các bất đẳng thức tích phân thuộc loại Ostrowski và các áp dụng của nó 2_2

luận văn trình bày một số dạng tổng quát của bất đẳng thức thuộc loại Ostrowski . Áp dụng các bất đẳng thức tìm được để nghiên cứu sự hộ tụ của công thức cầu phương tổng quát và đánh giá sai số trong một công thức tính tích phân số

f!lJal drb(? flute fk./' jtluiu h,i (j)~t~()rtOki Trang 5

CIIUdNG I

CAC DANG THUC TicH PHAN

Mt,lcdkh cua chlidng ngy la tr'inh bay mQt s6 cac d~ng thuc tkh phan bi~u di~n theo gia trj ham va cac dC;lO ham cua no tren cac khming tlidng ung C6ng ct,lchu ye'"ula vi~c sa dt,lng chung minh qui nl,lpva mQt .

s6 c6ng thuc trong phep tinh vi tkh phan

Tnioc he'"t, ke'"tqua sau day

})jnh IX 1.1

Cho Ik : a=Xo < XI < ",Xk-I < Xk=b lamQt phep pluln ho{Zeh eua do{Zn

~

[a,b], aj (i=O, ,k+l) La "k + 2" diem saG eho ao =a,a; E[X;_"X;] (i=l, ,k)

va ak+1 =b Ntu /: [a,b] ~ IR c() d{Zo ham dtn dip 11-I va /(11-1)lien t1;le

tuy~ t di;'i trerl [a,b] Khi de) ta 'eel dang tlu/e:

(1.1)

b

fU-])

( x ) - (x-a ) fU-I) ( X )}

b

a

trong do nluln Peano dU:(/eeho b(Ji:

(t-alr, t E [a, x,),

11!

11!

( 1.2) KII,k(t) =

11!

11!

vcJi n,k E IN va f(°>Cx)= lex).

Chung minh.

Ta chung minh b~ng qui n(;lptoaD hQC

Vc3i11p-1, chung to can cht1ngminhdangtl1l1c .~.

(1.3)

h

=- JKI,k (t)/I (t)dl,

a

trong do

(I - al), (t-a2),

IE [O,XI)'

IE[X"X2)' K1,k(I) =

(I-ak-I)' IE[Xk-2,Xk-')'

D~ chung minh (1.3), ta dung tich phan tUng phfin nhtl'sau

f K1,k (/)( (/)dt ==L f K1,k (t).t (I)dl

k-Ix'+1

= L J(t -a;+I)f (t)dl

;=0 x;

= *i [(t- al+1 )/(t) Ix;'1 - xJ :l<I)dl

]

~ ~[(x", - a",)[ (x"' ) - (x,-a", )[(x, ) - '[f(t)dt ]

= I [(X;+I-a,+I)/(x;+I)-(x; -a;+I)/(x;)]- I Jf(t)dl

= L [(X'+I - a,+1)/(X;+I ) - (XI - a,+1)f(x;)] - Jf(t)dl

Do do

I> I> k -1

V~y d~ng thuc (1.3) dtl<Jcc~!lng minh

(j). 'f,- olZ - ,if "fIJ

,JU,!n {/an ,H/(!(' ');'/ ,(Jan 7If)('

'

I f'- ::- 'fIJ".,

(1),-(11(11dd!1,? 11t(f'c Iklt la'trill, UJq;;CMt()l~ti Trang 7

Giel sli' r~ng (1.1) du ng vdi "n" va ta c~n chung minh r~ng (1.1) dung vdi

"n+ 1", tuc la, ta chung minh d£ng thuc sau day dung:

(1.4)

( ) ( )1f U-')

( )}

, t t + ~ ~ ~~ X;+I -ai+I, X;+I - X; -a;+1 X;

"

= (-1)"+1 JKI1+I,k(t)f(n+I)(t)dt,

a

trong do nhan Peano Kn+l,k(t) du'<;1ccho bdi:

(t - al )"+1

(t - a2 )"+1

(11+1)1 ' tE[XpX2)'

K n+l,k (t) =

(t -a k-l )"+1

(n+l)! ' tE[Xk-2,Xk-I)'

(t -a k ) /1+1 (n+1)!' tElXk-j,bj.

Xet tich phan

JKI1+I,k(t)f"+1\t)dt = I JKI1+I,k(t)f(I1+1) (t)dt

=I xJ (t ~a;+t)"+1 f(n+')(t)dt

;=0 x, (n+l)!

sau do dung tich phan tUng ph~n ta du'<;1c:

It

JK//+I,k(t)P/HI) (t)dt

a

k-I

[(

]

= I -a;+I. f(//)(t) IX'" - J -a;+1 f(I1) (t)dt

k-l

[(

)//+1 ( )"+1

]

- IxJ (t -a;+I)" f(I1)(t)dt

;=0 Xi n,

[(

)'1+1 ( )"+1

]

=L

,

Xi+l-ai+" j(I1)(XJ+I)- Xi-ai+l, j(I1)(X,)

, I i i i

"

-J K I1.k(I) (I1)'. ( t )dt.

(1

Ta vie't l~i d~ng thuc ri§y

I

[(

'

),,->1

]

fK (t)j (I1)

(t) dt ='," Xi+l-tXi+1

j (I1)

(x ) - X, -ai+1 j (") ( X )

( 1)' 1+1 ( 1)1 '

(1.5) 0

I"

- JKI1+lok(t)/,,+1)(t)dt.

,0

Theo gia thie't qui n~p, ta co:

Jj(t)dt+ 2::~2::{<Xi+l-ai+I)JfU-I)(Xi+l)-(x,-ai+I)J jU-I) (xJ}

(1 /=I.J i=O

"

= (-1)" JKl1ok(t)f(I1)(t)dt.

0

hay

( 1.6)

fK".k(t)j(I1)(t)dt = (-1)" Jj(t)dt

"

( 1)./ k-I ( 1)/1" - "f

)./j U-I)( ) ( )1j U-1)( )}

+ - L ~I /," .1I L ~ X'+I -a'+1'IOn Xi+1 - Xi -ai+1 X, Tir (1.5) va (1.6), ta co:

(-1)" fj(t)dt + (-lrI (-?/ I (<Xi+1 -ai+lf jU-I\Xi+')

-(Xi'-ai+I)'/ fU-')(X;)}

k-I

[(

]

=2:: Xi+l-a'+I, f(I1)(Xi+I)- x,-ai+l, f(/1)(xJ

,=0 (n+l) (n+l).

"

-J K11+I.k(t)f (I/+') ( t)dt, a

hay

" k-I 11

( 1)1 S

r

( )d "" - f

)./

f'U-')

( ) ( ) 1 f U-') ( )}

J t {+ L L ~~ X'+I -ai+1 0 X'+I - Xi -ai+1 Xi

a ,=0 /=1 J.

(fl. .,'- 'j7/ - Ijl' '{/}

If"qll NTlI /Tlqr '),r 1'(JfT.1I.-nfll' ,A:9'-"J/%n ,m«, 9/!r'hlfl

1 J.2 //, /, / Ii / - / 41 / /

k-I

[(

' ,

)'1+1

]

=(-1)"2: xi+l-ai+\ f(II)(Xi+I)- x;-a;+I, l")(X,)

i=O (n+l). (n+l).

h

- (-1)/1 JK,1f1,k(l)f(II+I)(l)dt

a

hay

( 1);

fr(. t)d t+ ~~~1Xi+l-ai+1"" - f( ) 1 f (;-') ( Xi+1 - ) (xi-ai+1 ) J f (;-I) (Xi)}

k-I [(

]

+ (_1)"+! L xi+,-a/+1 j(II)(Xi+1)- Xi -ai+1 j(II)(Xi)

(n+l)!-h

=(-1)"+1 JKII+l.k (I )j(I1+I) (t)dl

a

hay.

(1 7)

( 1);

Jr( )d "" - f( ) Jf(;-I)

( ) ( )1f (;-I)

( )}

+(-l)"!'~ [(x -a ) "+' f (/I) ( x )-(x-a ) "+I f (II) ( X )]

( 1)'£ i Ii" 1 1 1+1 1 1+1 1

n+ i=O

h

=(_1 )11+1J KII! l,k.( I ) f (II+I) ( I )dl

a

sO' hc:lllgthu hai va thu ba cua v~ tnii vi~t gom l~i thanh mOt sO'he.mg,do

do ta thu duQc:

(1.8)

( 1)1

f (;-I) ( ) ( ) 1 f (;-I) ( )}

t t+ ~~~tXi+l-ai+l. Xi+1 - xi-ai+1 Xi

a i=O}=1 /.

h

= (-1)"+1 f~:11+1,k(1)/(11+1) (t)dl,

a

nghla la, d~ng thuc (1.1) la dung va Dinh ly 1.1 duQc chung

minh.-H~ qua sau day cho mOt d~ng thuc tich phan khac vdi (1.1) se heru

ich trong cac ph§n sau

ryMT dr/,,//- 1/,,((' Iff-/' A/'rln f(J(d (il.}t;(J(t)k; Tran~lQ

Htimi 1.1.

Vdi ClJnggid thief cua djnh (v 1.1, ta co :

ff({)d{ +~ (-;n~ tx, - a,)J - (x, -a",)J }f(j-I) (X,)]

=(_1)11JKn,k (l)f(l1) (I)di.

a

Chung minh.

Tli (1.1) ta xet s6 hC;lngthil hai va vie't no thanh tc5ng cua hai s6 hC;lng

n ( I) J k-I

f u-IJ ( ) ( ) J f U-') ( )}

1 + 2 - ~ -: , ~ ~Xi+1 - ai+1 < Xi+1 - Xi - ai+1 Xi

/=1 J. i=O

k-l

= L{-(Xi+' -a'+I)f(Xi+I)+(Xi -ai+l)f(Xi)}

1=0

(1.10)

II

( I ) J k-I

{( ) / f U-J) ( ) ( ) J f U-IJ ( )}

+ ~ -: , ~ Xi+) - ai+1 < Xi+l - Xi<- ai+1 Xi

/=2 } i=O

k-I

=:L {-(X'+I -£Xl"!)f(Xi-l') + (x, - £x/<tt)/(x/)}

k-I /I

( I)'

"" - . f

+L L ~~X1+1-ai'l. i=OJ=2 J. xi+l-x,-ai+1 X;.

Bay gio 81 du'<Jcvi€t l,;ti

k-I

81 = (a-a,)f(a)+ L(xi -a'+I)f(xi)

i=1

k-2

i=O

k-I

;=1

k-I + L{-(xi -aJf(xi)}-(b-ak)f(b)

;=1

Cling v~y voi 82 du'<Jcvi€t lC;li

k-I /I

<; ,,- f( ) ; [ (/-I)

( ) ( ) ; f U~') ( )}

L 2 =L JL J = ,t Xi+1 -a,+1 ' X;+I - X, -a;+1 . X;

1=0/=2 J.

k-I /I (-1)/

= II = ,{(Xi+1 -a;+YfU-I)(Xi+I)}

i=0)=2 J. .

- II (-~r {ex;-al+YfU-l)(xJ}

i=O)=2 J.

- ~ (-;({(XU-aYIU-I)(xU)}- ~~ (-Jr{(Xi-a'~I)' IU-I)(x,)}

= I(-~r{(Xk -ak») fU-I)(Xk)}+II(-~r (eX,-a,») fU-I)(Xi)}

- I (-~r{(Xu-al») fU-I) (Xo)}- II (-~r {(Xi-ai+I») fU-') (X;)}

)=2 J. ;=1)=2 J.

= I (-~r {(Xk-ak») fU-')(Xk)}- I (-~r (exo-a,») fU-l)(xo)}

+~[~ (-X {(x,~a,)i ~(x, ~a",)j IF"IX,)]

= t(-?/ {(Xk -ak»)fU-I)(Xk)-(XO -al») fU-')(XO)}

)=2 J.

+ ~[~ (~;t {<x,-ay -(x, -aH,Y V"-l) (X,)]

=I (-~r {(b-akY fU-I)(b)-(a-a,y fU-')(a)}

)=2 J

+I[I (-~r{(X; -aY -(X; -a,+I») }JU-I) (X;)

]

.

'=1 )=2 J.

Tli (1.10), ta co

k-I

8, +S2 =(a-a,).!(a)- I(a,~,-a,).!(x;)-(h-ak).!(h)

)=2 J.

-'1:)~.kH.T(JN~IEN

THLT\lIEN

[1M1 rlrt/Ift (/",,(' (fr/'Ji/uin !.Jai @.)b(J,~t,,: Trang 12

~ -{(a, -a)f(a) + ~(a", -a,)f(x,) + (b -a,)f(b)}

+I (eXi ~a,)} -(Xi -ai+I)}}JU-I)(xJ+(b-akf fU-')(b)]

,=1

Chli Y r~ng Xo= a, ao = a, Xk = b Vaak+1 =b ta co th€ vie"t

{

k-I

}

8, +82 =- (al-a)/(a)+ t;(a,+,-a,)/(xJ+(b-ak)f(b)

+ I(-?i [-(a-al)}/C!-I)(a)

}=2 J.

+I {(x, -aJi -(x, -a1+,)}}fU-')(x,)+(b-ak)}fU-')(h)}

i=1

k

=-I(a'+1 -a,Jf(x,J

i=O

+I (-~r [-(a-at)} fU-')(a)

}=2 J.

k-I

;=1

k

=-I(ai+1 -a;)f(xi)

;=0

+I(-~ri=2 J. [-(xo-a,YfCH)(xo)

k-I

+I (ex,-aJ/-(x, -aj+,y}rU-')(x;)+(Xk -akYfi-I)(xd]

,=1

= - I(aj+1 -aj)f(x;)

,=0

11 ( 1)i

[

k

] +I~ I {(Xj-aJl -(x; -a;+,f}fu-')(xJ

1=2 J. ,=0

[1],;-1tUfJ~rIluf-clicit /tltalt kat (lM~()(':-)/{i Trang 13

~t (-;:J [tkx, -a,)' - (x, -a", Ji}J"O"(X,)}

Thay 8, + 82 vao so' h~lOgthU'hai cua (1.1) ta tho duQc d~ng thU'c(1.9)

Bay gio ta gia Slt rang cac di~m chiaXi cua phan ho~ch lk laceSdinh, ta

tho duQc h<$ qua gall.

He !loa 1.2.

Cho lk :a=xo <XI <",Xk-I <Xk =b La melt phiin ho(}ch cua do(}n [a,b].

Ne'u f: [a,b] +IR gi(fng nhll tron!? dinh Ly 1.1 Khi d6 ta c6 &lng thac:

(1.11)

hill

[

k

]

ff(t)dt+ ~2Jj! ~{-h/ +(-l)Jh/-I}rU-I)(x;)

h

= (-1)" fKII,k (t)f(lI) (t)dt,

a

trong d6 hi = X'+I- x" h_1= 0 Va hk = O.

Chung minh.

Ch<;>ncac di~m a; U=O, ,k+l) nht!sau:

, a+xl Xi-I +X, .

ao=a,al=-,a;= , (z=l, ,k),

Xk-I + Xk '

ak = va ak+' = b.

2

D~ng thU'c (1.9) vi€t l~i

(1.12)

vdi

[i:kXi - ai)/ - (Xi - ai+I)/ }fU-'J (Xi )

]

.

/=1 }! i=O

Ta chia so' h~ng thil hai 81+ 82 cua (1.9) thanh 3 so' h~ng nhu gall:

~Jnlitj{( 'uoAl1I!iJJ&ui.f1JlI}(L(~ - "",,-," "'-~

11

(]) ./

[

k

]

=-2:{(Xi -a;)-(Xi -ai+I)}[(Xi)

;=0

./=2 1.

+~[t I-~:j ¥X;- aY - (X;-a;+I)' Ir(JI)(X;) J sO' h~ng thil nha't 8, +82 cua (1.9) vdi 0 ~ i ~ k, i =1:

k

(i) :L{(x,-a,)-(x,-a,+,)}f(xJ

'=0

k-I

1=1

}

(ii) I(-~r}=2 1. (cb-ak)} fU-')(b)-(a-a,)J fU-I)(a)}

=~(_])I L 'I 2 J {hl- k I rU-I) (b) - ( -I ) JhllU-I)(aO. )}.

J=2.1.

(iii) ~ [~( -;r {(x,-aY - (x,- a,.,)' }Iii "(x,)]

=I[i:,

(~! '

2

]

'

f)~t ba sO'h~ng (i ), (ii ) va (iii ) vaa 81 + 52 ta thu du'Qc:

(Ll3) S, + S, = ~ (-;n~«X;-a,J' -(X, -a",)' }/U"'(X,J]

k

=- I {(x, -aJ - (x, -a;+,)}f(x;)

,=0

_.Imugjj

iYJril rI,j:'~11/ut(' lid" j'/uiu (wi f~JIt()r(;jt; Trang 15

+i (-?J {(b-adlf(l-I)(b)-(a-al)J fU-') (a)}

J=2 ./.

k-I

[

n

( 1) J

]

+~~fl(Xi-ai) -(xi-ai+i) / (Xi)

1

{

}

= -2 ho/(a) +~(hi + hi-I).I (x;) + hk-d(b)

+ ~L (-l)J '1 2 l {hJ- k 1 fU-I)(b)-(-l)JhJ 0,fU-') (a) } J=2./.

+ I1=1 l=2[i (~;:J. {h;~~1 - (-l)Jh/ lr(j-1) (x/) ]

[

~ (-l)l

{ h~ -(-l)J hJ } jU-1)(X)

]

1=0 J=I J.

=~ (-1)J ~{ h~ -(-l)JhJ }rU-1)(x).

J=I J. 1=0

Cu6i cling ta thu dt(Qc (1.11) b~ng cach thay 81+82 vao (1.12)

Tn(on g hop. ta la y cac di~m ehia x- eua I cach d~u ta tlm duoe he q1 k , Ua salt:

He qua 1.3 Cho

(1.14) I k : Xi =a +:{b~a).i = 0, , k

li@nt\,lc tl1yi$td6i tr~n [a,b] Khi d6 ta c6 dAng thd'c:

(1.15) J/(t)dt +t (b -a

)

.I

x;,[ - Iu "(0) +~I<-I)' -I }I"-"(x,) + (-1)1IU-'>(b)]

h

= (-1)" J KII,k (t).T(II) (t)dt.

Chu yr~ng sO'hi;lng thli hai cua (1.15) chI chlia cac di;lo ham cffp Ie ti;li tfft ca cac diem trong Xi' i=1, ,k-1.

Chung minh. Sa d\,lng (1.14), ta chu y r~ng

:YM1 d(j,~1 (I"f'(' (ir-/' ;'/,(j,J/ !oa; (iM~()fr:Jt; TranR-16

ho =x! -x() =-, hi =xi+!

hi! =Xi -XI-I ""k,(i=t, ,k-l)

( 1.16)

va the' vao (1.11), ta co:

(

)

.1

fI(t)dt +I ~

x;![-f'HI(a)+ ~k-l)' -1)I'HI(x,)+(-I)J fU-"(b)]

"

=(-1)" fKII,k(t)f(II)(t)dt.

a

Tinh tacin don gian, tU day ta suy tu d~ng thuc (1.15).8

Cong thuc gi6ng Taylor sail day voi phfin du tich phan cling dung

He gmi 1.4.

Cho X : [a,y] ~ 1R OJ d~w helm din clip n sao cho g(lI) lien t~lCtuy~t

at)'l tren ju,y] Khl d6, wYi mql XiE [a,y) fa c6 dilng tlllJC:

(1.17)

g(y) ~ g(a) - ~ [t(-;r (lx", - a",)' g"'(x",) - (x, - a,.,)' g(j)(X,)}]

y

+(~1)" fKII,k(y,t)g(II+I)(t)dt

(1

hay

[

k

]

y

+(-1)" JKII,k(y,t)g(II+)(f)dt.

a

luvt b~ng each clwn f = gl, b::::: y