6–202008AFBI6BA
C5C6“C7C8C9CAC3CB”CCCDATAUCEC7ALCFCG
BGBCBCBCBGBJCQBXB6B7B8AGBZAGBBCRCSCTAFAGAGACBEBCBCBJCUAHAIBDB2CPCV
AHAIA4BVBWA2?A3B6:“AHAIB0BXBYCXCYASBZ
AOCUC0DGC1B0BXBYCXCYAVB2,BPCSB7AVBUBOAXBB
C2,C3C2C4BDACC5D3CPC9BUBOAXB0B1B2,BPC6C7
BXCPCID8CSB7BBC2,C8C2DHC9CAB4ACC6CBACCCCS
AGD7CPB0C4BD.”BPBQAHAIC4BDDHAKDEAHAICDDF
B0AVA4CEAIBZCR,CDA1C9DEAHAIC5CFB0AVA4ATB8
C0AH,B1ASAHAIC4BDAPB0A4A4CGCHCIDECHA5CWAB
D7CJA7CKCQA6B0CLCM.A6AFC3BHAVBUDHBLAHB0
CSB7D7D8.
CN1B9aACb∈R
+
,CSB7:
a
2
b
+
b
2
a
greaterorequalslanta+b.
CSB7:?BBCNBGDHBLAHCGC2
a
2
b
+bgreaterorequalslant2a,
b
2
a
+agreaterorequalslant2b,A3DHBLAHCABFCGCSCP
a
2
b
+
b
2
a
greaterorequalslanta+b.
AVBVCOBCC8C4BD:(1)BLCPAOBSB0B5C0DEa=
b,C4CSB7CHCQCRDE?(2)B9CHaACbACc∈R
+
,AT
BDC2DJCSB2B0DHBLAH?
DDCN1BNCPC4BDB0DHBLAH:
a
2
b
+
b
2
c
+
c
2
a
greaterorequalslanta+b+c;···················A1
a
2
b+c
+
b
2
c+a
+
c
2
a+b
greaterorequalslant
a+b+c
2
.·····A2
C4CCBIAVBUDHBLAHCTCUCN1B0BZCRCGBNCPCV
CWB0CSB7,CXB8CZAYBIAKBUDHBLAHB0CSB7,CGB8
BBC2AMB0CSB7BZCRCY?CWABDHCZALAVAF.
CUBQC3C4CNBGDHBLAHCGC2A6AFD5BUDHBLAH:
a
2
b+c
+(b+c)greaterorequalslant2a,
b
2
c+a
+(c+a)greaterorequalslant2b,
c
2
a+b
+(a+b)greaterorequalslant2c.
D5AHCABFC2
a
2
b+c
+
b
2
c+a
+
c
2
a+b
greaterorequalslant0,
BHD0AYCWABD1D6D2D3:DEC4BDD4D5,C8DECS
B7D4D5?CUB9C5B8CAB4AVAFCSB7CXCYAP,BLCPAO
BSB0B5C0BTCGA6DDC9CPAB,BGBQa=b+c,b=
c+a,c=a+bB1ASAOBSC5CHDEa=b=c=
0,ASA6CKDHD6.D3BPCWABC5B8CCCSAVAFCSB7,D7
a=b=c>0AOBSA0CG.CSD8DAAF:
a
2
b+c
+
b+c
4
greaterorequalslanta,
b
2
c+a
+
c+a
4
greaterorequalslantb,
c
2
a+b
+
a+b
4
greaterorequalslantc,
D5AHCABFC2
a
2
b+c
+
b
2
c+a
+
c
2
a+b
greaterorequalslant
a+b+c
2
,
BDB0CAB4AVAFBLCPAOBSB0B5C02a=b+c,2b=
c+a,2c=a+b,BGBQCGA6C1a=b=c>0AO
BS.C3C4A6BSD9B4,DADBC4BDCSB7:B9aACbACcAC
d>0,AJ
a
2
b+c+d
+
b+c+d
9
greaterorequalslant
2a
3
,
b
2
a+c+d
+
a+c+d
9
greaterorequalslant
2b
3
,
c
2
a+b+d
+
a+b+d
9
greaterorequalslant
2c
3
,
d
2
a+b+c
+
a+b+c
9
greaterorequalslant
2d
3
,BTAHCABFCGC2
AVDHBLAH
a
2
b+c+d
+
b
2
a+c+d
+
c
2
a+b+d
+
d
2
a+b+c
greaterorequalslant
a+b+c+d
3
(BLCPAOBSB5C0a=
b=c=d>0).···········································A3
C3C4A1ACA2ACA3DHBLAH,CWABCGA6BNCPBE
AVCTB0C4BD:B9x
1
,x
2
,···,x
n
∈R
+
,A1x
1
+
x
2
+···+x
n
=a,CGC2CP
x
2
1
a?x
1
+
x
2
2
a?x
2
+
···+
x
2
n
a?x
n
greaterorequalslant
a
n?1
(BLCPAOBSB5C0x
1
=x
2
=
···=x
n
>0).············································A4
CSB7:
x
2
1
a?x
1
+
a?x
1
(n?1)
2
greaterorequalslant
2x
1
n?1
,
x
2
2
a?x
2
2008AFBI6BA6–21
+
a?x
2
(n?1)
2
greaterorequalslant
2x
2
n?1
,···,
x
2
n
a?x
n
+
a?x
n
(n?1)
2
greaterorequalslant
2x
n
n?1
,nBUDHBLAHCABFCGCSB7DHBLAHA4,BLCP
AOBSB0B5C0x
1
=x
2
=···=x
n
>0.
BMCXA6BSB0C4BDCXCYC8CSB7D9B4,CWABC5D3
CGA6BPCSB7C2DCC3C4BDBLCPAOBSB0B5C0,BQBCC3
C4CNBGDHBLAH,DDBYCACCB0B6AHAJDEAHAH,B4BP
CSB7B0CXCYAPD8DFDGDHDHBLAHB0BLCPAOBS,DHBL
CPB0BZDIBYDJ,BTCGA6A0BBC9A4BZCRA7CKAVAWDH
BLAHB0CSB7CQA6AB.
CN2BXDGBYAHaACbAWAXa+b=1,BWCS:
(a+2)
2
+(b+2)
2
greaterorequalslant
25
2
.
A7A6CSCRB4CI,A1A2C5BBA6BSAGA3B9B0BCBD:
DD(a+2)
2
=(b+2)
2
=
25
4
C4BDBLCPAOBSB0B5
C0DEa=b=
1
2
.B4CSC4BDAOBS:?BBDHBLAH(a+
2)
2
+
25
4
greaterorequalslant5(a+2),B1AX(b+2)
2
+
25
4
greaterorequalslant5(b+
2),BLCPAOBSA4CBa=b=
1
2
,
A2(a+2)
2
+(b+2)
2
+
25
2
greaterorequalslant5(a+b+4)=
25,CGCSC2(a+2)
2
+(b+2)
2
greaterorequalslant
25
2
.
CN3(1990AFBI1A5“A6A7A8”)BXDGxACyAC
z>0,B4A1
x
2
1+x
2
+
y
2
1+y
2
+
z
2
1+z
2
=2,BW
CS:
x
1+x
2
+
y
1+y
2
+
z
1+z
2
lessorequalslant
√
2.
CSB7:C4BDBLCPAOBSB0B5C0A5x=y=z=
√
2.
A1
x
2
1+x
2
+
y
2
1+y
2
+
z
2
1+z
2
=2
??
1
1+x
2
+
1
1+y
2
+
1
1+z
2
=1,
A1
x
1+x
2
=
√
2
2
√
2x
1+x
2
lessorequalslant
√
2
2
?
?
?
2+x
2
2
1+x
2
?
?
?
=
√
2
4
parenleftbigg
1+
1
1+x
2
parenrightbigg
(BKASBLCPAOBSB5C0x=
√
2).
B1AXCGC2
y
1+y
2
lessorequalslant
√
2
4
parenleftbigg
1+
1
1+y
2
parenrightbigg
,
z
1+z
2
lessorequalslant
√
2
4
parenleftbigg
1+
1
1+z
2
parenrightbigg
.
AUD5BUDHBLAHCABFC2
x
1+x
2
+
y
1+y
2
+
z
1+z
2
lessorequalslant
√
2
4
parenleftbigg
3+
1
1+x
2
+
1
1+y
2
+
1
1+z
2
parenrightbigg
=
√
2(DGDHBLCPAOBS).
CN4(BI36A5IMOALA6)D9aACbACc>0,A1
abc=1,BWCS:
1
a
3
(b+c)
+
1
b
3
(c+a)
+
1
c
3
(a+b)
greaterorequalslant
3
2
.
CSB7:C4BDBLCPAOBSB0B5C0A5a=b=c=
1.
1
a
3
(b+c)
+
1
b
3
(c+a)
+
1
c
3
(a+b)
greaterorequalslant
3
2
??
bc
a
2
(b+c)
+
ca
b
2
(c+a)
+
ab
c
2
(a+b)
greaterorequalslant
3
2
.
C3C4CNBGDHBLAHCGC2
bc
a
2
(b+c)
+
b+c
4bc
greaterorequalslant
1
a
,
ca
b
2
(c+a)
+
c+a
4ca
greaterorequalslant
1
b
,
ab
c
2
(a+b)
+
a+b
4ab
greaterorequalslant
1
c
.
BLCPAOBSB0B5C0A52bc=ab+ca,2ca=bc+ab,
2ab=ca+bc,D5AHB1ASAOBSCHa=b=c=1.
AUD5BUDHBLAHCABFC2
bc
a
2
(b+c)
+
ca
b
2
(c+a)
+
ab
c
2
(a+b)
+
a+b
4ab
+
c+a
4ca
+
b+c
4bc
greaterorequalslant
1
a
+
1
b
+
1
c
,
A0
bc
a
2
(b+c)
+
ca
b
2
(c+a)
+
ab
c
2
(a+b)
+
c(a+b)
4
+
b(c+a)
4
+
a(b+c)
4
greaterorequalslantbc+ca+ab,
A2
bc
a
2
(b+c)
+
ca
b
2
(c+a)
+
ab
c
2
(a+b)
greaterorequalslant
1
2
(bc+ac+ab)greaterorequalslant
3
2
3
√
a
2
b
2
c
2
=
3
2
(DGDH
BLCPAOBS).
AGA6BMDHBLAHAOBS.
CN5(2005AFAPA9A9BWAAAAABCHALA6)D9
aACbACc>0,ab+bc+ca=
1
3
,BWCS:
1
a
2
?bc+1
+
1
b
2
?ca+1
+
1
c
2
?ab+1
lessorequalslant3.
CSB7:C4BDBLCPAOBSB0B5C0A5
a=b=c=
1
3
.
A1a
2
?bc+1=a
2
?bc+3(ab+bc+ca)=
a(a+b+c)+
2
3
,APq=a+b+c,
1
a
2
?bc+1
+
1
b
2
?ca+1
+
1
c
2
?ab+1
6–222008AFBI6BA
=
1
aq+
2
3
+
1
bq+
2
3
+
1
cq+
2
3
=
3
3aq+2
+
3
3bq+2
+
3
3cq+2
,
A2
3
3aq+2
+
3
3bq+2
+
3
3cq+2
lessorequalslant3
??
3aq
3aq+2
+
3bq
3bq+2
+
3cq
3cq+2
greaterorequalslant1(BL
CPAOBSB5C0a=b=c=
1
3
A1q=1).
C3C4CNBGDHBLAH
3aq
3aq+2
+
aq(3aq+2)
3
greaterorequalslant
2aq(DGCSaq=
1
3
ASBLCPAOBS).
B1AXCGC2CP
3bq
3bq+2
+
bq(3bq+2)
3
greaterorequalslant2bq,
3cq
3cq+2
+
cq(3cq+2)
3
greaterorequalslant2cq,(bq=cq=aq,B1ASD6AUBL
CPAOBSB0B5C0),AGA6D5BUDHBLAHCABFC2
3aq
3aq+2
+
3bq
3bq+2
+
3cq
3cq+2
greaterorequalslantaq
parenleftbigg
4?3aq
3
parenrightbigg
+bq
parenleftbigg
4?3bq
3
parenrightbigg
+cq
parenleftbigg
4?3cq
3
parenrightbigg
=
q
2
[4?3(a
2
+b
2
+c
2
)]
3
=
q
2
3
parenleftbigg
4?3
parenleftbigg
q
2
?
2
3
parenrightbiggparenrightbigg
=q
2
(2?q
2
)
=?(q
2
?1)
2
+1.
B7?(q
2
?1)
2
+1lessorequalslant1,BQBKCSB7ACAD.
A6BSCSB7CXCYCJ?AWAXBLCPAOBSB5C0,B7AZ
DFCUCPD3AFC2CXCCAGB0AHAI,BKAS,CGBQCO?BB
AJAKB0DEAHAH,C1BBALAMDHBLAHDADBC6AOCSB7.
D9a
1
=
3aq
3aq+2
,b
1
=
aq(3aq+2)
3
,
a
2
=
3bq
3bq+2
,b
2
=
bq(3bq+2)
3
,
a
3
=
3cq
3cq+2
,b
3
=
cq(3cq+2)
3
.
A2(a
1
+a
2
+a
3
)(b
1
+b
2
+b
3
)
=
parenleftbigg
3aq
3aq+2
+
3bq
3bq+2
+
3cq
3cq+2
parenrightbigg
×
parenleftbigg
aq(3aq+2)
3
+
bq(3bq+2)
3
+
cq(3cq+2)
3
parenrightbigg
greaterorequalslant(
√
a
1
b
1
+
√
a
2
b
2
+
√
a
3
b
3
)
2
=(aq+bq+cq)
2
=q
4
.
BLCPAOBSB0B5C0
1
3aq+2
=
1
3bq+2
=
1
3cq+2
,
A0aq=bq=cq,CUAP
aq(3aq+2)
3
+
bq(3bq+2)
3
+
cq(3cq+2)
3
=
3(a
2
+b
2
+c
2
)q
2
+2(a+b+c)q
3
=
3
parenleftbigg
q
2
?
2
3
parenrightbigg
q
2
+2q
2
3
=q
4
A2
3aq
3aq+2
+
3bq
3bq+2
+
3cq
3cq+2
greaterorequalslant1.
BHA6BSB0CSB7CWABC5D3,ANBQ?BBAOD4DHBL
AHBPCSB7B0CXCYAPCHAVBOAPAQAZ,B7DEC3C4BLCP
AOBSB5C0C4DEAHAHB0AJAKC8ARAS,CUBQBPCSB7B0
CXCYAPBLDJCXB8BXBB.DACN5,CWABBPACADB0CN
ATBSDHBTBDB0AJAKCACCDEAHAH,BQCOC1BBALAMDH
BLAHBTCHDADBC6AOCSB7.
BMCXA6BSANCNCGA6C6CP,C4BDB4B4CSCPBLCP
AOBSB0B5C0DED0ATB8B0,B1ASC4DHBLAHB0AUBPC8
C0C8ASAX,A6AVBPCSB7CXCYAPB0ARAS,B4DGDHDH
BLAHB0BLCPAOBSDECSB7B0CCAW.BYDAAXAYAGAZ
BMB0“CGCHB0B1B0C4BD,BTCGCHB2B0B0C5D3”,AH
AIBWABC5D3AHAIB3B4B0CXCY,B5B5DEC3CHAVBUC4
BD,CDBCA0DEC4C4BDD8COB4CSAJCCBYB0CXCY.
B6B7B8B9
[1]BABBBCACBDBEBF.A2?A3BGBXBHA7.BIBJ
A4BVCPARBK.1998.
[2]BABLBMACBABNA2.AHAIC4BDB0CVBOACBZ
CRAVCUC4AHAIC5CFB0BPBQ.C6BRB0AIAIBS(CYBQ
CEAIAR).2007.6.
[[3]BABTBU.DDBYDIAYCSB7DHBLAH.AHAXCV
BR(A9APAR).2006.5.
[4]BVB7BWACBXBY.BBDIAYB1C2CRCSB7AVCV
BJAHDHBLAH.BZC0AXCEA4AICDDF.2007.1.A0A0
～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～～
D1D2
BGAABI4BABI23B1D0ADAZ“BMCXA6BSCGCT,CG
A6C6CPBVACBRACAMACBP”BCAMD3C1“AGB0CCB6,B1
ASC9D8AVA4CZC1DJAHAGB0BGBCAVAHAIBCBDBZCR
B0BAD0.D5BUDGAHBJC1DECCCCnB0AVB0ACAKB0AC
D5B0B0D3AHCCB6,ASBPAGB0DHAHCAB1,A5AUD4D8
AVA4AIATAHAIDGC1ACBZCRAVA0BBAHAIBCBDB1BO
ABCNAT.”CMBKBEBY,B4DIBXA2ASBUA2CUD4.
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