các bất đẳng thức tích phân thuộc loại Ostrowski và các áp dụng của nó 6_2

TranK..:1Q

-{if],;, r!rf,'fI flutf'. fir/' p/'rl}/, !off; fi.fflo,~l;

PH{) L{)C I

HAM LIEN TUC TUYET DOl
.
.
Tinh cha't lien t\,ICtuy~t d6i c':ia mQt ham xac dinh tren mQt do<;ln
dil du'<,iC dl,Jng trong su6t lu~n van nay.
Sl(
Dinh nghia 1.1: MQt ham

f: [a,b] --+IR c1tfgQi la lien tl;lc tuy~t c1{;'i

tren ra,b] ne-u:
'\1[;> 0,38>

"
0: I:1.l(p;);=1

.l(a;)1 < £

voi mQi n va mQi hQ cac khoang roi nhau (app,),...,(a",p,,) trong [a,b] c6
"

t6ng cac c1Qdai

I:(p; 1=1

a; ) < 8

.

-Hi~n nhien mQt ham lien tl;}C
tuy~t d6i tren [a,b] thllien

tl;}c(don gian ta

la'y n =1).
Dinh Iv 1.1: Giel su /: [a,b] --+lR lien tl,lc va khong giam. Khi d6 hai
di~u ki~n sau la tu'ong dtfong:
i) f: [a,b] --+IR lien tl,lCtuy~t d6i.
ii) f kha vi hfiu he-t tren [a,b], f'E L1
([a,bD, va
x

/(x)-

lea) = J/(t)dl,(a
a

S x s b)

Chung minh Dinh 19 1.1 c6 th~ fim tha'ytrong [12,W. Rudin, p.146-147].
Dinh IS'1.2: Gia su

/: [a,b]--+IR lien tl;}C
tuy~t d6i. Dinh nghla

N

F(x) = Slip I:IIU;);=1

.lU;-l)I,(a s x S b)

trong d6 SUpla'y tren ta't ca N Va Hit ca cach chQn {l;}sao cho:
a = 10 < II < -.. < IN = X.

Khi d6 cac ha m F, F+f, F-fla khong giam va lien tl;}C
tuy~t d6i tren
[a,b].
Ch" thich 1.1: Ne-u F(b) < 00, tIll ta n6if co bien phan bi chqn tren [a,bj.
Gia trj F(h) dlf
[a,b].

Chang minh Dinh 19 1.2 c6 th~ fim tha'y trong [12, W. Rudin, p.148].


Trang 37

-mal dJII(j flute fir/' ;'/'((11 foa; (l/tJ6(Jf,:)ki

Dinh IV 1.3: Gicl stY f: [a,b] ~ IR lien t\le tuy~t do"j, khi d6 fkhcl vi hftu
he"t tren fa,h], f'E LI([a,bD,

va

x

(1.1)

f(x)-f(a)

= aJl(t)dt, (as:xS:b}

Chung minh: GQi F la bie"n phan loan phftn euaf, theo dinh ly 1.2, ta d~t
1
1
1; = 2(F +f), f2 = 2(F - I).
Ap d\lI1g soy dlin i) => ii) eua dinh ly 1.1 eho fj, h va VI f=frh ta suy ra
(1.1).
Dinh lv 1.4: Gicl stY f: [a,b~~ IR khcl vi t(,limQi x E[a,b]va f'E L1aa,bD,
khid6
x

(1.2)

/(x)-/(a)

= Jl(t)dt,
a

(as:xS:b).

Chang minh Dtnh ly I.4 co th~ Om tha"ytrong [12, W. Rudin, p.149-150].

w

mril d~I-(/"t(;

Trang 38

(fro/l,/!/uJn foat" f~jt(o,t)ti

~

PHI) L I) C II
"

K.2

VE BA T DANG THUC OSTROWSKI
Trong mQt bai baa [9] (Comment. Math. Helv. 10 (1938), p. 226227) A. Ostrowski aa chung minh mOt bat a~ng thuc sail day:
(ILl)

h

~
b-a

lex) -

(X-~)2

fI(I)dl

~+

4

~ (b - a) supII'(I)1

a

a
(b-a)

2

J

l

vai mQi X E [a,h],

trong do

l: [a,b ]

'

2

IR co d?o ham tren (a,b) va

~

f': (a, b) ~ JR bj ch~n trong (a,b), tuc la

Ilf'IL = asuplf'(t)\ < +00 va h~ng s6

~ la t6t nhflt thea nghIa ding kh6ng th~ thay the' no b~ng mQt s6 nho
hdn.

Phac ho(,lcluIng minh cua Ostrmvski [9 J.
h

Chu

y rhng

1

ff(x)d\: = (b - a)flea + (b - a)l)dl.
0

(I

f)~t

h(t) =f(a+(b-a)t)
1

I

Ta co

=f(x),

fh(l)dl

0

tE[O,I].

h

= -b-a

va h'(t)

fI(x)d\:

= (b-a)f'(a+(b-a)t).

a

V~y (ILl) tucfng du'(jng vdi
I

(II.2)

1

her) - fh(s)ds

1

~ - + (t - _)2

[4

0

suplh' (1)1,\it E [0,1}

2 ] 0<1<1

i) Bat d~ng thuc (IL2) hi~n nhien dung ne'u Ilh'IL 0<1<1
= suplh'(t)1= O.

ii) Giii sa

Ilh'll", suplh'(I)1 ;/:o.
=

Bang cach thay h bdi

0<1<1

sa ding

~

Ilh't

ta co th~ giii

Ilht = 1.Ta chI din chung minh bat d~ng thlj'c'
I

(11.3)

her)-

fh(s)ds ~ ~+ (t - ~)2, vai
4
2

0

iii) D~t

1

get) = her)- fh(s)ds,
0

\if E [O,I}

suplh'(t)1= 1,\it E [0,1}
0<1<1


-:1],;1 r/,fl1//jhf{f'

(fr!t./lhrlu

Trang)

(oa; (!:!.(;O(rtJ!:;

9

I

ta co

ff;(S)ds = 0, suplg'(t)1 = 1.

0

(kl<1

V~y ta se chung minh bfft d~ng thuc
(II A)

1
Ig(t)1 ::; (t

-

1

2)2 + 4' \It

E [0,11

voi mqi ham g: [O,I]~ IR thaa cac di~u ki~n
I

(II.S)

fg(s)ds = 0, Ig'(t)! ::; I, \It E (0,1),
()

va hhng s6

~~

xufft hi~n trang (IIA) la t6t nhfft thea nghla r~ng khong the§;

thay the' no bhng m0t s6 nho hall.

* Chung minh (11.4).
Cho t,sE[0,11taco

Ig(t)-g(s)I=lg'(c)(t-s)I::;lt-sl.

V~y
(11.6)

g(t) -It -sl::; g(s)::; get) +Ir -051, \It,s E [0,1}

Tich phan bfft c1~ng thuc cua (11.6) thea s E [0,11 ta duQc
J

g(t)

-

J

]t

-

J

slds ::; fg(s)ds = o ::;get) + ]t

()

()

1

0

1

flt-slcts'=

f(t-s)ds+

f(s-t)ds

()

hay

1

0

Ig(t)l::;

=(1--)

V~y (11.4) dung.

1

1

2

+-

2

4

* H~ng sf) ~ xuflt hi~n trang (II A) 13 tf)t nhfft.
Gia slt C E IR thoa bfft c1~ng thuc:
(11.7)

Ig(t)j::; C +(t

-~J,

\It

E

[0,1]

-

slds


IfJrl( rlrfJ~'1-_ffl-,!~("II, Alld~

Tran~~9

!r)(/[f{J(/O(,:}ki

I

t£1co

= 0,

fr;(s)ds
0

suplg'(t)1 = 1.

(kl<1

V~y t£1se chilng minh bi{t d~ng thilc
(11.4)

1
Ig(t)I~(t-2Y+4'

1
VtE[a,11

voi mqi ham g: [a,l] ~ IR thaa cac di~u ki~n
1

(II.S)

fr;(s)ds = a, Ig'(t)1 ~ 1, Vt E (a,I),
()

vil h~ng s6

.~

xui{t hi~n trang (11.4)lil t6t nhi{t thea nghTa r~ng khong thc§

thay the' no b~ng mQt s6 nh6 hall.
* Chung minh (11.4).

Cho t,sE[0,11t£1co

Ig(t)-g(s)I=lg'(c)(t-s)I~lt-sl.

V~y
(11.6)

g(t)-I/-sl~g(s)~g(/)+It-sl,

VI,S E [a,l}

Tich phan bi{t d~ng th(fc cua (11.6) the a
I

g(t) -

]1 -

0

0

V~y (II.4) d\Jng.

2

= a ~ g(t)

I

Ig(t)1 ~ Jlt

1

]

slds ~ fg(s)ds'

1

=(t--)

[a,11 ta du<;Jc

I

()

hay

S E

2

+

]t -

0

I

- slds = J(t - s)ds + I(s -/)d~
0

+-

1

4

* I-I~ng s6 ~ xui{t hi~n trang (rIA) lil t6t nhi{t.

Gia su C E lR th6a bi{t d~ng thilc:
(11.7)

Ig(t)I~C+(t-~J,

VtE[a,l]

slds


Trang 40

3],;1 rfrf1l/l-111ft,.Iff-/' /t/'rill (Off; fJ.ibo((:}J.i

voi mQi ham g: [0,1]---).R thoi! cac di~u ki~n (11.5). Ta se chung minh
I
~

rangC~-.

1

4

Th~t v~y, chQn ham g(t) = t -~, ta co g tho a cac di~u ki~n (11.5)va co
It - ~I ,;

c + (t- ~ J,

'if

t

E

[0,1]

hay
2

max t -

(11.8)

Oslsl

(l

~
2

I

-

(

~

t-

2) J

= Os,sl/2
max

(.'I -

.'12) ~ s C.
=

4

Th~t fa, dtfa vao c15ngthuc (11.3), bftt c15ngthuc Ostrowski (ILl) co th~
chung minh kha don gian bon so vdi [9] nht( sau
1
f(x)--

h

h-a

a

1

ff(t)dt

x

h

=- b--a f(t-a)f'(t)dt+

f(t-b)f'(t)dt

0

X

I

s- b -a
=

x

[

h

f(t-a)f'(t)dt+

f(t-b)f'(t)dt

0

x

]

suplf'(t)1
o
~

(x - a)2 + (x - b)2
sup!f'(t)1
b- a[
2
] o.
(x

= ~+
4

a+b2
(b-a)

)

22

(b-a)suplf'(t)l,
o
[

VxE[a,b],

1

* Nghi~m 1<;1
i hhng sef ~ xu5t hi~n trang (ILl) 13 testnhftt.

.

Gii! SU- C

JR thoa bftt dAng thuc

E

(II.9)

I

lex) - -

h

b-aa

voi illQi X E [a,b],

ff(t)dt

trongdo

I
S - +
4

(

x--

2

a+b

2 )

(b-a)

f: [a,b] ~ IR

2

(b - a) osup If'

co dC;lOam tren (a,b) va
h

1': (a,6) ---). bi ch~n trong (a,b). Ta se chung minh r~ng C ~ ~.
IR


'{1M? (/rim/llutf'

Trang 41

1ft-/' j,/,rfJl lorn' (MirlftJl;

Th~t v~y, chQn ham f(x) = x- a;h,

(11.10)

x-I

a+b
2

,(x-ll;~r

c+

~

I

va thay vao (11.9), ta thu du<;1C

2

(h- a)

l(b-a),'vfxE[a,b].

V~y

I

C~ sup x-2

a+bl

a5,x5,h b-a

-

(

x-~

a+b)

(h-a)

= sup (t-t2)=!,.
095,1/2

(II

4


Trang 42

Mal d;h~? (lute (fellfllriJ/ (oai fM;o(tiki

PUT) LT)C III
'J'\.".2

"

..

BAT DANG THUC GRUSS
Nam 1935 (xem[7], Math. Z., 39,(1935), p.215- 226), GRUSS da
chung minh b§t d~ng thuc sail day:
/

?

Dinh Iy 111.1: (Bat dang thuc Gruss).
ClIo f,R: [a,h]~ IR khd ({ch sao clIo:
l11r

::; f(x)::;

Mr,l11J.!::; g(x)::; Mg,Vx
.

Khi eM'

(IlL

I~Jf(x)g(X)dX-~Jf(X)dx~Jg(X)dx
b-a"
b-a

I)

H

-

,..,2

I

'

?

kl

,(//1fILla, uang t 111e
xay ra
va M .r =M g =1.
Chung minh:

E>~t

](1)

= lex) = f(a

.
E [a,b].

b-aa

l

::;~(M
4

.

-n1 )(M,-m).
f

=

R(X) = sign ( x-2

~

) ' VOl

1
n1f = n1g=-

+ (b - a)/), 0::; t ::;1,a::; x::; b

-

1 h
ff(t)dt = fJ(x)dx
b-a a
0

ta co,

1

.

Do do ta co th~ gia sli'r~ng a = 0, b =1. Khi d6 (111.1) vie't lC;li
I

(III.2)

"

Jf(x)g(x)dx
0

-

1

Jf(x)dxJ g(x)dx ::;_ (Mf - n1f XMg - n1J.!
1
4
0
0
I

: [0,1] ~ 1R kha tfch thoa:

trong d6, f,g

l11/::;/(x)::;M/,

mg::;g(x)::;Mg,VXE[0,1]

Hon nG'a, (IIL2) xay ra d~ng thuc khi
1
l(x)=g(x)

= sign ( x-2' )

/'

VOl

".

n1f =mg =-1

va

Mf =Mg ::'::1.

Trude he't ta din b6 d~ sau.

RG d~ 111.1:Gia sli' l,g
ta c6:
(III.3)

,
I

~

E L2 =

L2(0,1),

"

I

'

2 ~

f(x)g(x)dx -! f(x)c[r! g(x)dx::;[ !f2 (x)dx - ( ! f(x)dx ) ] Ilglll}'
I

g

,.

a+b

u lex)

f


$,;1 mill//- 111ft£" pltfill
Iklt

I

(IlIA)
I

I

Jf(x)g(x)dx

0

Chung minh B6 d~ 111.1:
Ta co I
I
(IlLS)

I

ff(x)g(x)dx

0

Trang 43

lord (Jd(;(J(t)k;

- ff(x)dx
0

~

[ Af(X)

I

- 0 f(x)d:J
I

I

I

fg(x)d:< =

0

- !f(X)dX

I

R

0

J

0

Jr

I

= [ ff2(X)d'C

lex) - ff(x)d:< g(x)d:<

(fg'(X)dx

)~
I

I

2

2

I

- 2 ff(X)d:< ff(X)dX + ( ff(X)d:< )

]

IlgIIL'

I

= [If'(X)dx-(jf(X)dx HI,g""
Ba"tdftng thuc (Ill.3) dt(Qc chung minh.Ba't dftng thuc (IlIA) Ut«;1c
chung minh nhC1
vao bfit dftng thuc:
~

[jf'(X)dx-(!f(X)dx n

I

s[1f'(X)dX]' ~llfll"

B6 d~ sau day cho ta b§t dftng th((c III.2, tuc la dinh 19 III.! duQc chung
minh.B6 d~ 111.2:

Gia suf, gEL 00= L00(0,1),

thoa:

nIJ::;; f(x)::;; MJ,nI~ ::;;g(x)::;; M~,\;jx E [0,1].

Khi do :
(IIL6)

I

I

I

0

0

0

1
fj(x)g(x)d:< - fj(x)d:< fg(x)d:< ::;; (M J - nIJ)(Mg - nIg).
I

4

Chung minh B6 d~ 111.2:Voi c, d E JR, trong ba"tdftng thlic (IlIA) thay
f, g baif -c, g -d, ta co:
(III. 7)

Ihf(x) 0

c ][g(x) - d]dx - hf(x)
0

v~ tnii cua (III.7) la:
(111.8)

VI'

- c ]d:0

I

I

= Hj(x)-c][g(x)-d]d:<0

Hj(x)-c]d:
I

I

1

0

1

= IIf(x)g(x)d:< - I f(x)d:(I g(x)dx
0

0

I

0

0


Trang 44

. mal d(/J~r fluff' ffrl!Ji.ll(iJi frl(/('fJ:}6()(~k;

Chli

Y

rhng, III-ell", ~llf -eIIL~' o do suy tll (III.7), (IIL8) rhng:
d
I

(III. 9)

1

ff(x)~(x)dx-

0

I

fI(x)dx

Ve,d E IR

fg(x)dxl ~lIf -4.~llg-dll/.~'

0

0

Chon,

c = ~(M f + 111} d = ~(M I: + 111 ta thu dudc:
}
2
J
2
I!
'
= ~ M f - 111r = ~(M f - 111r)

III - ell ~ M I - ~(M f + m1)
I,
.
2
2
2
.'
TltOng ttf, ta (1uQc:
I
jig- d c ~ -2 ( M I: - 111I! )
,

'0

I

1

.

1

.

1

ll

.

Cu6i clIng,
(III.I0)

I

I

I

1

f I(x)g(x)dx - f f(x)dxf 0 g(x)d~ ~ -(M
0
0
4

XMg -l11g 1

f -l11f

I

V~y b6 de (JIL2) dl(QCcl1lIng minh,lI
B6 de sail day cho ta ba"tdfing thuc (4,3). (Dinh Iy 4.2, chuang IV).
B6 d~ 111.3:
Giil St( fEl},~Er' thoa
Khi d6
(111.11)

I

I

I

fI(x)g(x)dx
0

I11I! g(x)~MI:,VxE[O,l].
~
I

I

- fI(x)dx

fg(x)dx

0

~ ~(Mg - I11g) ff2(X)dx 2
[0

0

Chung minh Be) d~ 111.3:
Trang ba"td~ng thuc (lII.3) thay g bdi g-d, voi dEIR ta co:
(III. 12)

II
ff(x)[g(x)-d]dx-

I
ff(x)dr

0

0

s

I
f[g(x)-d]dx

0

[1 ['(x)<1<

-(1

f(x)dx

I

)}Ig - d1!"

ma ve"tnii cua (IIL12) la:
I

(III. 13)

VT = Iff(x)[g(x)

I'
-

d]dx - ff(x)dx

0

I

= I fJ(x)g(x)dx
0

I

0
I

~g(x) - d]dx
0

I

- fJ(x)dx fg(x)dxl.
0

0

ChQn d=~(MI! +1111!)3 thu dl(QC VPcua

t
2

(III. 12) la

I

( 0ff(x)dx

2

J ]

~
,


Trang 45

$fil dtfl'{f /lute licit jtlui4t kat @oAouMi

1

(III. 14 )

rr jig-dilL'

Vp= [!f2(X)dx-(Jf(X)dX

1

J Ilg-4..

~ [fJ2(X)dx-(fJ(X)dx)'

1

~[Jf2(X)dx-(Jf(X)dx)']'

~(Mg

-mg)
1

JT.

= ~ (Mg -mg{ fi2(X)dx-(fi(x)dx

V~y bftt d~ng thuc (III.It) duQc SUr tit (III. 12) - (III. 14).Chu thich 111.1: f)~ng thuc tfong (III.2) Kay fa khi:
(III.I5)

.

". M

l

.

)

I

Th~tv~y

!(x)=g(x)=slgnx-2",vOl
(

I

f=

g=

va

I

0

I

0

I

I

f/(x)dx

0

=

fg(x)dx

0

,-

f/(X)dx

(0

)

=
v€ tnii cua (III.2) Hi:
I

VT = If/(x)g(x)dx
0

vfi phai cua (III.2) vdiMf
(III.I8)

-

I

= f/(x)dx + f/(x)dx.

Ii

[0

2

f(-I)dx+

(0

I

2

Ii

2

Ii

(III. 17)

mf=mg=-l.

f/(x)g(x)dx = f/2(X)dx = fIdx = 1,

0

(III.I6)

1 ,

M

JIdx

Ii J

=0.

I

f/(x)dx fg(x)dxl = 1.

0

=M~ =1 va

0

m/ =mg=-1

1
1
VP= 4(Mf -mf )(Mg -mg) =4(1 + 1)(1 1)= 1.
+

v~y VT=VP va do d6 (III.2) Kay fa d~ng thuc.-

la

J


£lJdl dil~'!I tluJ:(; wit jtluin /au:
t

Trang 46

(!J)tiOf4Id

Chu thich III.~:
Theo Chu thich (III. 1) thl hang sf) ~trong bfft dAng thuc (IIL2) la
4
tf)t nhfft. Th~t v~y, ne"ubfft dAng thuc (UL2) dung vdi hang sf) C thay cho
~

""

1
4

h ang so -:

1

(IlL 19)

1

fl(x)g(x)dx

0

vdi ffiQi ham f, g : [0,1]

1

- fJ(x)dx fg(x)dxl
0

0

5.

C(M j - mj )(M

g

- mg)

IR khii tich thoa:

~

mj
Ta chQn lex) = g(x) = Sign(x - ~). vdi

Mj

= Mg = 1 va mj = mg

=-1.

Khi d6 (IlL 19) tra thanh
1

(IIl.20)

I

1

fJ(x)g(x)dx - fJ(x)dx fg(x)dxl = 15.C(M j -

0

0

0

mj

)(M g - mg)

= 4C

hay 15. 4C hay C ~ 1/4.
V~y hang sf) C = 1/4 trong (IIl.2) la tf)t nhfft.8

Chu thich III.3:
BAng thuc trong (III. 11) xiiy ra khi:
(IIl.2l)

l(x)=g(X)=sign(x-~).

vdi Mj =Mg=1 va mf =mg=-1.

Hon nlla, hang sf) 1/ 2 trong bfft dAngthuc (IlLll) la tf)t nhfft.8

w