論文の公開元へGeneralizations of Bebiano-Lemos-Providência inequality (Theory of function spaces and related topics)
概要
96
Generalizations of Bebiano-Lemos-Providencia inequality
iii# :iE~
::l,JU:!(~:k~ :g=,H:!!:~
Masatoshi Fujii Osaka Kyoiku University
mfujii@cc.osaka-kyoiku.ac.jp
:::. :::. c' 1i, ~ J].,,,-;:J].,, 1-- ~ r.JJ H J:: O)fl° JH,'lU~{'Fffl * >a:: _ljt f:: {'F JlHrU:: 115¥,~~:::. I:'.. f:: L, 19, {'F ffl * A iJ,
positive (A 2: 0) c'Ji:> Q /:'.. 1i,
(Ax,x)2:0
xEH
t ~, A> 0 c'~ L, 19, 1T:7Uc'g ;it_
t iJ, positive (A 2: 0) /::, iEJEfi!l:iJ' A> 0 t:~t:: fJ
-/J>J¾ fJ fl~:=:./:'../:'.. L, 19, if';¥/:, A iJ, positive iJ,~ invertible (1)
1:f, A iJ, '1-Fffe!.J.Efi!'l: (positive semidefinite) "c'Ji:> Q:::.
19, it::, §2:./:H~fFm* A,B t::J\"tVC A-B 2: 0 1::J:-::i··c {'Fffl*)l[a]i; A 2: B -/J>§~t=~A
~ tit 9, JR+ J::c'JE~ ~ tttdE~OO~
f
f:: J: Q functional calculus -/J>:::. (1)/ilf[Ji;>a::f!¥:B'9 Q, 9t~;b1:,
===>
A 2: B 2: 0
(!)/:'..~,
f
f(A) 2: f (B)
>a::fFffl*ljtfil.lJt.1,,,1,,,19, ¥00~/::~1,,,-c/i,
t ➔ t°' 11, a E
i'JZ(!)m~t~!ji:~iJ,§;□ Gti-c1,,,t9,
[O, 1] O)c 'f50)J-J., {'Jcffl~*~-Z..iv.>~o
imm', :=:.tttiLi:iwner-Heinzinequality /:'..115¥1:ftl,-C\,,'19, PJT(LH)c'~;bl,19, [29], [25], [32] ((LH)
/::)\"f9 Q Pedersen (1) Jlf.f!:t~lfiEa_ij /i, Appendix c'if./Hr L, 19,)
~-C, (LH) 1i/ Jv.l.;f~~f::J:Qf<,Jf[!l:t~~:EJl>a::lsJ-::i-C\,,'i9o [12], [22] {-l;;~i¥]td.,(1)iJ,, '/7{(1) ArakiCordes ;f~~ (AC) c'9 :
IIBtAtBtll::; IIBABllt for O::; t::; 1.
(AC) 1i, '/7{(1) J: -5 /:: t ~%-r:- ~ 19 :
(AC-1) IIBtAtBtll 2: IIBABW fort 2: 1.
(AC-2) IIAtBtll::; IIABllt for 0::; t::; 1.
(AC)
*lli1ic'1i, norm ;f~~(!):JJl"l-/J> G, Li:iwner-Heinz ~~;i:\;(1)~:!ll'>a:: Jll. -C 1,,, ~ t:: \,,' /:'.. ,W,1,,, 19o f:(!)f::
o/
7
'a::= =1:: lie l,-C ;l=:i' ~ t:: \,, \ t ,w, \, \ *9o
Ando-Hiai inequality
Li:iwner-Heinz inequality
Grand Furuta inequality
Furuta inequality
BLP inequality --+ Grand BLP inequality
97
f(x)
ii{'Fffl •-¥t!,J
"'*9:
=
u O)~l31,00~lU:: 1Jl¥iln--c 1,, t 9, tJ: B,
C, B
~
{'Fffl •-¥t!,J ii, l'JZO) 3 ~{lf'~i11lJtc: 9
2 J][jjfi~~ g
D =;- A u B ~ C u D
=;- An u Bn +Au B
Transformer inequality: T* (A u B)T ~ (T* AT) u (T* BT) for all T
A
Monotonicity:
~
l u x
Upper semi-continuity: An+ A, Bn
(Normalization:
+B
1 u l)
A#B =
t
tJ:
../AB
I? i9,
~;B,~£~~~
~~0)~5~9~~1'.)t~•~·~~-~~:¾'l,~O)~~~<,
m•fi~~~~L~-~ntL~:~B20~ML--C,
A#B=max{X20;
S:. S:. ~, A iJ! invertible ~ Ji) 7-i
(1 !)
20}
t 9 7-i t,
{c}
A-½BA-½ 2 (A-½xA-½)2
⇒ (A-½BA-½)½ 2 A-½xA-½
{c}
A½(A-½BA-½)½A½ 2 X
A# B = A½(A-½BA-½)½A½
t, Tm'.
a=½
O)~f::::Ji!l,3-L--c1,,t9,
*~•~·~#~~
98
{'PJ'lHMlfiiJ-¥:1:$J~f!£-::i t::;f~AO)!!!rnY.fliJt~:t 0) t VC ~ii-Bi'i-;f~Ailtl_lHf Gnt 9o Ando-Hiai [2]
/::: J::. 7.:, log-majorization theorem ii, 1XO) J::. -5 /:::*~ti -C ~, ;l': 9: a E [0, 1] t iDE'fiB'.fl'JU A, B /:::xt
l.,-C'
(A#a.BY >-(log) X #a.Br (r 2: 1)
i,i!J¾ IJ "fl--:;o t::. 6ilt~~/:::luEBJl~n-c~,7.:,0)/i, 1XO){'f'J'lHli;f~A
-C, Ando-Hiai inequality (AH) tl1'¥iin-c~,t9o (LH) 0)3iffi1,1!-tO)~iEBJlO)~~-C90
Proof of (AH).
ii\ aef.[0,1] /:::xtl.,-C, 2~}'.li[.
Aqa.B
= A½(A-½BA-½)a.A½ (A,B > 0)
~~,,\L.,190 C#a. t~i:l:Fil!~-C91,1\ ::.t,G0):15/:J:, {'pffl~-¥t~/:::/:J:t~IJ ii±lvo) .rJr-c, 1XO)~
~0:i:\~1!£ffi L, ;l': 9: AqaB = BQ1-a.A = B(B- 1Qa.-1A- 1)B
~-Cf.ixJEi:l:, 1XO)J::. -5 /:::~~~tl,;l':9: A #a.B s 1 {cc} ca.= (A-½BA-½)°' s A- 1 {cc} c-a 2 A
::_ O)r-C:i.R9-"q~- ::_ t /:J:, r = l + E (E E [0, l]) /:::xt l.,-C,
Ar#a.Br = A½(A' #a. A-½Br A-½)A½ S 1
(1) c-a 2 A /::: (LH) ~~ffl l.,-C, A' S c-a, 1,1!:fri,i> I) i 9o
(2) J::112.0)0:i:\~1!£~',
A-½Br A-½= A-½(A½cA½)" A-½= A- 1 Qr C = C(c- 1 #r-1 A)C
=
C(c- 1 #,
:/Jt-::i-C, A' #a A-½Br A-½ S c-m #a
A)c
s C(c-1 #, c-a.)c = c< 1-a.J,+1
C{l-a)<+l
= ca. S A- 1 J::. I)
Ar#a.Br = A½(A' #a. A-½Br A-½)A½ S A½A- 1A½ = 1
{'Fffl*~{iiJ-,P:t~ ~{!£-j ::. t /::: J::. I)' J::. I) ~~,:frtfi1,1!·c-~ 7.:,~J t l.,-C' ifal;f~;i:\~~tf Q::. t i,)!-C ~ Q
tJ~~'i9o iral;f~Ai:l:, 1XO)J::. -5 t~:t,O)-C91,1! (LH) O)~.)Jt~~f.llt'.{r.-Ctb 1J 190
(1 +r)q
Furuta Inequality (FI)
If A 2 B 2 0, then for each r 2 0,
(i)
and
(ii)
hold for p 2 0 and q 2 1 with (1
+ r)q 2 p + r.
= p+ r
99
Furuta inequality t::::009 Q ::,tilkii, [20], [21], [10], [11], [36], [16] f.t: t''1$,ttl:::;bt:: IJ 19o
(FI) ii, N*~flf:l::::t:,1t,-c~%~:ft0) t ~iJ,.t!ft:~-C.\ -t":h'a:'. a-geometric mean 'a:'.Jtllt''"C-t":hHi
9t~OJ~5~m~tt~G:h;Jc:9:
If A :::> B :::> 0, then for each r :::> 0
A-r #1+r BP::=; A
p+r
holds for p :::, L
Satellite of (FI) (SFI) If A :::> B :::> 0, then for each r :::> 0
A-r #1±:. BP:::; B (:::; A)
p+r
holds for p :::> L
If log A :::> log B for A, B > 0, then
A-r #Hr BP::=; B
p+r
holds for p :::, 1 and r :::, 0,
log A:::> logB ii, chaotic order t~i£:h-c1t,;Jc:9iJ>, logt 'h>{'Pffl*lj!w,llt.t:0)-C, A:::> B(> 0) ~ IJ ~
*
*
ma%
9
/i, A » B t ~ ~ :h 9
(SFI) 'a:'. ffi9f:: ~ O)Jj)j:~iJ', 1JZOJii'l:EmlOJ/f~~c-9-C
It' /i!ftff ,:::: t.t:-::, L It'
0
0
)
(FI) for chaotic order. If log A :::, log B for A, B > 0, then
holds for p :::> 0, q :::> 1 and r :::> 0 such that rq:::, p
+ r,
q
rq
______,_ ____,,__ _ _ _
1
-r
(CFI) If A» B for A, B > 0, then
~
= p+ r
_ __. p
100
holds for p 2 0 and r 2 0.
logx)n
( 1+-----;;-+x
(n-+oo)
O):;j;UJ'fH:::ib f? 19, A~ B r:::X'f"L.,-c
log A B _
log B
A n -_ 1 + --,
n - 1 + -n
t :B< t, An 2 Bn (n = 1,2, · · ·)
ti G r:::Jt Ve (FI) 'a'.Jfi1!ffl9 Qt,
/:::t~ f) 19, .'IIBJE-C9;1Ji', (FI)
~
-c fiJ,;zj!O)
~
n
Gr:, +51::kt.i:-Q n /:'.:-:Jl.'1~/i, An 2 Bn > 0 ;iJi'fiJtv:L.,19, ::.
t (CFI) /ilm{ii-CibQ::. t ;IJi',t;IJ, f) 19,
(SFI) O)fi'J*lffit
> 0, then
If log A 2 log B for A, B
A-r #1+r BP:::; B
p+r
holds for p ;::,: 1 and r 2 0.
Transposition: A#aB
= B#1-aA
Multipicativity: A#af3B
= A#a(A#(3B)
::.0)$-0liiO)--rc-_ uamJ:.r.,ffO)~f)~1,,tO)-C9:
by (Mp)
by (Tp)
by (CFI)
= J #1p BP= B.
101
BLP /:J:, Bebiano-Lemos-Providencia O)jffiJ'.("J'.i: .I& ·:d: {, 0)-c', flit G /:J: [4] /:: J'>v'--C{'j{O) J
Jv L. ~~:ft
H~~G* Gt::o
Bebiano-Lemos-Providencia inequality (BLP)
holds for A, B 2 0 and s 2 t 2 0.
Theorem 4.1. If A, B > 0, then
holds for p 2 1 ands 2 r 2 0.
...E...±...'.'. = 1
p(l+r)
'
r(p + s)
---=s
l+r
(BLP) iJ'fi G tl,Q;: t J: fJ 9;!l G tL* To
~B, {, -5 ~ I., p;j•fwi:~ x./;:f, {'.J{O){fpffl*~~A i: ~ I.,, -f O) norm ~~Alf& t I., --C Theorem 4.1 ;/J'{ft
ii:Mlt G ti,* To
Theorem 4.2. If A, B
> 0 satisfy A # 1p
8
BP+s S Al+• for some p 2 1 ands 2 0, then Bl+s S Al+s
and so Bl+r S Al+r for OS r S s.
;:O)~~:ft/:J:, (FI) 7J''?~iJ,ti,1To ~~- {&;E: A8 #1 BP+s
tt--c, 1&JE i:
s Al+s O)jjljJill/::fi!IT{)laiJ>G
<'XI::, (FI) t (AH) O)[m~:/'J!5i"t'ibQ Grand Furuta inequality (GFI) t:J:,
v'iT
Grand Furuta inequality (GFI) If A 2 B > 0 and t E [0, 1], then
holds for r 2 t and p, s 2 1.
A-½ i:ltl-
p
<'.J\O)J:
-5 ~~"t'~tf:\~tl,--C
102
(GFI) If A 2':: B > 0 and t E [O, 1], then
holds for r 2':: t and p, s 2':: 1.
;,)Zr:, (GFI) /::~J;t't7.i BLP ,f~ci'tO)-J!lH~~cjlf.:z -CJ:i-*'t, :Z:O)f:::.~/:, 1JZO){/pffl*,f~ci't~®L,
* 't, 'M~O).:. t. t.i::il• G, .:.ti/i Theorem 4.2 0) (GFI) IIBU:'Mt:::.-:, -c~, * 't,
Theorem 4.3. Suppose that A, B > 0 and t E [O, 1]. If Ar-tq1 (Arq1B(p-t)s+r) <:'. Al-t+r for some
p
P, s 2':: 1 and r 2':: t, then Bl-t+r <:'. A 1 -t+r.
'
Corollary 4.4. Suppose that A, B > 0 and t E [O, 1]. Then
holds for p, s 2':: 1 and r 2':: t.
Theorem 4.3 lJz r.J Corollary 4.4 --C, t = 0, s = 1 t. T 7-i t., Theorem 4.2 lJz r.J Theorem 4.1 -/J>f~ G ti
*T, *t:::.,
(GFI) t. (AH) 0)00~/i,
~
(GFI) fort= 1, r = s
(AH)
c'ib7.i.:. t. -/J>;b-/J>-:, -C ~' * TO)--C:, Theorem 4.3 --C, t = 1, s = r t. T 7-i t.,
Let r 2':: 1 be given. If Ar-l#1(Ar#1Bpr) <:'. Ar for some p 2':: 1, then Br<:'. Ar.
p
r
A- 1 #1(I
#1
A-~Bpr A-~)<:'. I
r
P
i,
°6 G /::, Cl'.=
B1 = (A-~Bpr A-~)~ t_ J:,
1961 -4:,
B1
<:'. I t.~~f!J!;z G:h* T, -1J,
Nakamura and Umegaki [31] ti,
S(A) = -AlogA
/:: ~-:, -C, {/pffi*.:r.:,, t- O !::"-~~Al,* l, t:::., :Z:0)1~, f'pffi*'Jlf;,JO)f_llli0)9£/:!!!0)r--C, 1989 -4:, J-I
Fujii and Kamei [9] /:: ~ iJ, {'Fffi*;j:lcf:Srt.:r.:,, t- O 1::"--/J>
S(AIB) = A½ log(K½BA-½)A½
103
S(AII) = S(A) = -AlogA
J:: fJ 9;11 G:hi 9o t~;B, f'FJlHltffixt.r.:,, J-. P
[38] Ii, Tsallis {/pffl*f§xt.r.:,, J-. P ~ -
~-0)
§~t~ili{.GU: L,--c, Yanagi, Kuriyama and Furuichi
E [0, 1]) ~~AL, i L,f;:: :
Ta(Pla) (a
Ta(Pla) = P #a a - P
a
Ta(Pla)
--1-
S(pla) (a
d(p #a a) I
da
:1/&f&r:::, fFJij*f§xj.r.:,, r- P
--1-
O);
- S( I )
a=O -
Pa
1:'.. -1:::0097., BLP inequality /:::--:J1,,--c,
ii'\ (CFI) O)}it,fflt L,--c, J'.XO)::f~AiJtfijG:hi9o
0
~x. --CJJ.f;::1,,c:,l[l.t1,,i90
> 0 and r > 0 be given. Then, if S(ArlAP+r) 2 S(A'IBP+r)
for some p > 0, then A' 2 Br.
Theorem 5.1. Let A, B
Proof.
ii\ ~S(ArlBP+r) = A~ log(A-~ BP+r A-~);; A~
t~O)""C,
{.&'.
J'.XO)
J:: 5 r::: 131,,~x_ G:h
i9o
r
r
1
log A 2 log(A-2BP+r A-2),.
~::.--c,
B 1 =(A-~BP+rA-~)it:B
Theorem 5.2. Let A, B > 0 and t, r 2 0 be given. Then, if
S(At+'IAP+t+r) 2 S(At+'IAr q¼ B(p+t)s+r)
holds for some p, s > 0 with (p
+ t)s 2 t, then
At+r
2 Bt+r.
If A» X for A,X > 0, then
A
(p+t)s+r
q
r
t
t
r
!
2[A2(A2XPA2) 8 A2]q
holds for p,t,r,s 2 0, q 2 l with (t+r)q 2 (p+t)s+r.
(CFI)J::fJ
104
.:.:ti,J: fJ, q = (p~2~+r and X
~~'a::ffi\i'Q t,
[A-'tr (Ar Q1 B(p+t)s+r)A-'trJ½ t l.,"Ll::1120) (GFI) j!i'l_O){/pffl~::f
=
s
t
r
t
r
-----1±.!:__
At+r 2". ...
