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-{if],;, r!rf,'fI flutf' fir/' p/'rl}/, !off; fi.fflo,~l; TranK :1Q

PH{) L{)C I

Tinh cha't lien t\,ICtuy~t d6i c':ia mQt ham xac dinh tren mQt do<;ln dil du'<,iCSl( dl,Jng trong su6t lu~n van nay

Dinh nghia 1.1: MQt ham f: [a,b] +IR c1tf<jcgQi la lien tl;lc tuy~t c1{;'i

tren ra,b] ne-u:

"

'\1[;> 0,38> 0: I:1.l(p;)- .l(a;)1 < £

;=1

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

t6ng cac c1Qdai

"

I:(p;- a;)< 8

1=1

-Hi~n nhien mQt ham lien tl;}Ctuy~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 S x s b)

a

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

N

F(x) =Slip I:IIU;)- lU;-l)I,(a s x S b)

;=1

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;}Ctuy~t d6i tren [a,b].

Gia trj F(h) dlf<jcgQi la bien phan ,toempluln cuaftren [a,b].

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

-mal dJII(j flute fir/' ;'/'((11 foa; (l/tJ6(Jf,:)ki Trang 37

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

(1.1)

x

f(x)-f(a) = Jl(t)dt, (as:xS:b}

a Chung minh: GQi F la bie"n phan loan phftn euaf, theo dinh ly 1.2, ta d~t

1; = 2(F +f), f2 = 2(F- I).

Ap d\lI1g soy dlin

(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

(1.2)

x /(x)-/(a) = Jl(t)dt, (as:xS:b).

a

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

w

mril d~I-(/"t(; (fro/l,/!/uJn foat" f~jt(o,t)ti Trang 38

PHI) L I) C II

VE BA T DANG THUC OSTROWSKI

Trong mQt bai baa [9] (Comment Math Helv 10 (1938), p

226-227) A Ostrowski aa chung minh mOt bat a~ng thuc sail day:

h

l

(X-~)2

J

lex) - ~ fI(I)dl ~ (b - a) supII'(I)1 ~+ 2 2 '

b-a a a<l<h 4 (b-a)

(ILl)

f': (a, b) ~ JR bj ch~n trong (a,b), tuc la Ilf'IL = suplf'(t)\ < +00 va h~ng s6

a<l<h

hdn

Phac ho(,l cluIng minh cua Ostrmvski [9 J.

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

f)~t h(t) =f(a+(b-a)t) =f(x), tE[O,I]

fh(l)dl = b-a fI(x)d\:

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

I

[

]

her)- fh(s)ds ~ - + (t - _)2 suplh' (1)1,\it E [0,1}

0 4 2 0<1<1

(II.2)

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

ii) Giii sa Ilh'll",= suplh'(I)1 ;/:o Bang cach thay h bdi ~ ta co th~ giii

I

her)- fh(s)ds ~~+ (t - ~)2, vai suplh'(t)1= 1,\it E[0,1}

0

1

get) = her) - fh(s)ds, \if E [O,I}

(11.3)

iii) D~t

-:1],;1 r/,fl1//jhf{f' (fr!t./lhrlu (oa; (!:!.(;O(rtJ!:; Trang) 9

ta co

I

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

V~y ta se chung minh bfft d~ng thuc

Ig(t)1 ::; (t - 2)2 + 4' \It E [0,11

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

(II.S)

I

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

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

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

hay

Ig(t)l::; flt-slcts'= f(t-s)ds+ f(s-t)ds

1 2 1

=(1 )

* H~ng sf) ~ xuflt hi~n trang (II A) 13 tf)t nhfft

Ig(t)j::; C +(t -~J, \It E [0,1]

(11.7)

IfJrl( rlrfJ~'1-_ffl-,!~("II, Alld~ !r)(/[f{J(/O(,:}ki Tran~~9

t£1co

I fr;(s)ds =0, suplg'(t)1 = 1.

0 (kl<1

Ig(t)I~(t-2Y+4' VtE[a,11

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

(II.S)

1

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

()

thay the' no b~ng mQt s6 nh6 hall

* Chung minh (11.4).

V~y

g(t) - ]1- slds ~ fg(s)ds' = a~ g(t) + ]t- slds

hay

Ig(t)1 ~ Jlt- slds=J(t - s)ds + I(s -/)d~

=(t )

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

Ig(t)I~C+(t-~J, VtE[a,l]

(11.7)

3],;1 rfrf1l/l-111ft,.Iff-/' /t/'rill (Off; fJ.ibo((:}J.i Trang 40

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

~ 1

rangC~-

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

(l

t - ~

I

-(t- ~ )

2

J

= max (.'I- .'12)=~s C.

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

f(x) h-a ff(t)dt =-b a f(t-a)f'(t)dt+ f(t-b)f'(t)dt

I

[

]

s-b -a f(t-a)f'(t)dt+ f(t-b)f'(t)dt o<l<hsuplf'(t)1

[

(x - a)2 + (x - b)2

] sup!f'(t)1

[

1

= ~+ 22 (b-a)suplf'(t)l, VxE[a,b],

4 (b-a) o<l<h

(II.9)

. Gii! SU- C E JR thoa bftt dAng thuc

2

a+b

lex) - - ff(t)dt S - + 2

b-aa 4 (b-a) (b - a) sup If' (t)l,

o<l<h

voi illQi X E [a,b], trong do f: [a,b] ~ IR co dC;lO ham tren (a,b) va

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

'{1M? (/rim/llutf' 1ft-/' j,/,rfJl lorn' (MirlftJl; Trang 41

I

x I

~ c+ 2 l(b-a),'vfxE[a,b].

V~y

a5,x5,h b-a (h-a) 095,1/2 4

(II

Mal d;h~? (lute (fellfllriJ/ (oai fM;o(tiki Trang 42

PUT) LT)C III

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 E [a,b].

(IlL I) I~Jf(x)g(X)dX-~Jf(X)dx~Jg(X)dx

l

::;~(M -n1 )(M,-m).

kl .

(

a+b

)

,.

1

,(//1fILla, uang t 111exay ra u lex) = R(X)=sign x-2 ' VOl n1f =

n1g=-va M.r =M =1.g

Chung minh:

E>~t ](1) =lex) =f(a + (b - a)/), 0::; t ::;1,a::; x::; b

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

0 a

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

(III.2)

I "

I

1

Jf(x)g(x)dx - Jf(x)dxJ g(x)dx ::;_4(Mf - n1f XMg - n1J.!1

trong d6, f,g : [0,1] ~ 1R kha tfch thoa:

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 E L2 = L2(0,1),

ta c6:

I

I

[

I

(

'

)

2

]

~

$,;1 mill//- 111ft£" Iklt pltfill lord (Jd(;(J(t)k; Trang 43

I

I

Jf(x)g(x)dx - J f(x)d:<J g(X)d:< ::;;Ilflll} Ilgll/."

Chung minh B6 d~ 111.1:

Ta co

(IlIA)

R

I

J

(IlLS) Iff(x)g(x)dx - ff(x)dx fg(x)d:< = lex) - ff(x)d:< g(x)d:<

~ [Af(X) - !f(X)dX Jr(fg'(X)dx )~

I

[

(

I

)

2

]

2

= ff2(X)d'C - 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«;1cchung minh nhC1vao bfit dftng thuc:

[jf'(X)dx-(!f(X)dx n 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, gEL00= L00(0,1), thoa:

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

Khi do :

(IIL6)

I

1

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

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 - c ]d:<J[g(x) - d]dxl::;; III - cllL,llg- dill,"0

v~ tnii cua (III.7) la:

I

(111.8) VI' =Hj(x)-c][g(x)-d]d:<- Hj(x)-c]d:<Hg(x)-d]d:<

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

mal d(/J~r fluff' ffrl!Ji.ll(iJi frl(/('fJ:}6()(~k; Trang 44

(III 9)

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

Ve,d E IR

Chon, c = ~(M + 111} d = ~(M + 111} ta thu dudc:

III - ell '0 ~

1

M I - ~(M f + m1)

1

I

M f - 111r1 = ~(M f - 111 r)

TltOng ttf, ta (1uQc:

I jig- dll ~ - ( M - 111)

Cu6i clIng,

(III.I0)

I

1

f I(x)g(x)dx - f f(x)dxf g(x)d~ ~ -(M f -l11f XMg -l11g 1

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:

Khi d6

I

[

I

(

I

J

2

]

~

(111.11) fI(x)g(x)dx - fI(x)dx fg(x)dx ~ ~(Mg - I11g) ff2(X)dx - ff(x)dx ,

Chung minh Be) d~ 111.3:

Trang ba"td~ng thuc (lII.3) thay g bdi g-d, voi dEIR ta co:

ff(x)[g(x)-d]dx- ff(x)dr f[g(x)-d]dx

s [1 ['(x)<1< -(1 f(x)dx )}Ig - d1!"

ma ve"tnii cua (IIL12) la:

(III 13)

VT= Iff(x)[g(x) - d]dx - ff(x)dx ~g(x) - d]dx

=IfJ(x)g(x)dx - fJ(x)dx fg(x)dxl.

ChQn d=~(MI! +1111!)t3 thu dl(QC VPcua2 (III 12) la

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

(III 14 )

Trang 45

1

Vp= [!f2(X)dx-(Jf(X)dX rr jig-dilL'

1

1

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

1

14).-Chu thich 111.1: f)~ng thuc tfong (III.2) Kay fa khi:

(III.I5)

Th~tv~y

(III.I6)

(

l

)

".

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

2

f/(x)dx fg(x)dx =

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

0 0 ,- 0 0 Ii

(

Ii

J

2

= f(-I)dx+ JIdx =0

v€ tnii cua (III.2) Hi:

(III 17)

VT= If/(x)g(x)dx - f/(x)dx fg(x)dxl = 1.

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

thuc.-£lJdl dil~'!I tluJ:(;twit jtluin /au: (!J)tiOf4Id Trang 46

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

h~ang so -:"" 1

4

(IlL 19)

fl(x)g(x)dx - fJ(x)dx fg(x)dxl 5.C(M j - mj )(Mg - mg)

mj <S.J(x) <S.Mj' mg 5 g(x) <S.Mg, \!XE [O,1}

Khi d6 (IlL 19) tra thanh

(IIl.20)

fJ(x)g(x)dx - fJ(x)dx fg(x)dxl = 15.C(M j - mj )(Mg - mg) =4C

Chu thich III.3:

BAng thuc trong (III 11) xiiy ra khi:

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

w