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LUẬN VĂN: Phương pháp sử dụng tính chất hàm lồi pot

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BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG…………………













LUẬN VĂN

Phương pháp sử dụng
tính chất hàm lồi


1
Mu
.
cLu
.
c
Mo
.


d¯ ˆa
`
u 2
Chu
.
o
.
ng 1. Phu
.
o
.
ng pha´p su
.

du
.
ng tı´nh chˆat ha`m lˆo
`
i (lo
˜
m) 5
1.1 Th´u
.
tu
.
.
s˘a
´
pd¯u
.

o
.
.
ccu

ada
˜
ybˆa
´
td¯˘a

ng th´u
.
c sinh bo
.

i ha`m lˆo
`
i (lo
˜
m) 5
1.2 Bˆa
´
td¯˘a

ng th´u
.
c Karamata 11
1.3 Gi´o
.

i thiˆe
.
umˆo
.
tsˆo
´
ha`m lˆo
`
iva` ha`m lo
˜
m 19
1.3.1 Mˆo
.
tsˆo
´
ha`m lˆo
`
i 19
1.3.1 Mˆo
.
tsˆo
´
ha`m lo
˜
m 19
1.4 Ba`i tˆa
.
p 20
Chu
.

o
.
ng 2 Phu
.
o
.
ng pha´p lu
.
.
acho
.
n tham sˆo
´
24
2.1 Ca´c da
.
ng toa´n ch´u
.
a tham sˆo
´
d¯ ˆo
.
clˆa
.
p 25
2.1.1 Tham sˆo
´
chı

thuˆo

.
cmˆo
.
tvˆe
´
cu

abˆa
´
td¯˘a

ng th´u
.
c 25
2.1.2 Tham sˆo
´
co´ trong hai vˆe
´
cu

abˆa
´
td¯˘a

ng th´u
.
c 30
2.2 Ca´c da
.
ng toa´n ch´u

.
a tham phu
.
thuˆo
.
cva`o tham sˆo
´
kha´c 36
2.3 Ba`i tˆa
.
p 42
Chu
.
o
.
ng 3 Phu
.
o
.
ng pha´p su
.

du
.
ng tı´nh chˆa
´
tcu

a ha`m d¯o
.

nd¯iˆe
.
u 45
3.1 Ha`m d¯o
.
nd¯iˆe
.
u 45
3.2 Tı´nh d¯o
.
nd¯iˆe
.
ucu

a ha`m ca´c d¯a
.
ilu
.
o
.
.
ng trung bı`nh . . . . . . . . . . . 49
3.2.1 Ca´c d¯a
.
ilu
.
o
.
.
ng trung bı`nh 50

3.2.2 Ca´c d¯a
.
ilu
.
o
.
.
ng trung bı`nh suy rˆo
.
ng 50
3.3 Tı´nh d¯o
.
nd¯iˆe
.
ucu

a ha`m ca´c d¯a th´u
.
cd¯ˆo
´
ix´u
.
ng so
.
cˆa
´
p 55
Chu
.
o

.
ng 4 Phu
.
o
.
ng pha´p hı`nh ho
.
c 62
4.1 Hı`nh ho
.
c ho´a ca´c d¯a
.
ilu
.
o
.
.
ng trung bı`nh 62
4.2 Mˆo
.
tsˆo
´
phu
.
o
.
ng pha´p kha´c 65
4.1 Ba`i tˆa
.
p 72

Kˆe
´
t luˆa
.
ncu

a luˆa
.
n v˘an 73
Ta`i liˆe
.
u tham kha

o 74
2
Mo
.

d¯ ˆa
`
u
Bˆa
´
td¯˘a

ng th´u
.
c (BD
-
T) la` mˆo

.
t trong nh˜u
.
ng nˆo
.
i dung quan tro
.
ng trong chu
.
o
.
ng
trı`nh toa´n phˆo

thˆong, no´ v`u
.
a la` d¯ˆo
´
itu
.
o
.
.
ng d¯ˆe

nghiˆen c´u
.
u ma` cu
˜
ng v`u

.
a la` mˆo
.
t
cˆong cu
.
d¯ ˘a
´
clu
.
.
c, v´o
.
inh˜u
.
ng ´u
.
ng du
.
ng trong nhiˆe
`
ulı
˜
nh vu
.
.
c kha´c nhau cu

a toa´n ho
.

c.
Trong ca´c d¯ˆe
`
thi cho
.
nho
.
c sinh gio

i toa´n o
.

ca´c cˆa
´
p, nh˜u
.
ng ba`i toa´n vˆe
`
ch´u
.
ng minh
BD
-
Tthu
.
`o
.
ng xuˆa
´
thiˆe

.
nnhu
.
mˆo
.
tda
.
ng toa´n kha´ quen thuˆo
.
c, nhu
.
ng d¯ˆe

tı`m ra l`o
.
i
gia

i khˆong pha

i la` mˆo
.
tviˆe
.
cdˆe
˜
da`ng.
Ly´ thuyˆe
´
tBD

-
Td¯a
˜
d¯ u
.
o
.
.
c kha´ nhiˆe
`
u ta`i liˆe
.
ud¯ˆe
`
cˆa
.
pva` ca´c ba`i tˆa
.
pvˆe
`
BD
-
Tcu
˜
ng
kha´ phong phu´, d¯a da
.
ng, trong d¯o´ ca´c phu
.
o

.
ng pha´p ch´u
.
ng minh BD
-
T la` phˆa
`
nnˆo
.
i
dung quan tro
.
ng thu
.
`o
.
ng g˘a
.
p trong nhiˆe
`
u ta`i liˆe
.
u.
Mˆo
.
t trong nh˜u
.
ng phu
.
o

.
ng pha´p ch´u
.
ng minh BD
-
T ho˘a
.
c sa´ng ta
.
o ra nh˜u
.
ng BD
-
T
m´o
.
ila`viˆe
.
c la`m ch˘a
.
tBD
-
T.
Gia

su
.

ta co´ (ho˘a
.

ccˆa
`
nch´u
.
ng minh) BD
-
T A<B(tu
.
o
.
ng tu
.
.
v´o
.
iBD
-
T A>B, A≤
B, A ≥ B). Nˆe
´
u tı`m d¯u
.
o
.
.
cbiˆe

uth´u
.
c C sao cho A<C<B, thı` ta no´i r˘a

`
ng BD
-
T
th ´u
.
nhˆa
´
td¯a
˜
d¯ u
.
o
.
.
c la`m ch˘a
.
t (nghiˆem ng˘a
.
t) bo
.

iBD
-
Tth´u
.
hai va`hiˆe

n nhiˆen, BD
-

T
th ´u
.
nhˆa
´
td¯u
.
o
.
.
c suy ra t`u
.
BD
-
Tth´u
.
hai. Viˆe
.
cch´u
.
ng minh d¯u
.
o
.
.
cBD
-
Tth´u
.
hai cho

ta mˆo
.
tca´chch´u
.
ng minh BD
-
Tth´u
.
nhˆa
´
tva`d¯ˆo
`
ng th`o
.
i sa´ng ta
.
o ra nh˜u
.
ng BD
-
Tm´o
.
i.
Do d¯o´, viˆe
.
c tı`m ra ca´c phu
.
o
.
ng pha´p d¯ˆe


la`m ch˘a
.
tBD
-
Tla`rˆa
´
t co´ y´ nghı
˜
a.
D
-
o´cu
˜
ng la` nˆo
.
i dung ma` luˆa
.
n v˘an na`y d¯ˆe
`
cˆa
.
p.
Luˆa
.
n v˘an da`y 74 trang, gˆo
`
m ca´c phˆa
`
nmu

.
clu
.
c, Mo
.

d¯ ˆa
`
u,4chu
.
o
.
ng nˆo
.
i dung, Kˆe
´
t
luˆa
.
nva`Ta`i liˆe
.
u tham kha

o.
Chu
.
o
.
ng 1: Phu
.

o
.
ng pha´p su
.

du
.
ng tı´nh chˆa
´
tcu

a ha`m lˆo
`
i (lo
˜
m) .
D
-
ˆay la` phu
.
o
.
ng pha´p co
.
ba

nva` quan tro
.
ng nhˆa
´

td¯ˆe

la`m ch˘a
.
tBD
-
Tma`mˆo
.
tsˆo
´
ta`i liˆe
.
uhiˆe
.
n ha`nh cu
˜
ng d¯a
˜
d¯ ˆe
`
cˆa
.
p, d¯˘a
.
cbiˆe
.
t la` ta`i liˆe
.
u [1]. Phˆa
`

nd¯o´ng go´p cu

a luˆa
.
n
v˘an, chu

yˆe
´
u la` viˆe
.
ccu
.
thˆe

ho´a ly´ thuyˆe
´
tcu

aphu
.
o
.
ng pha´p na`y b˘a
`
ng nh˜u
.
ng vı´ du
.
va` ba`i tˆa

.
pcu
.
thˆe

, co´ thˆe

ta´ch riˆeng tha`nh nh˜u
.
ng ba`i tˆa
.
pvˆe
`
BD
-
T kha´ phong phu´.
Kha´ nhiˆe
`
uBD
-
T quen thuˆo
.
c, la` tru
.
`o
.
ng ho
.
.
p riˆeng cu


a ca´c BD
-
Td¯a
˜
d¯ u
.
o
.
.
cta
.
orat`u
.
nh˜u
.
ng minh ho
.
a na`y. Trong phˆa
`
n cuˆo
´
i chu
.
o
.
ng, luˆa
.
n v˘an cu
˜

ng d¯a
˜
d¯ u
.
a ra d¯u
.
o
.
.
c kha´
3
nhiˆe
`
u ha`m lˆo
`
i (lo
˜
m) d¯ˆe

ba
.
nd¯o
.
c co´ thˆe

a´p du
.
ng sa´ng ta
.
o ra nhiˆe

`
uBD
-
T kha´c.
Chu
.
o
.
ng 2: Phu
.
o
.
ng pha´p lu
.
.
a cho
.
n tham sˆo
´
.
Co´ thˆe

minh ho
.
ay´tu
.
o
.

ng cu


aphu
.
o
.
ng pha´p na`y bo
.

imˆo
.
t vı´ du
.
sau d¯ˆay: Gia

su
.

a, b, c la` 3 sˆo
´
khˆong ˆam co´ tˆo

ng b˘a
`
ng 3. Dˆe
˜
da`ng ch´u
.
ng minh d¯u
.
o

.
.
cbˆa
´
td¯˘a

ng th´u
.
c

a +

b +

c ≥ ab + bc + ca.
Nhu
.
vˆa
.
y, v´o
.
i k ≥
1
2
thı` BD
-
T sau d¯ˆay luˆon d¯u´ng
a
k
+ b

k
+ c
k
≥ ab + bc + ca.
Mˆo
.
t cˆau ho

itu
.
.
nhiˆen d¯u
.
o
.
.
cd¯˘a
.
t ra, v´o
.
i k<
1
2
thı` khi na`o BD
-
T trˆen vˆa
˜
n d¯u´ng?
Viˆe
.

c tı`m d¯u
.
o
.
.
csˆo
´
k (k<
1
2
) nho

nhˆa
´
t sao cho BD
-
T trˆen vˆa
˜
n d¯u´ng cho ta mˆo
.
t
phu
.
o
.
ng pha´p d¯ˆe

la`m ch˘a
.
tBD

-
T.
D
-
o´cu
˜
ng la` nˆo
.
i dung ma` luˆa
.
nv˘and¯ˆe
`
cˆa
.
p trong chu
.
o
.
ng na`y, trong d¯o´ tham sˆo
´
k d¯ u
.
o
.
.
cxe´to
.

hai da
.

ng, la` tham sˆo
´
d¯ ˆo
.
clˆa
.
p ho˘a
.
c co`n phu
.
thuˆo
.
cva`o mˆo
.
t tham sˆo
´
kha´c.
Chu
.
o
.
ng 3: Phu
.
o
.
ng pha´p su
.

du
.

ng tı´nh chˆa
´
tcu

a ha`m d¯o
.
nd¯iˆe
.
u.
Phu
.
o
.
ng pha´p na`y cu
˜
ng d¯a
˜
d¯ u
.
o
.
.
cmˆo
.
tsˆo
´
ta`i liˆe
.
ud¯ˆe
`

cˆa
.
p, d¯˘a
.
cbiˆe
.
t la` ta`i liˆe
.
u [1].
Phˆa
`
nd¯o´ng go´p cu

a luˆa
.
nv˘ano
.

chu
.
o
.
ng na`y chu

yˆe
´
u la` viˆe
.
chˆe
.

thˆo
´
ng ho´a mˆo
.
tsˆo
´
phu
.
o
.
ng pha´p s˘a
´
pth´u
.
tu
.
.
ca´c d¯a
.
ilu
.
o
.
.
ng trung bı`nh va`cu
.
thˆe

ho´a ly´ thuyˆe
´

tcu

a
phu
.
o
.
ng pha´p b˘a
`
ng nh˜u
.
ng vı´ du
.
va` ba`i tˆa
.
pcu
.
thˆe

. Kha´ nhiˆe
`
uBD
-
Tm´o
.
id¯u
.
o
.
.

c luˆa
.
n
v˘an sa´ng ta´c, thˆong qua viˆe
.
c la`m ch˘a
.
tBD
-
Tb˘a
`
ng ca´ch su
.

du
.
ng phu
.
o
.
ng pha´p na`y.
Chu
.
o
.
ng 4: Phu
.
o
.
ng pha´p hı`nh ho

.
c.
Nˆo
.
i dung chu
.
o
.
ng na`y d¯ˆe
`
cˆa
.
pd¯ˆe
´
nmˆo
.
tsˆo
´
phu
.
o
.
ng pha´p la`m ch˘a
.
tBD
-
Td¯a
.
isˆo
´

thˆong qua nh ˜u
.
ng u
.
´o
.
clu
.
o
.
.
ng tru
.
.
c quan t`u
.
hı`nh ho
.
c, v´o
.
inh˜u
.
ng vı´ du
.
minh ho
.
a kha´
cu
.
thˆe


.
Luˆa
.
n v˘an d¯u
.
o
.
.
c hoa`n tha`nh du
.
´o
.
isu
.
.
hu
.
´o
.
ng dˆa
˜
n khoa ho
.
ccu

aTiˆe
´
nsy
˜

Tri
.
nh
D
-
a`o Chiˆe
´
n - Ngu
.
`o
.
i Thˆa
`
yrˆa
´
t nghiˆem kh˘a
´
cva`tˆa
.
n tˆam trong cˆong viˆe
.
c, ngu
.
`o
.
i Thˆa
`
y
khˆong chı


giu´p d¯˜o
.
, cung cˆa
´
p ta`i liˆe
.
u, go
.
.
imo
.

cho ta´c gia

nhiˆe
`
uy´tu
.
o
.

ng hay va`
truyˆe
`
nd¯a
.
t nhiˆe
`
ukiˆe
´

nth´u
.
c quı´ ba´u, cu
˜
ng nhu
.
nh˜u
.
ng kinh nghiˆe
.
m nghiˆen c´u
.
u khoa
ho
.
c ma` co`n chı

ba

o cho ta´c gia

trong ta´c phong la`m viˆe
.
c, thˆong ca

m, khuyˆe
´
n khı´ch
d¯ ˆo
.

ng viˆen ta´c gia

vu
.
o
.
.
t qua nh˜u
.
ng kho´ kh˘an trong chuyˆen mˆon va` cuˆo
.
csˆo
´
ng. Chı´nh
vı` vˆa
.
y ma` ta´c gia

luˆon to

lo`ng biˆe
´
to
.
n chˆan tha`nh va`su
.
.
kı´nh phu
.
c sˆau s˘a

´
cd¯ˆo
´
iv´o
.
i
thˆa
`
ygia´ohu
.
´o
.
ng dˆa
˜
n-Tiˆe
´
nsy
˜
Tri
.
nh D
-
a`o Chiˆe
´
n.
Nhˆan d¯ˆay, ta´c gia

cu
˜
ng xin ba`y to


lo`ng biˆe
´
to
.
n chˆan tha`nh d¯ˆe
´
n Ban Gia´m Hiˆe
.
u
4
tru
.
`o
.
ng D
-
a
.
iho
.
c Quy Nho
.
n, Pho`ng d¯a`o ta
.
oD
-
a
.
iho

.
cva` sau D
-
a
.
iho
.
c, khoa Toa´n, quı´
Thˆa
`
y cˆo gia´o tru
.
.
ctiˆe
´
p gia

ng da
.
yd¯a
˜
ta
.
omo
.
id¯iˆe
`
ukiˆe
.
n thuˆa

.
nlo
.
.
i trong th`o
.
i gian ta´c
gia

tham gia kho´a ho
.
c.
D
-
ˆo
`
ng th`o
.
i ta´c gia

cu
˜
ng xin ba`y to

lo`ng biˆe
´
to
.
nd¯ˆe
´

n UBND Tı

nh Gia Lai, So
.

Gia´o du
.
cva` d¯a`o ta
.
oTı

nh Gia Lai, Ban Gia´m Hiˆe
.
u tru
.
`o
.
ng THPT Ia Grai, d¯a
˜
d¯ ˆo
.
ng
viˆen va`ta
.
omo
.
id¯iˆe
`
ukiˆe
.

n thuˆa
.
nlo
.
.
id¯ˆe

ta´c gia

co´ nhiˆe
`
u th`o
.
i gian nghiˆen c´u
.
uva`
hoa`n tha`nh d¯ˆe
`
ta`i.
Trong qua´ trı`nh hoa`n tha`nh luˆa
.
n v˘an na`y, ta´c gia

co`n nhˆa
.
nd¯u
.
o
.
.

csu
.
.
quan tˆam
d¯ ˆo
.
ng viˆen cu

ame
.
,vo
.
.
, ca´c anh chi
.
em trong gia d¯ı`nh, ca´c ba
.
nd¯ˆo
`
ng nghiˆe
.
p, ca´c anh
chi
.
em trong l´o
.
p cao ho
.
c kho´a VII, VIII, IX cu


a tru
.
`o
.
ng D
-
a
.
iho
.
c Qui Nho
.
n. Ta´c gia

xin chˆan tha`nh ca

mo
.
ntˆa
´
tca

su
.
.
quan tˆam va`d¯ˆo
.
ng viˆen d¯o´.
D
-

ˆe

hoa`n tha`nh luˆa
.
n v˘an, ta´c gia

d¯ a
˜
rˆa
´
tcˆo
´
g˘a
´
ng tˆa
.
p trung nghiˆen c´u
.
u, song do
ı´t nhiˆe
`
uha
.
n chˆe
´
vˆe
`
th`o
.
i gian, cu

˜
ng nhu
.
vˆe
`
n˘ang lu
.
.
cnˆen ch˘a
´
cch˘a
´
n trong luˆa
.
n v˘an
co`n nhiˆe
`
uvˆa
´
nd¯ˆe
`
chu
.
ad¯ˆe
`
cˆa
.
pd¯ˆe
´
nva` kho´ tra´nh kho


inh˜u
.
ng thiˆe
´
u so´t nhˆa
´
td¯i
.
nh.
Ta´c gia

rˆa
´
t mong nhˆa
.
nd¯u
.
o
.
.
csu
.
.
chı

ba

ocu


a quı´ thˆa
`
ycˆova`nh˜u
.
ng go´p y´ cu

aba
.
n
d¯ o
.
cvˆe
`
luˆa
.
n v˘an na`y.
Quy Nho
.
n, tha´ng 02 n˘am 2008
Ta´c gia

5
Chu
.
o
.
ng 1
Phu
.
o

.
ng ph´ap su
.
˙’
du
.
ng t´ınh chˆa
´
t
h`am lˆo
`
i (l˜om)
1.1 Th´u
.
tu
.
.
s˘a
´
pd¯u
.
o
.
.
ccu

ada
˜
ybˆa
´

td¯˘a

ng th´u
.
c
sinh bo
.

i ha`m lˆo
`
i (lo
˜
m)
Tru
.
´o
.
chˆe
´
t, v´o
.
i hai sˆo
´
thu
.
.
c a ≥ b, ta su
.

du

.
ng kı´ hiˆe
.
u I(a; b)d¯ˆe

ngˆa
`
md¯i
.
nh mˆo
.
t
trong bˆo
´
ntˆa
.
pho
.
.
p(a; b), [a; b), (a; b]va`[a; b].
Trong [1], hai kˆe
´
t qua

sau d¯ˆay d¯a
˜
d¯ u
.
o
.

.
cch´u
.
ng minh:
D
-
i
.
nh ly´ 1.1.1. Gia

su
.

cho tru
.
´o
.
c ha`m sˆo
´
y = f(x) co´ f

(x) ≥ 0 (ha`m lˆo
`
i) trˆen
I(a; b) va` gia

su
.

x

1
,x
2
∈ I(a; b) v´o
.
i x
1
<x
2
. Khi d¯o´, v´o
.
imo
.
ida
˜
ysˆo
´
t˘ang dˆa
`
n {u
k
}
trong

x
1
;
x
1
+ x

2
2

:
x
1
= u
0
<u
1
<u
2
< < u
n
<
x
1
+ x
2
2
(1.1)
va` da
˜
ysˆo
´
gia

mdˆa
`
n {v

k
} trong

x
1
+ x
2
2
; x
2

:
x
1
+ x
2
2
<v
n
<v
n−1
< <v
1
<v
0
= x
2
(1.2)
sao cho
u

j
+ v
j
= x
1
+ x
2
, ∀j =0, 1, , n (1.3)
ta d¯ˆe
`
uco´
f(u
0
)+f(v
0
) ≥ f (u
1
)+f(v
1
) ≥ ≥ f (u
n
)+f(v
n
). (1.4)
No´i ca´ch kha´c: Da
˜
y

f(u
j

)+f(v
j
)

, j =0, 1, , n, la` mˆo
.
tda
˜
y gia

m.
6
D
-
i
.
nh ly´ 1.1.2. Gia

su
.

cho tru
.
´o
.
c ha`m sˆo
´
y = f(x) co´ f

(x)  0 (ha`m lo

˜
m) trˆen
I(a; b) va` gia

su
.

x
1
,x
2
∈ I(a; b) v´o
.
i x
1
<x
2
. Khi d¯o´, v´o
.
imo
.
ida
˜
ysˆo
´
t˘ang dˆa
`
n {u
k
}

trong

x
1
;
x
1
+ x
2
2

:
x
1
= u
0
<u
1
<u
2
< < u
n
<
x
1
+ x
2
2
va` da
˜

ysˆo
´
gia

mdˆa
`
n {v
k
} trong

x
1
+ x
2
2
; x
2

:
x
1
+ x
2
2
<v
n
<v
n−1
< <v
1

<v
0
= x
2
sao cho
u
j
+ v
j
= x
1
+ x
2
, ∀j =0, 1, , n,
ta d¯ˆe
`
uco´
f(u
0
)+f(v
0
)  f (u
1
)+f(v
1
)   f (u
n
)+f(v
n
). (1.5)

No´i ca´ch kha´c: Da
˜
y

f(u
j
)+f(v
j
)

, j =0, 1, , n, la` mˆo
.
tda
˜
y t˘ang.
Nhˆa
.
n xe´t r˘a
`
ng, d¯ˆe

co´ d¯u
.
o
.
.
cnh˜u
.
ng kˆe
´

t qua

t`u
.
D
-
i
.
nh lı´ 1.1.1 ho˘a
.
cD
-
i
.
nh lı´ 1.1.2,
d¯ i ˆe
`
u quan tro
.
ng tru
.
´o
.
chˆe
´
t la` pha

i xˆay du
.
.

ng trˆen I(a; b) hai da
˜
y {u
k
} va` {v
k
} thoa

ma
˜
nnh˜u
.
ng d¯iˆe
`
ukiˆe
.
ncu

ad¯i
.
nh lı´. Sau d¯o´ la` viˆe
.
c tı`m nh˜u
.
ng ha`m sˆo
´
y = f (x)co´
f

(x) ≥ 0 ho˘a

.
c f

(x)  0 trˆen I(a; b)d¯ˆe

a´p du
.
ng.
Du
.
´o
.
i d¯ˆay la` mˆo
.
tva`i minh ho
.
a cho hai d¯i
.
nh lı´ trˆen, v´o
.
inh˜u
.
ng da
˜
ysˆo
´
va` ha`m
sˆo
´
d¯ o

.
n gia

n nhˆa
´
t. Ba
.
nd¯o
.
c co´ thˆe

tı`m ra nh˜u
.
ng kˆe
´
t qua

kha´c, phong phu´ho
.
n.
V´o
.
i hai sˆo
´
thu
.
.
c cho tru
.
´o

.
c x
1
<x
2
, hı`nh a

nh cu

aca´cd¯iˆe

m u
j
va` v
j
lˆa
`
nlu
.
o
.
.
t
”tiˆe
´
nd¯ˆe
`
u” vˆe
`
trung d¯iˆe


mcu

a d¯oa
.
n[x
1
x
2
]la`
x
1
+ x
2
2
trˆen tru
.
csˆo
´
giu´p ta xˆay du
.
.
ng
d¯ u
.
o
.
.
c hai da
˜

y {u
k
} va` {v
k
} thoa

ma
˜
nnh˜u
.
ng d¯iˆe
`
ukiˆe
.
ncu

aD
-
i
.
nh lı´ 1.1.1 va`D
-
i
.
nh lı´
1.1.2 nhu
.
sau:
Vı´ du
.

1.1.
u
0
= x
1
,u
1
= x
1
+
x
2
− x
1
2.(n +1)
, ,u
n
= x
1
+ n
x
2
− x
1
2(n +1)
=
(n +2)x
1
+ nx
2

2(n +1)
;
v
0
= x
2
,v
1
= x
2

x
2
− x
1
2.(n +1)
, ,v
n
= x
2
− n
x
2
− x
1
2(n +1)
=
nx
1
+(n +2)x

2
2(n +1)
.
Bˆay gi`o
.
, xe´t ha`m sˆo
´
f(x)=x
2
; x ∈ R.
Ta co´
f

(x)=2> 0; ∀x ∈ R.
Do d¯o´, theo D
-
i
.
nh lı´ 1.1.1, ta co´
7
Bˆa
´
td¯˘a

ng th´u
.
c 1.1.
x
2
1

+ x
2
2


(2n +1)x
1
+ x
2
2(n +1)

2
+

x
1
+(2n +1)x
2
2(n +1)

2


2nx
1
+2x
2
2(n +1)

2

+

2x
1
+2nx
2
2(n +1)

2
···


(n +2)x
1
+ nx
2
2(n +1)

2
+

nx
1
+(n +2)x
2
2(n +1)

2



x
1
+ x
2
2

2
; ∀x
1
,x
2
∈ R.
Tiˆe
´
ptu
.
c, nˆe
´
u xe´t ha`m sˆo
´
f(x)=
1
x
; x>0.
Ta co´
f

(x)=
2
x

3
> 0; ∀x>0.
Do d¯o´, theo D
-
i
.
nh lı´ 1.1.1, ta co´
Bˆa
´
td¯˘a

ng th´u
.
c 1.2.
1
x
1
+
1
x
2

2(n +1)
(2n +1)x
1
+ x
2
+
2(n +1)
x

1
+(2n +1)x
2

2(n +1)
2nx
1
+2x
2
+
2(n +1)
2x
1
+2nx
2
≥···

2(n +1)
(n +2)x
1
+ nx
2
+
2(n +1)
nx
1
+(n +2)x
2

4

x
1
+ x
2
; ∀x
1
,x
2
> 0,n≥ 1.
Bˆay gi`o
.
, xe´t ha`m sˆo
´
f(x)=

x; x>0.
Ta co´
f

(x)=−
1
4x

x
> 0; ∀x>0.
Do d¯o´, theo D
-
i
.
nh lı´ 1.1.1, ta co´

Bˆa
´
td¯˘a

ng th´u
.
c 1.3.

x
1
+

x
2


(2n +1)x
1
+ x
2
2(n +1)
+

x
1
+(2n + 1)3x
2
2(n +1)



2nx
1
+2x
2
2(n +1)
+

2x
1
+2nx
2
2(n +1)
 ···

(n +2)x
1
+ nx
2
2(n +1)
+

nx
1
+(n +2)x
2
2(n +1)


x
1

+ x
2
2
; ∀x
1
,x
2
> 0 n ≥ 1.
Tiˆe
´
ptu
.
c, nˆe
´
u xe´t ha`m sˆo
´
f(x)=
sinx
1+sinx
; x ∈ (0; π).
Ta co´
f

(x)=−
sinx +1+cos
2
x
(1 + sinx)
3
< 0; ∀x ∈ (0; π).

Do d¯o´, theo D
-
i
.
nh lı´ 1.1.1, ta co´
8
Bˆa
´
td¯˘a

ng th´u
.
c 1.4.
sinx
1
1+sinx
1
+
sinx
2
1+sinx
2

sin
(2n +1)x
1
+ x
2
2(n +1)
1+sin

(2n +1)x
1
+ x
2
2(n +1)
+
sin
x
1
+(2n +1)x
2
2(n +1)
1+sin
x
1
+(2n +1)x
2
2(n +1)
 ···

sin
(n +2)x
1
+ nx
2
2(n +1)
1+sin
(n +2)x
1
+ nx

2
2(n +1)
+
sin
nx
1
+(n +2)x
2
2(n +1)
1+sin
nx
1
+(n +2)x
2
2(n +1)
≤ 2
sin
x
1
+ x
2
2
1+sin
x
1
+ x
2
2
; ∀x
1

,x
2
∈ (0; π),n≥ 1
Bˆay gi`o
.
, tro
.

la
.
iv´o
.
iD
-
i
.
nh lı´ 1.1.1 va`D
-
i
.
nh lı´ 1.1.2. Co´ thˆe

ch´u
.
ng minh d¯u
.
o
.
.
c

r˘a
`
ng kˆe
´
t qua

(1.4) va` (1.5) vˆa
˜
nd¯u´ngnˆe
´
u thay (1.3) bo
.

imˆo
.
t gia

thiˆe
´
tma
.
nh ho
.
n.
Ta co´ ca´c kˆe
´
t qua

sau d¯ˆay:
D

-
i
.
nh ly´ 1.1.3. Gia

su
.

cho tru
.
´o
.
c ha`m sˆo
´
y = f(x) co´ f

(x) ≥ 0 (ha`m lˆo
`
i) trˆen
I(a; b) va` gia

su
.

x
1
,x
2
∈ I(a; b) v´o
.

i x
1
<x
2
. Khi d¯o´, v´o
.
imo
.
ida
˜
ysˆo
´
t˘ang dˆa
`
n {u
k
}
trong

x
1
;
x
1
+ x
2
2

:
x

1
= u
0
<u
1
<u
2
< < u
n
<
x
1
+ x
2
2
va` da
˜
ysˆo
´
gia

mdˆa
`
n {v
k
} trong

x
1
+ x

2
2
; x
2

:
x
1
+ x
2
2
<v
n
<v
n−1
< <v
1
<v
0
= x
2
sao cho
x
1
+ x
2
= u
0
+ v
0

≥ u
1
+ v
1
≥···≥u
n
+ v
n
, (1.6)
ta d¯ˆe
`
uco´
f(u
0
)+f(v
0
) ≥ f (u
1
)+f(v
1
) ≥···≥f(u
n
)+f(v
n
).
No´i ca´ch kha´c: Da
˜
y

f(u

j
)+f(v
j
)

, j =0, 1, ··· ,n, la` mˆo
.
tda
˜
y gia

m.
Ch´u
.
ng minh. V´o
.
imˆo
˜
i j ∈{0, 1, ··· ,n},t`u
.
ca´c gia

thiˆe
´
t, ta co´
u
j
<u
j+1
<

u
j+1
+ v
j+1
2

u
0
+ v
0
2
=
x
1
+ x
2
2
<v
j+1
<v
j
.
9
Bˆay gi`o
.
,v´o
.
imˆo
˜
i j ∈{0, 1, , n},d¯˘a

.
t



u
j+1
−u
j
= 
j+1
v
j
− v
j+1
= δ
j+1
.
Thˆe
´
thı`
0 <
j+1
 δ
j+1
; ∀j ∈{0, 1, , n}.
Bˆay gi`o
.
,v´o
.

imˆo
˜
i j ∈{0, 1, , n}, theo D
-
i
.
nh lı´ Lagrange, ta co´
f(u
j+1
) −f (u
j
)=f

(c
j+1
)(u
j+1
− u
j
)=f

(c
j+1
)
j+1
,v´o
.
i c
j+1
∈ (u

j
; u
j+1
);
f(v
j
) −f (v
j+1
)=f

(d
j+1
)(v
j
−v
j+1
)=f

(d
j+1

j+1
,v´o
.
i d
j+1
∈ (v
j+1
; v
j

).
Ho
.
nn˜u
.
a, vı` c
j+1
<d
j+1
; ∀j ∈{0, 1, , n} va` f

(x) ≥ 0, nˆen ta co´
f

(c
j+1
)  f

(d
j+1
); ∀j ∈{0, 1, , n}.
Do d¯o´, ta co´
f(u
j+1
) −f (u
j
)  f (v
j
) −f (v
j+1

); ∀j ∈{0, 1, , n},
hay
f(u
j
)+f(v
j
) ≥ f(u
j+1
)+f(v
j+1
); ∀j ∈{0, 1, , n}.
Ta co´ d¯iˆe
`
u pha

ich´u
.
ng minh.
Tu
.
o
.
ng tu
.
.
, ta co´
D
-
i
.

nh ly´ 1.1.4. Gia

su
.

cho tru
.
´o
.
c ha`m sˆo
´
y = f(x) co´ f

(x)  0 (ha`m lo
˜
m) trˆen
I(a; b) va` gia

su
.

x
1
,x
2
∈ I(a; b) v´o
.
i x
1
<x

2
. Khi d¯o´, v´o
.
imo
.
ida
˜
ysˆo
´
t˘ang dˆa
`
n {u
k
}
trong

x
1
;
x
1
+ x
2
2

:
x
1
= u
0

<u
1
<u
2
< ···<u
n
<
x
1
+ x
2
2
va` da
˜
ysˆo
´
gia

mdˆa
`
n {v
k
} trong

x
1
+ x
2
2
; x

2

:
x
1
+ x
2
2
<v
n
<v
n−1
< ···<v
1
<v
0
= x
2
sao cho
x
1
+ x
2
= u
0
+ v
0
≥ u
1
+ v

1
≥···≥u
n
+ v
n
,
ta d¯ˆe
`
uco´
f(u
0
)+f(v
0
)  f (u
1
)+f(v
1
)  ··· f(u
n
)+f(v
n
).
No´i ca´ch kha´c: Da
˜
y

f(u
j
)+f(v
j

)

, j =0, 1, ··· ,n, la` mˆo
.
tda
˜
y t˘ang.
10
Bˆay gi`o
.
,v´o
.
i hai sˆo
´
thu
.
.
c cho tru
.
´o
.
c x
1
<x
2
, hı`nh a

nh cu

a ca´c d¯iˆe


m u
j
va` v
j
lˆa
`
n
lu
.
o
.
.
t ”tiˆe
´
nchˆa
.
mdˆa
`
nd¯ˆe
`
u” vˆe
`
trung d¯iˆe

mcu

a d¯oa
.
n[x

1
x
2
]la`
x
1
+ x
2
2
trˆen tru
.
csˆo
´
giu´p ta xˆay du
.
.
ng d¯u
.
o
.
.
c hai da
˜
y {u
k
} va` {v
k
} thoa

ma

˜
nnh˜u
.
ng d¯iˆe
`
ukiˆe
.
ncu

aD
-
i
.
nh
lı´ 1.1.3 va`D
-
i
.
nh lı´ 1.1.4 nhu
.
sau:
Vı´ du
.
1.2.
u
0
= x
1
,u
1

= x
1
+
x
2
− x
1
2
2
, ,
u
n
= x
1
+
x
2
− x
1
2
2
+ ···+
x
2
− x
1
2
n+1
=
(2

n+1
− 2
n
+1)x
1
+(2
n
−1)x
2
2
n+1
;
v
0
= x
2
,v
1
= x
2

x
2
− x
1
2
2
, ··· ,
v
n

= x
2

x
2
− x
1
2
2
−···−
x
2
− x
1
2
n+1
=
(2
n
− 1)x
1
+(2
n+1
− 2
n
+1)x
2
2
n+1
.

Ngoa`i ra, co´ thˆe

phˆo
´
iho
.
.
p ca´c ca´ch ta
.
oda
˜
ynhu
.
trˆen, ta thu d¯u
.
o
.
.
cca´cc˘a
.
pda
˜
y
{u
k
} va` {v
k
} thoa

ma

˜
nnh˜u
.
ng d¯iˆe
`
ukiˆe
.
ncu

aD
-
i
.
nh lı´ 1.1.3 va`D
-
i
.
nh lı´ 1.1.4, ch˘a

ng
ha
.
n:
Vı´ du
.
1.3.
u
0
= x
1

,u
1
= x
1
+
x
2
− x
1
2(n +1)

x
2
− x
1
2
2
(n +1)
, ··· ,
u
n
= x
1
+ n
x
2
− x
1
2(n +1)



x
2
−x
1
2
2
(n +1)
+
x
2
− x
1
2
3
(n +1)
+ ···+
x
2
− x
1
2
n+1
(n +1)

=

(n + 1)2
n+1
−(n −1)2

n
−1

x
1
+

(n −1)2
n
+1

x
2
(n + 1)2
n+1
;
v
0
= x
2
,v
1
= x
2

x
2
− x
1
2(n +1)

, ··· ,v
n
= x
2
− n
x
2
− x
1
2(n +1)
=
nx
1
+(n +2)x
2
2(n +1)
.
Cuˆo
´
i cu`ng, v´o
.
iviˆe
.
ccho
.
n ca´c ha`m sˆo
´
y = f(x)co´f

(x) ≥ 0 ho˘a

.
c f

(x)  0
trˆen I(a; b), ta se
˜
thu d¯u
.
o
.
.
c kha´ nhiˆe
`
u vı´ du
.
phong phu´.
D
-
ˆo
´
iv´o
.
i ca´c ha`m sˆo
´
lˆo
`
i ho˘a
.
clo
˜

m, ngoa`i ca´c d¯i
.
nh lı´ nˆeu trˆen, ca´c da
.
ng cu

aBˆa
´
t
d¯ ˘a

ng th´u
.
c Karamata co`n cho ta nh˜u
.
ng phu
.
o
.
ng pha´p la`m ch˘a
.
tbˆa
´
td¯˘a

ng th´u
.
crˆa
´
t

hiˆe
.
u qua

. Sau d¯ˆay la` ca´c kˆe
´
t qua

cˆo

d¯ i ˆe

n, d¯a
˜
d¯ u
.
o
.
.
c trı`nh ba`y trong [1], ma` ta co´
thˆe

mˆo ta

thˆong qua mˆo
.
tsˆo
´
vı´ du
.

.
11
1.2 Bˆa
´
td¯˘a

ng th´u
.
c Karamata
D
-
i
.
nh ly´ 1.2.1. (Bˆa
´
td¯˘a

ng th´u
.
c Karamata)
Cho ha`m sˆo
´
y = f (x) co´d¯a
.
o ha`m cˆa
´
p hai ta
.
imo
.

i x ∈ (a; b) sao cho f

(x) > 0
v´o
.
imo
.
i x ∈ (a; b).
Gia

su
.

a
1
,a
2
, ··· ,a
n
va` x
1
,x
2
, ··· ,x
n
la` ca´c sˆo
´
thuˆo
.
c [a;b], thoa


ma
˜
nd¯iˆe
`
ukiˆe
.
n
x
1
≥ x
2
≥···≥x
n
,
a
1
≥ a
2
≥···≥a
n
va`




















x
1
≥ a
1
x
1
+ x
2
≥ a
1
+ a
2

x
1
+ x
2
+ + x
n−1

≥ a
1
+ a
2
+ + a
n−1
x
1
+ x
2
+ + x
n
= a
1
+ a
2
+ + a
n
Khi d¯o´, ta luˆon co´
n

k=1
f(x
k
) ≥
n

k=1
f(a
k

).
Nhˆa
.
nxe´tr˘a
`
ng, ca´c gia

thiˆe
´
tcu

a hai da
˜
y {x
k
} va` {a
k
} la` kha´ nhiˆe
`
u. V´o
.
inh˜u
.
ng
kiˆe
´
nth´u
.
cco
.

ba

nvˆe
`
d¯ a
.
isˆo
´
tuyˆe
´
n tı´nh, ta co´ thˆe

ch´u
.
ng minh kˆe
´
t qua

sau d¯ˆay
D
-
i
.
nh ly´ 1.2.2. (I.Schur)
D
-
iˆe
`
ukiˆe
.

ncˆa
`
n va` d¯u

d¯ ˆe

hai bˆo
.
da
˜
ysˆo
´
d¯ o
.
nd¯iˆe
.
u gia

m {x
k
,a
k
; k =1, 2, ···,n},
thoa

ma
˜
nca´c d¯iˆe
`
ukiˆe

.
n



















x
1
≥ a
1
x
1
+ x
2
≥ a

1
+ a
2

x
1
+ x
2
+ ···+ x
n−1
≥ a
1
+ a
2
+ ···+ a
n−1
x
1
+ x
2
+ ···+ x
n
= a
1
+ a
2
+ ···+ a
n
la` gi˜u
.

a chu´ng co´mˆo
.
t phe´p biˆe
´
nd¯ˆo

i tuyˆe
´
n tı´nh da
.
ng
a
i
=
n

j=1
t
ij
x
j
; i =1, 2, ··· ,n,
12
trong d¯o´
t
kl
≥ 0,
n

j=1

t
kj
=1,
n

j=1
t
jl
=1;k,l =1, 2, ···,n.
Co´ thˆe

mˆo ta

ma trˆa
.
n(t
ij
) qua mˆo
.
t vı´ du
.
sau d¯ˆay:
Vı´ du
.
1.4. Xe´t da
˜
ysˆo
´
khˆong ˆam bˆa
´

tky`α
1

2
, ··· ,α
n
co´tˆo

ng b˘a
`
ng α>0.
V´o
.
imˆo
˜
i i =1, 2, ···,n, ta d¯˘a
.
t
α
i
α
= a
i
Thˆe
´
thı` ma trˆa
.
n (a
ij
); i, j =1, 2, ···,n,co´thˆe


xa´c d¯i
.
nh nhu
.
sau
a
ij
= a
i+j−1
; nˆe
´
u i + j  n +1
a
ij
= a
i+j−n−1
; nˆe
´
u i + j>n+1.
Vı´ du
.
1.5. Gia

su
.


1
, 

2
, 
3
la` 3 sˆo
´
du
.
o
.
ng co´tˆo

ng b˘a
`
ng 1. Cho
.
n k thoa

ma
˜
n
0  k  min{
1

1
(1 −
1
)
;
1


2
(1 − 
2
)
;
1

3
(1 − 
3
)
}.
Thˆe
´
thı` ma trˆa
.
n (a
ij
); i, j =1, 2, ···,n,co´thˆe

xa´c d¯i
.
nh nhu
.
sau
a
ij
= k
2
i

−k
i
+1 ;nˆe
´
u i = j
a
ij
= k
i

j
;nˆe
´
u i = j.
Tu
.
o
.
ng tu
.
.
D
-
i
.
nh lı´ 1.2.5, ta co´
D
-
i
.

nh ly´ 1.2.3. Cho ha`m sˆo
´
y = f(x) co´d¯a
.
o ha`m cˆa
´
p hai ta
.
imo
.
i x ∈ (a; b)
sao cho f

(x) < 0 v´o
.
imo
.
i x ∈ (a; b).
Gia

su
.

a
1
,a
2
, ··· ,a
n
va` x

1
,x
2
, ··· ,x
n
la` ca´c sˆo
´
thuˆo
.
c [a;b], thoa

ma
˜
nd¯iˆe
`
ukiˆe
.
n
x
1
 x
2
 ··· x
n
,
a
1
 a
2
 ··· a

n
va`



















x
1
 a
1
x
1
+ x
2
 a

1
+ a
2

x
1
+ x
2
+ ···+ x
n−1
 a
1
+ a
2
+ ···+ a
n−1
x
1
+ x
2
+ ···+ x
n
= a
1
+ a
2
+ ···+ a
n
Khi d¯o´, ta luˆon co´
n


k=1
f(x
k
) 
n

k=1
f(a
k
).
13
Tuy nhiˆen, khi gia

thiˆe
´
t cuˆo
´
i cu`ng
x
1
+ x
2
+ ···+ x
n
= a
1
+ a
2
+ ···+ a

n
trong D
-
i
.
nh lı´ 1.2.1 va`D
-
i
.
nh lı´ 1.2.2 bi
.
pha´ v˜o
.
,cˆa
`
n pha

i co´ nh˜u
.
ng kˆe
´
t qua

ma
.
nh ho
.
n
d¯ ˆe


thay thˆe
´
. Ta co´ hai kˆe
´
t qua

sau d¯ˆay
D
-
i
.
nh ly´ 1.2.4. Cho ha`m sˆo
´
y = f(x) co´d¯a
.
o ha`m cˆa
´
p hai ta
.
imo
.
i x ∈ (a; b)
sao cho f

(x) ≥ 0 v´o
.
imo
.
i x ∈ [a; b] va` f


(x) > 0 v´o
.
imo
.
i x ∈ (a; b).
Gia

su
.

a
1
,a
2
, ··· ,a
n
va` x
1
,x
2
, ··· ,x
n
la` ca´c sˆo
´
thuˆo
.
c [a;b], d¯ˆo
`
ng th`o
.

i thoa

ma
˜
n
ca´cd¯iˆe
`
ukiˆe
.
n
a
1
≥ a
2
≥···≥a
n
,
x
1
≥ x
2
≥···≥x
n
va`














x
1
≥ a
1
x
1
+ x
2
≥ a
1
+ a
2

x
1
+ x
2
+ ···+ x
n
≥ a
1
+ a
2

+ ···+ a
n
Khi d¯o´, ta luˆon co´
n

k=1
f(x
k
) ≥
n

k=1
f(a
k
).
D
-
i
.
nh ly´ 1.2.5. Cho ha`m sˆo
´
y = f(x) co´d¯a
.
o ha`m cˆa
´
p hai ta
.
imo
.
i x ∈ (a; b)

sao cho f

(x) ≥ 0 v´o
.
imo
.
i x ∈ [a; b] va` f

(x) < 0 v´o
.
imo
.
i x ∈ (a; b).
Gia

su
.

a
1
,a
2
, ··· ,a
n
va` x
1
,x
2
, ··· ,x
n

la` ca´c sˆo
´
thuˆo
.
c [a;b], d¯ˆo
`
ng th`o
.
i thoa

ma
˜
n
ca´cd¯iˆe
`
ukiˆe
.
n
a
1
 a
2
 ··· a
n
,
x
1
 x
2
 ··· x

n
va`













x
1
 a
1
x
1
+ x
2
 a
1
+ a
2

x
1

+ x
2
+ ···+ x
n
 a
1
+ a
2
+ ···+ a
n
Khi d¯o´, ta luˆon co´
n

k=1
f(x
k
) 
n

k=1
f(a
k
).
14
Ta thˆa
´
yr˘a
`
ng, d¯ˆo
´

iv´o
.
i ca´c da
.
ng cu

abˆa
´
td¯˘a

ng th´u
.
c Karamata, viˆe
.
c tı`m ra ca´c
c˘a
.
pda
˜
y {a
k
} va` {x
k
} thoa

ma
˜
nd¯iˆe
`
ukiˆe

.
ncu

ad¯i
.
nh lı´ la` rˆa
´
t quan tro
.
ng. Sau d¯ˆay
la` mˆo
.
tsˆo
´
vı´ du
.
vˆe
`
viˆe
.
c xˆay du
.
.
ng ca´c da
˜
y na`y.
Vı´ du
.
1.6. Gia


su
.

cho tru
.
´o
.
cda
˜
ysˆo
´
gia

m
x
1
≥ x
2
≥ ≥ x
n
.
Khi d¯o´, luˆon tˆo
`
nta
.
ida
˜
ysˆo
´
khˆong ˆam α

1

2
, ··· ,α
n−1
sao cho
x
1
− α
1
≥ x
2
+ α
1
− α
2
≥···≥x
n−1
+ α
n−2
−α
n−1
≥ x
n
+ α
n−1
.
Thˆa
.
tvˆa

.
y, ta chı

cˆa
`
ncho
.
nda
˜
y α
1

2
, ··· ,α
n−1
nhu
.
sau
0  α
1

x
1
− x
2
2
, 0  α
2

x

2
− x
3
2
, ··· , 0  α
n−1

x
n−1
−x
n
2
.
Ch˘a

ng ha
.
n, xe´t da
˜
ysˆo
´
{x
n
},v´o
.
i x
n
= −n, n =1, 2, ···
Khi d¯o´, ta co´ x
1

≥ x
2
≥···≥x
n
.
Ngoa`i ra, v´o
.
imo
.
i n ≥ 2, ta co´
x
n−1
− x
n
2
=
1
2
.
Vˆa
.
y, nˆe
´
ucho
.
nda
˜
ysˆo
´
α

1

2
, ··· ,α
n−1
, trong d¯o´
α
n
=
1
2n
; n ≥ 2
thı` ta co´
0 <α
n

1
2
; ∀n ≥ 2
va`
α
n−2
− α
n−1
=
1
2(n − 2)(n −1)
; ∀n ≥ 3.
Thˆe
´

thı`, ta co´
x
1
− α
1
≥ x
2
+ α
1
− α
2
≥···≥x
n−1
+ α
n−2
−α
n−1
≥ x
n
+ α
n−1
.
Bˆay gi`o
.
, xe´t ha`m lˆo
`
i f(x)=x
2
; x ∈ R.Thˆe
´

thı`, theo nhˆa
.
n xe´t trˆen, ta co´ kˆe
´
t
qua

sau d¯ˆay
Bˆa
´
td¯˘a

ng th´u
.
c 1.5.
x
2
1
+ x
2
2
+ ···+ x
2
n


x
1

1

2

2
+

x
2
+
1
4

2
+ ···
+

x
n−1
+
1
2(n −2)(n −1)

2
+

x
n
+
1
2(n − 1)


2
v´o
.
imo
.
isˆo
´
thu
.
.
c x
1
,x
2
, ··· ,x
n
.
15
Vı´ du
.
1.7. Gia

su
.

a
1
,a
2
, ··· ,a

n
la` ca´c sˆo
´
thu
.
.
cdu
.
o
.
ng.
Ta xe´t bˆo
.
b =

b
1
,b
2
, ··· ,b
n

la` bˆo
.
hoa´n vi
.
cu

ada
˜

y

lna
1
,lna
2
, ··· ,lna
n

xˆe
´
p theo th´u
.
tu
.
.
gia

mdˆa
`
n. V´o
.
imˆo
˜
i i ∈{1, ···,n},co´thˆe

coi b
i
= lna
k

i
,v´o
.
i

k
1
,k
2
, ··· ,k
n

la` hoa´n vi
.
na`o d¯o´cu

a (1, 2, ··· ,n).
Bˆay gi`o
.
, ta la
.
i xe´t bˆo
.
c =

c
1
,c
2
, ··· ,c

n

la` bˆo
.
hoa´n vi
.
cu

ada
˜
y

ln
a
2
1
a
2
,ln
a
2
2
a
3
, ··· ,ln
a
2
n−1
a
n

,ln
a
2
n
a
1

xˆe
´
p theo th´u
.
tu
.
.
gia

mdˆa
`
n. V´o
.
imˆo
˜
i i ∈{1, ···,n},co´thˆe

coi c
i
= ln
a
2
k

i
a
k
i
+1
,v´o
.
i

k
1
,k
2
, ··· ,k
n

la` hoa´n vi
.
na`o d¯o´cu

a (1, 2, ··· ,n).
Dˆe
˜
da`ng kiˆe

m tra d¯u
.
o
.
.

cr˘a
`
ng c˘a
.
pda
˜
y {c
k
} va` {b
k
} thoa

ma
˜
nd¯iˆe
`
ukiˆe
.
ncu

aD
-
i
.
nh
lı´ 1.2.1.
Bˆay gi`o
.
, xe´t ha`m lˆo
`

i f(x) =ln

1+e
x

, x ∈ R.Thˆe
´
thı`, theo D
-
i
.
nh lı´ 1.2.1, ta
co´
Bˆa
´
td¯˘a

ng th´u
.
c 1.6.

1+a
1

1+a
2

···

1+a

n



1+
a
2
1
a
2

1+
a
2
2
a
3

···

1+
a
2
n
a
1

v´o
.
imo

.
isˆo
´
thu
.
.
cdu
.
o
.
ng a
1
,a
2
, ··· ,a
n
.
Hˆe
.
qua

1.2.1.

1+a
1

1+a
2

···


1+a
n



1+
a
2
1
a
2

1+
a
2
2
a
3

···

1+
a
2
n
a
1




1+
a
4
1
a
3
a
4
2

1+
a
4
2
a
4
a
4
3

···

1+
a
4
n
a
2
a

4
1

 ···
v´o
.
imo
.
isˆo
´
thu
.
.
cdu
.
o
.
ng a
1
,a
2
, ··· ,a
n
.
Ta thˆa
´
yr˘a
`
ng, v´o
.

ic˘a
.
pda
˜
y {c
k
} va` {b
k
} trˆen, nˆe
´
ucho
.
n ha`m sˆo
´
phu` ho
.
.
p, ta
se
˜
thu d¯u
.
o
.
.
c nhiˆe
`
ubˆa
´
td¯˘a


ng th´u
.
c kha´c. Ch˘a

ng ha
.
n, xe´t ha`m lˆo
`
i f(x)=

1+e
x
,
x ∈ R, ta d¯u
.
o
.
.
c
Bˆa
´
td¯˘a

ng th´u
.
c 1.7.

1+a
1

+

1+a
2
+ ···+

1+a
n


1+
a
2
1
a
2
+

1+
a
2
2
a
3
+ ···+

1+
a
2
n

a
1
,
v´o
.
imo
.
isˆo
´
thu
.
.
cdu
.
o
.
ng a
1
,a
2
, ··· ,a
n
.
16
Hˆe
.
qua

1.2.2.


1+a
1
+

1+a
2
+ ···+

1+a
n


1+
a
2
1
a
2
+

1+
a
2
2
a
3
+ ···+

1+
a

2
n
a
1


1+
a
4
1
a
3
a
4
2
+

1+
a
4
2
a
4
a
4
3
+ ···+

1+
a

4
n
a
2
a
4
1
 ···
v´o
.
imo
.
isˆo
´
thu
.
.
cdu
.
o
.
ng a
1
,a
2
, ··· ,a
n
.
Vı´ du
.

1.8. Tru
.
´o
.
chˆe
´
t, ta co´ nhˆa
.
n xe´t r˘a
`
ng: Nˆe
´
u hai da
˜
ysˆo
´
{x
k
,y
k

I(a; b); k =1, 2, ···,n} thoa

ma
˜
nca´c d¯iˆe
`
ukiˆe
.
n

x
1
≥ x
2
≥···≥x
n
,
y
1
≥ y
2
≥···≥y
n
va`
x
i
x
j

y
i
y
j
; ∀i<j,
thı` chu´ng thoa

ma
˜
nd¯iˆe
`

ukiˆe
.
ncu

aD
-
i
.
nh lı´ 1.2.1.
Ch´u
.
ng minh. Thˆa
.
tvˆa
.
y, xe´t hai bˆo
.
sˆo
´
(x
1
,x
2
, ··· ,x
n
)va`(y
1
,y
2
, ··· ,y

n
).
V´o
.
i i lˆa
`
nlu
.
o
.
.
tb˘a
`
ng 1, 2, ··· ,n, ta co´
x
i
x
1

y
i
y
1
.
Cˆo
.
ng ca´c bˆa
´
td¯˘a


ng th´u
.
c na`y theo vˆe
´
, ta co´
x
1
+ x
2
+ ···+ x
n
x
1

y
1
+ y
2
+ ···+ y
n
y
1
.
Suy ra
x
1
≥ y
1
.
Bˆay gi`o

.
,tiˆe
´
ptu
.
c xe´t hai bˆo
.
sˆo
´
(x
1
+ x
2
,x
3
, ··· ,x
n
)va`(y
1
+ y
2
,y
3
, ··· ,y
n
).
Ch´u
.
ng minh tu
.

o
.
ng tu
.
.
, ta co´
x
1
+ x
2
≥ y
1
+ y
2
.
Tiˆe
´
ptu
.
c qua´ trı`nh tu
.
o
.
ng tu
.
.
, ta co´
x
1
+ x

2
+ ···+ x
n−1
≥ y
1
+ y
2
+ ···+ y
n−1
17
Nhu
.
vˆa
.
y, cu`ng v´o
.
inh˜u
.
ng gia

thiˆe
´
t ban d¯ˆa
`
u, nhˆa
.
n xe´t trˆen d¯a
˜
d¯ u
.

o
.
.
c kh˘a

ng d¯i
.
nh.
Bˆay gi`o
.
,xe´ta
1
,a
2
, , a
n
la` ca´c sˆo
´
thu
.
.
cdu
.
o
.
ng. V´o
.
imˆo
˜
i i ∈{1, , n}, ta d¯˘a

.
t
y
i
=
a
i
a
1
+ a
2
+ ···+ a
n
,
x
i
=
a
2
i
a
2
1
+ a
2
2
+ ···+ a
2
n
.

Khi d¯o´
x
1
+ x
2
+ ···+ x
n
= y
1
+ y
2
+ ···+ y
n
=1.
Khˆong mˆa
´
t tı´nh tˆo

ng qua´t, gia

su
.

a
1
≥ a
2
≥ ≥ a
n
. Khi d¯o´

x
1
≥ x
2
≥···≥x
n
,
y
1
≥ y
2
≥···≥y
n
.
Ngoa`i ra, v´o
.
imo
.
i i ≥ j, ta co´
x
i
x
j
=
a
2
i
a
2
j


a
i
a
j
=
y
i
y
j
.
Nhu
.
vˆa
.
y, theo nhˆa
.
n xe´t trˆen, c˘a
.
pda
˜
ysˆo
´
{x
k
} va` {y
k
} thoa

ma

˜
nd¯iˆe
`
ukiˆe
.
ncu

a
D
-
i
.
nh lı´ 1.2.1 va`dod¯o´, v´o
.
i ha`m sˆo
´
lˆo
`
i
f(x)=
x
1 −x
; x>0,
ta co´ kˆe
´
t qua

sau
Bˆa
´

td¯˘a

ng th´u
.
c 1.8.
a
1
a
2
+ a
3
+ ···+ a
n
+ +
a
n
a
1
+ a
2
+ ···+ a
n−1

a
2
1
a
2
2
+ a

2
3
+ ···+ a
2
n
+···+
a
2
n
a
2
1
+ a
2
2
+ ···+ a
2
n−1
,
v´o
.
imo
.
isˆo
´
thu
.
.
cdu
.

o
.
ng a
1
,a
2
, ··· ,a
n
.
Hˆe
.
qua

1.2.3.
a
1
a
2
+ a
3
+ ···+ a
n
+···+
a
n
a
1
+ a
2
+ ···+ a

n−1

a
2
1
a
2
2
+ a
2
3
+ ···+ a
2
n
+ +
a
2
n
a
2
1
+ a
2
2
+ + a
2
n−1

a
4

1
a
4
2
+ a
4
3
+ ···+ a
4
n
+ ···+
a
4
n
a
4
1
+ a
4
2
+ ···+ a
4
n−1
 ···,
v´o
.
imo
.
isˆo
´

thu
.
.
cdu
.
o
.
ng a
1
,a
2
, ··· ,a
n
.
18
Lu
.
u y´: Ngu
.
`o
.
i ta d¯a
˜
ch´u
.
ng minh d¯u
.
o
.
.

cr˘a
`
ng, ca´c kˆe
´
t qua

cu

aD
-
i
.
nh lı´ 1.2.1
va`D
-
i
.
nh lı´ 1.2.4 vˆa
˜
n d¯u´ng ma` khˆong cˆa
`
nd¯ˆe
´
n gia

thiˆe
´
t
x
1

≥ x
2
≥···≥x
n
.
D
-
iˆe
`
u na`y cu
˜
ng tu
.
o
.
ng tu
.
.
d¯ ˆo
´
iv´o
.
i gia

thiˆe
´
t
x
1
 x

2
 ··· x
n
trong ca´c D
-
i
.
nh lı´ 1.2.3 va`D
-
i
.
nh lı´ 1.2.5.
Khi d¯o´, ta quy u
.
´o
.
cgo
.
i ca´c d¯i
.
nh lı´ tu
.
o
.
ng tu
.
.
lˆa
`
nlu

.
o
.
.
tla`D
-
i
.
nh lı´ 1.2.1a, D
-
i
.
nh lı´
1.2.3a, D
-
i
.
nh lı´ 1.2.4a va`D
-
i
.
nh lı´ 1.2.5a.
Ngoa`i ra, trong [1] cu
˜
ng d¯a
˜
trı`nh ba`y mˆo
.
tsˆo
´

kˆe
´
t qua

vˆe
`
ca´c da
.
ng D
-
i
.
nh lı´
Karamata mo
.

rˆo
.
ng ma` ba
.
nd¯o
.
c co´ thˆe

tham kha

o.
Ho
.
nn˜u

.
a, kha´ nhiˆe
`
ukˆe
´
t qua

vˆe
`
d¯ ˆo
.
gˆa
`
nd¯ˆe
`
uva`th´u
.
tu
.
.
s˘a
´
pd¯u
.
o
.
.
ccu

amˆo

.
tda
˜
y
ca´c tam gia´c cu
˜
ng d¯a
˜
d¯ u
.
o
.
.
cd¯ˆe
`
cˆa
.
p trong [1]. D
-
ˆay chı´nh la` mˆo
.
tphu
.
o
.
ng pha´p kha´
h˜u
.
uhiˆe
.

ud¯ˆe

la`m ch˘a
.
t ca´c bˆa
´
td¯˘a

ng th´u
.
clu
.
o
.
.
ng gia´c cu

a tam gia´c. Vı´ du
.
sau d¯ˆay
se
˜
cho ta mˆo
.
t minh hoa
.
d¯ o
.
n gia


nvˆe
`
vˆa
´
nd¯ˆe
`
na`y.
Vı´ du
.
1.9. Xe´t tam gia´c ABC. Khˆong mˆa
´
t tı´nh tˆo

ng qua´t, co´thˆe

gia

su
.

A ≥ B ≥ C.
D
-
˘a
.
t A

=2A − B, B

=2B − C, C


=2C − A.
Ro
˜
ra`ng A

> 0 va` B

> 0. Do d¯o´, nˆe
´
u thˆem gia

thiˆe
´
t C

> 0 (t´u
.
cla`A<2C),
thı` A

, B

, C

cu
˜
ng la` 3 go´c cu

amˆo

.
t tam gia´c. Ho
.
nn˜u
.
a, ta co´
A

≥ A
A

+ B

≥ A + B
A

+ B

+ C

= A + B + C
Do d¯o´, hai bˆo
.
da
˜
ysˆo
´
{A, B, C} va` {A

,B


,C

} thoa

ma
˜
nca´c d¯iˆe
`
ukiˆe
.
ncu

a
D
-
i
.
nh lı´ 1.2.1a.
Bˆay gi`o
.
,nˆe
´
u xe´t ha`m sˆo
´
lˆo
`
i f (x)=sinx; x ∈ (0; π), thı` ta co´ kˆe
´
t qua


sau
Bˆa
´
td¯˘a

ng th´u
.
c 1.9. Gia

su
.

tam gia´c ABC co´ go´c l´o
.
n nhˆa
´
t nho

ho
.
n hai lˆa
`
n
go´c nho

nhˆa
´
t. Thˆe
´

thı`, ta co´
sin(2A − B)+sin(2B − C)+sin(2C − A) ≥ sinA + sinB + sinC.
Phˆa
`
n na`y se
˜
d¯ u
.
o
.
.
c khe´p la
.
iv´o
.
iviˆe
.
c gi´o
.
i thiˆe
.
umˆo
.
tsˆo
´
ha`m lˆo
`
i, lo
˜
md¯ˆe


ba
.
nd¯o
.
c
co´ thˆe

a´p du
.
ng.
19
1.3 Gi´o
.
i thiˆe
.
umˆo
.
tsˆo
´
h`am lˆo
`
iv`ah`am l˜om
1.3.1 Mˆo
.
tsˆo
´
ha`m lˆo
`
i

f(x)=x
α
; x>0, α<0.
f(x)=(S − x)
α
; S>0, x ∈ (0; S), α<0.
f(x)=

x +
1
x

α
; x>0, α>1.
f(x)=
x
2
x + S
; S>0, x ≥ 0.
f(x)=
1
x
2
; x>0.
f(x)=
x
S − x
; S>0, x ∈ (0; S).
f(x)=e
αx

; x ∈ R, α>0.
f(x)=
1
1+e
αx
; x ≥ 0, α ≥ 1.
f(x)=

e
x
+
1
e
x

2
; x ∈ R.
f(x) =ln

1+
1
x

; x>0.
f(x) =ln

1+e
αx

; x ∈ R, α>0.

f(x) =ln

e
x
+
1
e
x

; x ∈ R.
f(x) =sin
k
x ; x ∈ (0; π), k<0.
f(x) =ln

1+
1
sin
2
x

; x ∈ (0; π).
f(x) =cos
k
x ; x ∈

0;
π
2


, k<0.
f(x) =tan
k
x ; x ∈

0;
π
2

, k ≥ 1.
f(x) =arcsinx ; x ∈ (0; 1).
f(x) =arctan(e
x
);x<0.
f(x) =arctan
x
S − x
; S>0, x ∈ (0; S).
1.3.2 Mˆo
.
tsˆo
´
ha`m lo
˜
m
f(x)=x
α
; x>0, 0 <α<1.
f(x)=(S − x)
α

; S>0, x ∈ (0; S), 0 <α<1.
f(x) =lnx
1/n
; x>1, n =1, 2,
f(x) =ln

e
αx
− 1

; x>0, α>0.
f(x) =lnx ; x>0.
f(x) =sin
k
x ; x ∈ (0; π), k ∈ (0; 1].
f(x) =ln(sinx);x ∈ (0; π ).
20
f(x)=
sinx
1+sinx
; x ∈ (0; π).
f(x) =cos
k
x ; x ∈

0;
π
2

, k ∈ (0; 1].

f(x) =ln(1- cosx);x ∈ (0; π).
f(x) =cosx tanx ; x ∈

0;
π
2

.
f(x) =arcsin
x
1+x
; x ≥ 0.
f(x) =arcsin
x
S + x
; S>0, x>0.
f(x) =arctanx ; x>0.
1.4 Ba`i tˆa
.
p
Ba`i tˆa
.
p 1.1. Cho a, b, c > 0.Ch´u
.
ng minh r˘a
`
ng v´o
.
i α>0,taco´:


a + b
2

α
+

b + c
2

α
+

c + a
2

α


2a + b
3

α
+

2b + c
3

α
+


2c + a
3

α


2(4a + b)
9

α
+

2(4b + c)
9

α
+

2(4c + a)
9

α
≤···
Hu
.
´o
.
ng dˆa
˜
n: Xe´t ha`m sˆo

´
f(x)=x
α
; x>0
Ba`i tˆa
.
p 1.2. Cho a ≥ b ≥ c ≥ 0. Ch´u
.
ng minh r˘a
`
ng
1
a
+
4
b
+
9
c

1

a + b
2

+
4

c + a
2


+
9

b + c
2


1

2a + b + c
4

+
4

a +2b + c
4

+
9

a + b +2c
4

≥···
Hu
.
´o
.

ng dˆa
˜
n: Xe´t ha`m sˆo
´
f(x)=
1
x
; x>0
Ba`i tˆa
.
p 1.3. Cho a, b, c khˆong ˆam.Ch´u
.
ng minh r˘a
`
ng v´o
.
i α ≥
1
2
,taco´:

1+a
2

α
+

1+b
2


α
+

1+c
2

α


1+ab

α
+

1+bc

α
+

1+ca

α
≥ 3

3+
3

a
2
b

2
c
2

α
.
Hu
.
´o
.
ng dˆa
˜
n: Xe´t ha`m sˆo
´
f(x)=

1+x
2
; x ≥ 0
21
Ba`i tˆa
.
p 1.4. Cho a, b, c ∈ (0; 1).Ch´u
.
ng minh r˘a
`
ng:

1 − a +


1 −b +

1 − c ≤

1 −

ab +

1 −

bc +

1 −

ca


1 −
4

ab
2
c +

1 −
4

bc
2
a +


1 −
4

ca
2
b ≤···
Hu
.
´o
.
ng dˆa
˜
n: Xe´t ha`m sˆo
´
f(x)=

1 −x; x ∈ (0; 1)
Ba`i tˆa
.
p 1.5. Cho a, b, c ∈ (0; 1).Ch´u
.
ng minh r˘a
`
ng

1 −a +

1 − b +


1 −c ≤

1 −

ab +

1 −

bc +

1 −

ca


a + b − 2ab
a + b
+

b + c − 2bc
b + c
+

c + a −2ca
c + a
Hu
.
´o
.
ng dˆa

˜
n: Xe´t ha`m sˆo
´
f(x)=

1 − x ; x ∈ (0; 1)
Ba`i tˆa
.
p 1.6. Cho a ≥ b ≥ c ≥ 0.Ch´u
.
ng minh r˘a
`
ng:
a +2

b +3
3

c ≥

ab +2
4

ca +3
6

ca

4


a
2
bc +2
8

ab
2
c +3
12

abc
2
≥···
Hu
.
´o
.
ng dˆa
˜
n: Xe´t ha´m sˆo
´
f(x)=e
x
; x ∈ R
Ba`i tˆa
.
p 1.7. Cho a, b, c > 0.Ch´u
.
ng minh r˘a
`

ng
a + b + c ≤
3

a
2
b +
3

b
2
c +
3

c
2
a

9

a
4
b
4
c +
9

b
4
c

4
a +
9

c
4
a
4
b ≥···
Hu
.
´o
.
ng dˆa
˜
n: Xe´t ha`m sˆo
´
f(x)=e
x
; x ∈ R
Ba`i tˆa
.
p 1.8. Cho a, b, c > 0.Ch´u
.
ng minh r˘a
`
ng v´o
.
i α, β > 0 va` α + β =1,taco´:
a + b + c ≥ a

α
b
β
+ b
α
c
β
+ c
α
a
β
≥ a
α
2
b
2αβ
c
β
2
+ b
α
2
c
2αβ
a
β
2
+ c
α
2

a
2αβ
b
β
2
≥···
Hu
.
´o
.
ng dˆa
˜
n: Xe´t ha`m sˆo
´
f(x)=e
x
,x∈ R
Ba`i tˆa
.
p 1.9. Choa, b, c du
.
o
.
ng. Ch´u
.
ng minh r˘a
`
ng

ab +


bc +

ca ≤
3

a
2
b +
3

b
2
c +
3

c
2
a

9

a
8
b
2
+
9

b

8
c
2
+
9

c
8
a
2
≤···
Hu
.
´o
.
ng dˆa
˜
n: Xe´t ha`m sˆo
´
f(x)=e
x
,x∈ R
22
Ba`i tˆa
.
p 1.10. Cho a, b, c khˆong nho

ho
.
n 1. Ch´u

.
ng minh r˘a
`
ng:
a
1+a
+
b
1+b
+
c
1+c

3

a
2
b
1+
3

a
2
b
+
3

b
2
c

1+
3

b
2
c
+
3

c
2
a
1+
3

c
2
a

9

a
4
b
4
c
1+
9

a

4
b
4
c
+
9

b
4
c
4
a
1+
9

b
4
c
4
a
+
9

c
4
a
4
b
1+
9


c
4
a
4
b
≤···.
Hu
.
´o
.
ng dˆa
˜
n: Xe´t ha`m sˆo
´
f(x)=
e
x
1+e
x
; x ≥ 0
Ba`i tˆa
.
p 1.11. Cho a, b, c khˆong nho

ho
.
n 1. Ch´u
.
ng minh r˘a

`
ng v´o
.
i α, β > 0 va`
α + β =1, ta co´:
a
1+a
+
b
1+b
+
c
1+c

a
α
b
β
1+a
α
b
β
+
b
α
c
β
1+b
α
c

β
+
c
α
a
β
1+c
α
a
β

a
α
2
.b
2αβ
.c
β
2
1+a
α
2
.b
2αβ
.c
β
2
+
b
α

2
.c
2αβ
.a
β
2
1+b
α
2
.c
2αβ
.a
β
2
+
c
α
2
.a
2αβ
.b
β
2
1+c
α
2
.a
2αβ
.b
β

2
≤···
Hu
.
´o
.
ng dˆa
˜
n: Xe´t ha`m sˆo
´
f(x)=
e
x
1+e
x
; x ≥ 0
Ba`i tˆa
.
p 1.12. Cho a, b, c l´o
.
nho
.
n 1. Ch´u
.
ng minh r˘a
`
ng
a
a − 1
+

b
b − 1
+
c
c −1

3

a
2
b
3

a
2
b − 1
+
3

b
2
c
3

b
2
c − 1
+
3


c
2
a
3

c
2
a − 1

9

a
4
b
4
c
9

a
4
b
4
c − 1
+
9

b
4
c
4

a
9

b
4
c
4
a − 1
+
9

c
4
a
4
b
9

c
4
a
4
b − 1
≥···.
Hu
.
´o
.
ng dˆa
˜

n: Xe´t ha`m sˆo
´
f(x)=
e
x
e
x
− 1
; x>0
Ba`i tˆa
.
p 1.13. Cho a, b, c l´o
.
nho
.
n 1. Ch´u
.
ng minh r˘a
`
ng v´o
.
i α, β > 0 va` α + β =1,
ta co´:
a
a − 1
+
b
b − 1
+
c

c − 1

a
α
b
β
a
α
b
β
− 1
+
b
α
c
β
b
α
c
β
−1
+
c
α
a
β
c
α
a
β

− 1

a
α
2
.b
2αβ
.c
β
2
a
α
2
.b
2αβ
.c
β
2
− 1
+
b
α
2
.c
2αβ
.a
β
2
b
α

2
.c
2αβ
.a
β
2
− 1
+
c
α
2
.a
2αβ
.b
β
2
c
α
2
.a
2αβ
.b
β
2
− 1
≥···
Hu
.
´o
.

ng dˆa
˜
n: Xe´t ha`m sˆo
´
f(x)=
e
x
e
x
− 1
; x>0
Ba`i tˆa
.
p 1.14. Cho a, b, c ∈ (0; 1).Ch´u
.
ng minh r˘a
`
ng:
(1 − a)(1 − b)(1 − c) ≤

1 −

ab

1 −

bc

1 −


ca



1 −
4

ab
2
c

1 −
4

bc
2
a

1 −
4

ca
2
b

≤···
23
Hu
.
´o

.
ng dˆa
˜
n: Xe´t ha`m sˆo
´
f(x)=ln(1 −x);x ∈ (0; 1)
Ba`i tˆa
.
p 1.15. Cho a, b, c ∈ (0; 1).Ch´u
.
ng minh r˘a
`
ng:
(1 − a)(1 − b)(1 − c) ≤

1 −

ab

1 −

bc

1 −

ca



a + b −2ab

a + b

b + c − 2bc
b + c

c + a − 2ca
c + a

Hu
.
´o
.
ng dˆa
˜
n: Xe´t ha`m sˆo
´
f(x)=ln(1 −x);x ∈ (0; 1)
24
Chu
.
o
.
ng 2
Phu
.
o
.
ng ph´ap lu
.
.

acho
.
n tham sˆo
´
Tru
.
´o
.
chˆe
´
t ta xe´t ba`i toa´n sau
Ch´u
.
ng minh r˘a
`
ng nˆe
´
u a, b, c la` 3 sˆo
´
khˆong ˆam co´tˆo

ng b˘a
`
ng 3, thı` ta co´

a +

b +

c ≥ ab + bc + ca

L`o
.
i gia

i cho ba`i toa´n na`y khˆong qua´ kho´.
Ta co´
2(ab + bc + ca)=(a + b + c)
2
−(a
2
+ b
2
c
2
).
Do d¯o´ ta cˆa
`
nch´u
.
ng minh r˘a
`
ng
a
2
+ b
2
+ c
2
+2(


a +

b +

c) ≥ 9
Su
.

du
.
ng bˆa
´
td¯˘a

ng th´u
.
cgi˜u
.
a trung bı`nh cˆo
.
ng va` trung bı`nh nhˆan (thu
.
`o
.
ng go
.
ila`
bˆa
´
td¯˘a


ng th´u
.
c AM-GM) cho 3 sˆo
´
, ta co´
a
2
+

a +

a ≥ 3a
b
2
+

b +

b ≥ 3b
c
2
+

c +

c ≥ 3c
Cˆo
.
ng ca´c vˆe

´
cu

aca´cbˆa
´
td¯˘a

ng th´u
.
c trˆen, ta d¯u
.
o
.
.
cd¯iˆe
`
ucˆa
`
nch´u
.
ng minh.
Nhˆa
.
n xe´t r˘a
`
ng, bˆa
´
td¯˘a

ng th´u

.
c trˆen co´ thˆe

viˆe
´
tla
.
idu
.
´o
.
ida
.
ng
a
1
2
+ b
1
2
+ c
1
2
≥ ab + bc + ca
Nhu
.
vˆa
.
y, v´o
.

i k ≥
1
2
thı` bˆa
´
td¯˘a

ng th´u
.
c sau d¯ˆay luˆon d¯u´ng
a
k
+ b
k
+ c
k
≥ ab + bc + ca.

×