash r)r\}p}}\lrcorner 4^{\frac{-(r\dashv.r\dashv 7)}{\underline{\backslash }}}$

. $\geq.4^{-(\prime}..-,\cdot \mathbb{C}_{A.B}[’\iota]^{\frac{r_{t_{-\vdash\ldots+}?l}+1}{q|n|}}A^{\frac{\text{二}\Omega\pm\vdash\text{・火}\vdash\vdash \text{π})}{}}\underline{I^{-|\cdot l\underline{9}+\ldots-\vdash J_{7l})}}.,.,$ (5.3)

$f\dot{o}r\cdot p_{1}$ $f$ holcls , $l”\underline{)},$ $\ldots.p_{7}$ , satisfy $ir|g(3.2)$ , th a is,

$p_{1} \geq 0_{l}.)\geq\frac{l1}{p_{1+?1}}$ . $l^{J_{3}}. \geq\frac{r_{1}.+r2}{(r))1}\cdots\cdots\prime 4^{y},$ $\geq\frac{l\downarrow+l_{\sim}+\ldots+_{n-1}}{r4[r|-1|}$ .

Corollary 5.4. Let $A\geq B\geq 0(\iota\gamma 71\cdot 2\cdot\cdots,\}?\geq 0f\dot{0}’\cdot$ a natural nuntber $l\cdot\cdot$

$Th_{C71}$.

$44 \geq B\geq A^{-.\underline{\tau}_{1}}-,\cdot(2)\lrcorner\frac{1-\vdash}{l^{l}1^{1}}\perp_{A^{\underline{-}}\sim}l.’$,

$\geq.4^{\underline{-(\dagger}}L^{\underline{+72)}}2^{\cdot}\{A^{\frac{r}{Q\sim}}(A_{-]\ni l\downarrow A^{\perp_{2}})^{I2}A^{\frac{r}{2}}\}^{\frac{1+7_{1}\vdash}{(t)\iota\dashv-r_{1})\rho_{\wedge^{\backslash }}+r\underline{\cdot\backslash }}}A^{\underline{-t\prime}}}^{7_{\underline{|}^{\Gamma}}}).,.’\mapsto^{o\sim+r\circ)}$

$\geq-,’.:\underline{\cdot}.\cdot a\lrcorner’.,-4^{\dot{2}}]^{p_{3}}A^{\frac{1:}{\underline{9}}}\}^{\frac{1\{-\tau_{1}\dashv-r\underline{o}+r_{3}}{\{(+)\nu\succ r_{\sim}o\},.a-\prime}}A\mapsto^{-(r+1’\vdash’:)2}$

$\geq A^{\frac{-(r_{1}+r\cdot\underline{r}+\ldots\dashv\prime n)}{2}\mathbb{C}_{j\downarrow.B}[rt]^{\frac{J\dashv r_{1}\dashv-r\underline{o}+\ldots+\prime r}{1|z|}}A^{\frac{-(r+r,|+r_{1})}{2}}}$ (5.4)

holds for $\rho\downarrow:l^{y_{2:}\ldots,\rho_{lt}.\backslash ^{-}\cdot atisf\dot{y}i_{7}\iota g}(3.7)$ . thut is.

$p_{1}\geq 1$ $p_{2} \geq\frac{\ovalbox{\tt\small REJECT}+\prime\iota}{\mathcal{P}\downarrow+\prime_{1}},$ . $p_{3} \geq\frac{1+,1+.\prime\underline{\backslash }}{(+l_{1}},\ldots\ldots\int)_{n}\geq\frac{1+\prime_{1+r_{2+\cdots+\prime_{n-1}}}}{rl[1\iota-1]}$.

$n;here\mathbb{C}_{A.B}[n]$ is $leﬁ\downarrow nedi\uparrow\iota(2.1)$ and $q[n]$ is deﬁ$ncd$ in (2.4).

Remark 5.1. Corollary 5.2 is a further extension of [25]. [17], [20]. [34] and Tlleorem

FKN-2 in [9]. Corollary 5.3 is $nlOl\cdot e$ precise estimation $t$ hau Corollary 3.2.

We would like to $(^{J}.n|,q)husi\sim e$ that $Co\prime ollu\gamma\eta/5.4i,4^{\backslash }a$ . $f\dot{u}r\cdot thc^{J}re\prime x.\cdot fensior|$. of Theorern 3.3

$ir\iota$ since (5.4) easily implies (3.6) Theorem 3.3 $a7|,d$ moreover the $ess$ ential part of (5.4) $i\uparrow\iota$ $Co$ rollary 5.4 on the usual order $(A\geq B\geq 0)$ is $der\cdot ive(lfi^{\sim}0\gamma n$ Corollary 5.2 on the $cl$ }, $aotic$ order $(A\gg B)$ . 96

$B$ $C$ $\phi()$ Further extensions of Theorem and Theorem

$-\backslash .oreInB$ $C$ $b.\prime v$ Further extensions of $T11$ ( and Theoreln are givcn tlsing the $oI$ )$erator$ function $\tilde{\theta}_{1}(p_{\iota}, r_{\gamma\}})=A^{\frac{-.\tau}{\underline{)}}\perp}\mathbb{C}_{A.B}[n]^{\frac{\dot{\delta}+\prime_{1}+l_{\sim^{)}}’\ldots+l\gamma\iota}{q|?\iota|}A^{-1}\underline{)}}-\dot{1}L$ in \S 4.

$\ldots.\gamma_{n}\geq 0$ Theorem 6.1. Let $A\gg B$ and $[\cdot\gamma,\underline{)}$ . for a natural number 11. For any

$ﬁ,\prime n()d\delta\geq()$ . le $tp1\cdot l)\underline{\cdot)}$ . $\ldots.p_{\gamma\downarrow}l)es$ atisﬁed by

$p_{j} \geq\frac{\dot{\delta}+1\iota+?^{r}:+..+r_{J-1}}{q|j-1]}$ for $j=1.2,$ $\ldots.n$ , (4.1) that is.

$|J_{1} \geq\delta_{:l^{J_{2}\geq}}\frac{\delta+\prime\iota}{p\downarrow+r_{1}},\ldots.l)k\geq\frac{\dot{\delta}\perp r_{1+\prime\underline{\cdot 1}+\cdots+\cdot k-1}\prime}{q[A\cdot-1]}\cdots\cdot\cdot\prime l)_{l}\geq\frac{\dot{\delta}+|1+|-+\ldots+\prime,i-\downarrow}{q|\prime|-L]}$.

$Tl\iota.e’\iota$

$\mathfrak{F}_{n}(jJ_{r\iota}, /\cdot,,.)=A^{=A’}2\mathbb{C}_{1.B},[n]^{\frac{\dot{\delta}+\cdot\iota+r_{-}+\ldots+r_{n}}{\mathfrak{g}|n|}}A^{\frac{-.r}{\underline{1}}A}$ (6.1)

is $(\iota$ decreasing function of both $r_{\iota}\geq 0$ and $p_{7l}\cdot u;l_{l},ic/|$. $satis^{\backslash }fies$

$C$ $l),,$ $\geq\frac{\delta\prime}{q[r1-1)}$ . ( .2)

$’ 2\cdots\cdot\cdot\uparrow,1\geq 0a\uparrow’.d$ Corollary 6.2. Le $tA\gg Bt7\iota l’|,$ also $p_{1},$ $\rho_{2,\ldots.l)_{?}},\geq 1$ for a

$Tl_{l}.er$ naturul number $\eta$ . },

$S_{l}(l^{J_{\gamma}}, , \uparrow r\iota )$ $=d\angle 1arrow’-’\cdot\underline{-}r_{1}$

is a $d\cdot.cxS\cdot g$ function of both $r_{1}\geq 0ar$ }$.dp_{\tau\iota}\geq 1$ .

$Theo\uparrow err\iota$ $Tl’,e7e$ ’ $4\cdot l\cdot uiaTh\epsilon orv^{J}r$ $|,$ $6.1$ Remark 6.1. is an alternutiv $\sim$ proof of . } .

Remark 6.2. Theoreni 6.1 is a further extensions of (ii) in Theorem C. In fact. (ii)

$(-\backslash 01^{\cdot}elll6.1$ $\cdot$ of Theorel$i_{1}C$ is just Th in the case 7 $l$. $=1$ . Moreover Theorein C.1 is a $fu$ ltIiei

$\cdot$ extension of $T1$ leorel$1iB$ since the $1_{1}ypot.hesis_{4}4\gg B$ in Theorem 6.1 is weaker than the hypotliesis,$4\geq B\geq 0$ in $T1_{1}eore\ln$ B. 97

REFERENCES

[1] T. $A_{1}iclo$ . $Or\iota\backslash \cdot 0\uparrow n$(’ operator in equality. AIath. $A_{1l}n..279(1^{(})87$), $157-159$ .

$A_{1}1clo$ $\cdot$ [2] T. and F.Hiai. $Log/n(l\dot{j}\cdot’.(\iota tior\iota$ and $(’0\prime npl\prime r,\uparrow.e7,$ tu , $yGolder\iota$-Thompson $t_{t}/I)cir\iota eq\prime n\iota lit?$, es, Linear Alg. and Its Appl..197, 198 (1994), 11.3-131. [3] R.Bhatia. $Positi’|)eDfin\uparrow t(:i\downarrow$ノ$[atr\cdot ice,s$ . $p_{I]nceton}$ Univ. Press. 2007.

$b$ $s$. [4] I.Fujii. Furutu inequality and its $rnear\iota$, theoretic $(\iota pp\gamma onch$. $.I$ . Operator Theory. 23 (lS 90), $()7-72$ .

$j\downarrow\prime Ic.n7t$ [5] M.Fujii amd E.Kirnlei, . $th.ei$) $r\cdot etic$ approctch to the $g\uparrow^{\alpha}un.(l$ Furufa $inc^{J}.(1^{\cdot}u(xlity$ . $p_{lO(}\backslash$ . Anier. Math. Soc.. 124 (1) $9())$ . $2751-2756$ .

[6] $hI.Ftljii_{:}T.Ft11^{\cdot}tlta$ and E.K $\dot{\epsilon}\iota nlei$ . $Fu\uparrow^{u}nt(\iota^{:}\backslash \cdot$ inequality $ar\downarrow dit$; application to

$ndo^{:}s$ A theorem. Linear Alg. $A_{I)[)}1.,$ $179(10^{(}):;),$ $1(^{\backslash })1-169$ .

[7] bI.Ftljii, A.Matsimioto and R.Nakamoto, A short proof of the best possibility $fo\cdot r$. the.$q\uparrow(\iota r’,(l$ Furutu inequa,lity. .1. of $I_{11}\epsilon^{\backslash }.(111a1.\dot{\epsilon}111d$ Appl.. 4 (1999). 339-344.

[8] M.Fujii. E.Kamei and R.Nakamoto. $Or|$. a que,$5^{\cdot}$ tion of $Fur\cdot n$ ta on chaotic order. Linear Algebra Appl.. 341 (2002). 11 $()$-127.

[9] M.Ftljii, E.Krmtei and $R..N_{\iota}\backslash k_{\dot{\epsilon}}\iota nloto$ . On a question of Furnta on chaotic $0\uparrow\cdot der\cdot.II$ . $NIat1_{1}$ . J. $Ok_{\dot{\epsilon}t.\}cT}’\cdot na$ Univ., 45 (2003). 123-131. [10] $T.F\iota ir\iota ta,$ $A\geq B\geq 0ass\cdot nre,s(B_{-}’4^{\nu}B^{r})lll\geq B^{(+2\prime\cdot)/q}\uparrow)$ for $r\geq 0.p\geq 0.q\geq 1u$ ) $ith$ $(1+2r\cdot)q\geq p+2r$ , Proc. Amer. Math. Soc.,101 (1987). 85-88. [11] T.Furuta. $Element_{(}\iota r^{\nu}\prime\prime$ proof of an orcler preserving inequa,lity. $P_{1}\cdot oc$ . Japan Acad.. 65 (1989). 126.

$\Gamma\prec tI^{\cdot}\iota lta$ [12] T. , Applicatio $7|,S$ of $07^{\cdot}d\cdot rcser^{}\{)$ opetator in$equalitie6$ . Operator Theory:

$A$ Advances and ]$)])]ic_{\dot{c}}\iota tior$l $:\backslash$ . $59(1^{(})^{(})2)_{:}180-190$ .

$l$ [13] T.Furuta. An e.ntension of the Furuta $in\alpha juality$ an Ando-Hiai $log\uparrow\gamma_{\text{ノ}}njo,.i^{\sim}at\cdot ion$.

Linear Alg. and Its Appl..219 $(1^{\langle}J95)$ . $1\backslash \}_{\backslash }^{(}J-155$ . [14] T.Furuta. $Si_{7}nplifi,ed$ proof of an order $prese\uparrow^{4}|)ingopera\iota to’$ . inequality, Proc. .$T_{41)_{\dot{C}}tI1}Ac_{\dot{c}}\iota d..74$, Ser. A(1998). 114. [15] T.Furuta. Results under $\log A\geq\log B:(l7l$ be deri $nc^{J}d$ from oncs urider $A\geq B\geq 0$

$Uchiya7na$ $C?ato7^{\cdot}$. by ’s method $ass$ ociated with Furutu and $Kanto7ovich$ type $0\iota$)

inequalitie,$\backslash \cdot$ . AIatli. Inequal. Appl.. 3 (2000). 423-436.

[16] T.Furuta. $Iitition$ to $Li_{7},.eo,r$ . Operators, Taylor & imd Frtmcis, 2001. LoIiclon.

$as\cdot\backslash \cdot ur\cdot esA^{1+2-\mathfrak{j}}\geq\{A^{\underline{\frac{\gamma}{\circ}}}(A^{-\prime}\overline{\underline{\backslash }}B^{p}A^{\frac{-t}{2}})A_{-}^{\underline{r}}\}^{\frac{1-1\cdot\prime-\prime}{(\rho-\iota)s+}}$. [17] T.Furuta, $A\geq B>0$ for $t\in[0.1]$ . $r\geq t.p\geq 1.s\geq 1$ and relatecl inequalities. Arcliives of Inequalities and Applications 2 (2004). 141-158.

$Fu\cdot r\cdot the.r\cdot e\tau:ter\iota sior’$ [18] T.Furuta, , of an $07de7^{\cdot}$ preserving operator $in\alpha 1^{\cdot}n\iota lity$, J.Math. Incqual.. 2 (2008). 465-472. 98

$7l(:huotic:or(ler\cdot\cdot in$ $opcrat_{07}\cdot ine$ [19] T.Furuta. $Opera\mathfrak{s}.tor$ functio 77. $S$ ) volving order preservinq qualitie.s. to appear in J.Matli. Iuequal.

[20] T.Furuta $imdh\cdot I.H_{\dot{\epsilon}}\llcorner^{\backslash }\backslash 1_{1}i_{111}oto$ and $b\prime I.Ito$ . $Eq_{1l}i_{1\prime}ulJ?\iota(:e,$’ relation between.$qc^{J}.\uparrow\iota e7t\iota lizr.(l$

$\cdot$ $e111atic\cdot\iota e^{\backslash }$ $1$ $Fur\cdot 1\iota tutn.c^{3Ij}$ uali $tya.r\iota del_{(}\iota ted$ opera.for $f\dot{n}r|,ctio7\prime S$ , Scicutiae LIath . (1998). 257-259.

$F_{1}n^{y}|\iota$ [21] $T.F$llruta, AI.Yanagida and T.Yamazaki. Operator function,5 $i_{7nl}$ lying ta $ir\}.C11?\iota$ ality. Math. Inequal. Appl., 1 $(1^{(}J98)$ . $12_{\iota}\}-1.30$ .

[22] E.Heinz. $Beitr\cdot\ddot{a}./e$ zur $Sﬁir\cdot t7’.g,;teorie$ der $6^{\urcorner}pekfml_{\tilde{4}}erlc^{J}//v.\uparrow\iota.g$ , Afath. Ami..123 (1951). 415-438.

[23] M.Ito and E. $K\dot{\iota}Il\iota ei$ . Ittfean theoretic approach to $a$ further ertensiOn of grand Furuta $irt$. equality. J. $M_{i}\iota tl1$ . Inequttl., 4 (2010). 325-333.

$\dot{\zeta}\iota 1\sigma_{\dot{f}}t\ln\iota u\cdot a$ [24] S.Izumino. N.N tmd M.Tomin $\mathfrak{c}\backslash g_{\dot{C}}\iota.$ , Mean th, eoretic operator functions for extension s

$irl.C/1^{ual\dot{r}ty}$ $I_{1}$ $72$ of the $9^{7ur\iota d}$ Furuto. . Sci. Matli. . ) $u.,$ (2010). 157-163.

[25] E.Kalilei. A satelite to $Furuta,$ $\cdot.\backslash \cdot$ in $equ$ality. $LI_{\dot{C}}\iota t,h$ . $I_{\ddot{\prime}}\iota poI1..33$ (1988). 883-886.

[26] E.Kamei. $Parvm$ ) $et_{7^{\sim}}ized$ grand $R\iota r\cdot nta$ inequality. Mat $h$ . Japon.. 50 (1999). 79-83.

[27] F.Kubo rmd T.Ando, $\Lambda!Iea\uparrow|_{\backslash }\backslash ^{\backslash }$ of positive linear opemtors, ItIath. Ann., 246 (1980), 205-224. [28] K. L\"owner, \"Uber monotone Matri,$\gamma f\iota\iota.n\lambda:tior|,en$ . Math. Z.. 38 (1934), 177-21(;.

$\cdot$ [29] G.K.Pedersen, Some opera tot 7”on oton. $e$ functions. Proc. Anler.Math. Soc., 36 (1972), 309-310. [30] K.Tanahashi, Bebt possibility of $tl_{l}.e$ Furuta inequality. Proc. Amer. Matli. Soc.,124 $(199())$ ,

141-14 $()$ .

$P_{1}\cdot oc$ $L^{J}I_{i}\iota t,1_{1}$ [31] K.Tal $1ah_{\delta_{L}^{(}\backslash }\backslash 1_{1}i$ . Tlie best possibility of the $gr(md$ Furuta in equality. . Amer. . Soc..128 (2000). 511-519.

$h$ $Uc:hi\}_{C}’\urcorner J\mathfrak{U}_{t}1,$ $So\uparrow n.e(^{J},a:\iota)one71$ ) $erato\uparrow$ . Ltat, [32] AI. . tiod $ot$ in.equalities. . Inequal. Appl., 2 (1999).

$46^{(})_{-}471$ .

$11az_{\dot{C}}\iota 1\dot{u}’$ $Tc\iota r\iota oJ’.asl_{1}.isre,\backslash \cdot ultor|$ [33] T.Yal . Simpliﬁed $p_{7}oof$ of , the best possibilify of.$qe?\iota$ eralizeil Furufa incquality, IS Iath. Inequal. $A_{P1)}1.,$ $2$ (1999), 473-477.

[34] J. Yuan and Z. Gao. Classiﬁed $c,\cdot 0$ nstruction of $g\cdot yp$ . operotor functions,Matli. InequaJ. Appl.. 11 (2008), 189-202. [35] C.. Yang \v{c}md Y. NViing. Fvrther extension of $Fum$ta inequality, J. Math. Inequal.. 4 (2010), $3^{(})1-:\}08$ . $bIa$ililig address:

1-4-19 $Ki\iota y_{rtII1}ad$l$O11$ Fuchu city Tokvo 183-0041 Japan

E-m $\iota il$ address: $fu$ltlta$(CJarrow 1r\backslash ,\backslash .k_{\dot{\epsilon}\mathfrak{X}}\iota\iota.t11\not\in;.\dot{\iota}c..|p$