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
)δ
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
,α
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
).
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