các bất đẳng thức tích phân thuộc loại Ostrowski và các áp dụng của nó 4_2

';1],;1 rlrf)~(jI/"ff: lfell flldJ/

Trang 24

!mi (il,J!tortJki

c HUONGIll

Stj HOI T{) eUA CONG THue
~

,:?

~

CAU PHUONG TONG QUA T
Ml;!Cdich cua cJllJ'ongnay la nhhm trtnh bay mOt s6 cac ap dl;!ng
vao vi<$cnghien CUllst,t hOi tl;!cua cong th((c c~u phuong t5ng quat va
danh gia cac sai s6 nay thong qua cac b§t d5ng th((c duQc phat tri~n d
trong cac chuang trudc.
Cho

tJ.nr: a = X6nr)< x~nr) < ... < X~;'~)1 X~;') = b
<

la mOtday cac phan ho~ch

cua do~n [a,b], va xet day cac cong th((c tich phan s6:
(3.1) Inr(f,f',...,f(II),tJ.III'wm)
III

=

Iwjm)f(xjlll»)/~o

I

(-1)'

r~2

x[

~ {(X)'"

-0-

r!

~w~""J+~""

-0-

~W;"")k.,"(xj""

J]

m

vui

wjnr)

(j = 0,..., m)

Djnh

la cac tn;mg c~u phuong thoa

I wjnr)= b - a .
)=0

19 sau day ch((a mOt di~u ki~n dii cho cac tn;>ng
b

wjm)(j = O,.."m)sao

cho

1m(f,f',...,f(II)

,tJ.m, WIll)

x§p xi tich phan ff(x)dx vdi
, a

mOt sai s6 duQc bi~u thj theo Ilf(II)IL.
l>inh If 3.1:
Clzof: [a,b] ~ IR Lien tf:lC
tren [a,h], saD choftUYft

d(/i (ren [a,bl. Ne'u cac trQng c£1uplU{dng wjm)t/1(3adiiu ki~l1:


Trang 25

.[llrfl rMJ~fJIlute lid, jil/(ln loaf f);j{lo,ttJi,.

I

(3.2)

X(m)
I

- a < "\' w(m) <
- L... J -

X (m)
1+1

-a
,

Vi=O,...,m-l.

./=0

Khi eM fa co danh gid
h

(3.3)

[1I/(/,/',...,/(I1),611/'WII/)-

ff(t)dt
a

'

I1+1
[(11)
II

II

:::;'

if) 'II

(n + I)! i=O

[(

a + :t w;m)

- X~III)

+

J

/=0

(

x~::) - a -

t

/=0

:::;IIJ(I1)L 'I1Vlilll)t'
(11+1)! i=O
[(11)
11

11

:::;'
if)
(n + I)!
trong

do

[(II)
.

E

D(ic biet, ne'u

L'" [a b], v(h(m)= i=O, m-1 { I
max h(m)
-'
/(11)

.

[v(h(III))J1(b-a),

11

11

<

'"

dJ u theo cae tron g
.

thE

CXJ

}
,

h(m)

I

=x(m)-x(m)
1+1

Jim 1 ([,/',...,/(11),6

"(11('0))--+0 m .

m

I'

h

,w ) = fJ(t)dt
m
a

w(m).
J

Chun,g minh:
I

Ta dinh nghTa day cae s6 thlfc ai(:?= a +

Lw;m),

i = 0,..., m.

)=0

m

Chti

Y

m+1= a + "\' w(m) =
L... J

r~ng

a(m)

)=0

a + b - a = b.

Do giii thie't (3.2), ta co:
(m)

,(III)

at+1 E [X,

D)nh nghla

at)

(mT

,Xi+1

Vi = 0,..., m-1.

]

= a va Hnh,

a I(m) -

(m) (m)
ao
- Wo
i

i-I

ai(:;)-a;m) =0+ Lw;m)-oVO

Ill-I

+ L.., w(l11)- a-'\' L.., w(m) = w(m)
'\'
J
Jill'
/=0

m

va do d6

"\'

=w;m),(i=o,,:.,m-l)

)=0
III

=0

a(m) - a (111)
III
+I
III

Lwt)

/=0

m

(

(m)

L.. a,+1
1=0

-

a,

(1I1)

(m)

)r(X,

,(111)
)="\' }I, /(
L..

1=0

(m)

X, )

"+1

w;m)

J

]

Trang 26

Wlril rlrfJlrjjl"tr f;,.;'-AI,,;)/, 4x.!i fMiOftJi;

m

va

I

w;m)f(xj"'»)

)=0

-

(-I)
-

I

II

r=2

r.

r

m

r

I

(m)

x.

[I

-a-

)

)=0 {(

I

)-I

(m)

W

S

s=o

-

(m)

x

(

]

t

r

.

-a-

)

(m)

W

s=o

s.)

(r-I)

f
]

(x.("'»
)1]

=lm(f,f',...,f(II),6.m'~Vm)'

Ap d~Jngba'"td~ng thuc (2.1) ta thu du<)cdanh gia (3.3),8
'
"
"
.b - a ,
h
T nieJng h <)p kl11Ph an h O':lC ]a d ell: E = Xi(m) = a + 1---'--, (1 = 0,...,111)
;"/11
111
l<

'

va djnh nghla day cac cong th((c c~u phuong Hnh sf):
1",(1,;

,...,f(n) ,6.111' /11)
w

= fwt)

-

Ja+}(b-a)

} ~

j=O

I (-Ir [ f {(
r.

}(b-a)

j=O

r=2

)

111

IW~/11»

111

r

(

)

s=O

r

_ }(b-a)

- tw.~II1»

111

fr-I)

) }

s=O

(

a+}(b--a)
m

l
)
Ij

Khi do ta thu du<)ch~ qua sau:

He qua 3.1:
ClIo f: [a,b] ~ IR lien t{lC tren [a,b], sao clIo fen-I) la lien t~lC
tuy~t d/;'i tren [a,b]. Ne'u cac trQng c6.u phl1ang w;/II) hoa diJu ki~n:
t
i

1 ' (/II) i+1
.
-
I
b - a j=O

<-

I

-

11j

111

,

- 0

T-

,...,111

- 1

,

Khi de) ta C()danh gia:

h

1/11(f,f',...,f(II),6./II,w/ll)-

ff(t)dt

a

"+1

.

Il

~.

f(II)

"II
twjm)-i
(11+ I). ;=0 [( j=O

~ Ilf(lI)I\"
(11+I)!

trong
.

b-~

11

eM fell) E L,,[a,b].

~
(

111

)

II+l,

(

111

)J

b-a
+ (i+l)

(

.

(

111

"+I

.

) ~twY")
)=0

J

]


Trang 27

f/JriZ rl
h

Dgc bi~t,
trQng

ne'll

11/(11)11",

< co, thi J,~~ImC/,f',...,/(II), Wm)= It
Jf(t)dt d~u theo cae

\I,~III).

Chung minh h~ qua (3.1) suy tn!c tie'p tu dinh 19 (3.1).-

w


:1],;1 mfl'flI/If"'~.Jf('f, hf",n

Trang 28

kx,; (if.)/;(J(t)/d

CHUaNG 4
"

?

BA.T DANG THUC THUQC LO~I GRUSS
BJt d~ng thuc thuQc lo~i GrUss la bfft d~ng thuc tich phan chos\.f
lien h~ giC'(atkh phan cua mOt tich hai ham sf{va tich clIa cac tich phan
clla tung ham sf{.Trudc he"tta co bfft d~ng thuc sail:

Dinh If 4.1:
ClIo h, g .. [a,b]~ IR Ld /wi helm khd tfch saD cha r/Js hex)

va y sg(x) s Tve!i 111Qi
x

E

s

CP

[a,b], r/J,
ta c6
(4.1)

]
IT(h,g)1 ~ -«1) - ~Xr - r)
4

trong dr5

(4.2)

Ih

T(h,g) =-b-a

Ih]h

fh(x)g(x)dx--

a

b-a

fh(x)dx- b-a

a

fg(x)dx

a

va hling s6' ~ trang bat dcing thac (4.1) La td( n/1{1'trhea nghfa rang kh6ng
4
thi thay the' no bang mQt s6' khac n/1() /1(}n.

Vi~c chung minh bfft d~ng thuc nay duQc tlm thffy trong [7].
Trong nhi~u tai li~u [4, 5, 6, 11] va cac tai li~u tham khao trong do, thl
btlt d~ng thuc (4.1) g9i la btlt c1~ngthuc GrUss. T6ng quat hdn, btlt d~ng
thuc GrUss c1uQc
phat bi~u nhu sail:
Dinh If 4.2:
Chofvd g La/wi ham khd (ich tren [a,b]
xE[a,h]. Khi dr5ta cd:

(4.3)

IT(f,g~~ r; r (T(f,f))i,

va y s g(x) s r wJinu/i


Tran&.J9

.3M, d(f/~r; flute fff'/" I!./'rin !rxr( (!jdtt(J(~k(

(rong de) T(f,f),

T(f,g)

dur;c xac dinh nhu (4.2).
2

Ta cGng chu

9 r~ng

T(f,f)

=~

b-a

ff2(X)dx

[

a

-~ ( a
ff(X)dx
b-a

~ O.
J ]

Dinh 19 4.2 dfi dU<;1c
chung minh bdi Malic, Pecaric, va Ujevic [8] va bfft
d~ng th((c nay cho ta mQt danh gia t6t han b§t d~ng thuc cua Gruss (4.1).
Th~t v~y, gia sa h, g thoa cac gia thj.e"t,cua Dinh 19 4.1. Ap dl;1ng(4.3)

voif= g = h, ta co:
T(h,h)

5.
-~
2 {T(h,h»)'I2,

hay
(4.4 )

(T(h,h)t25.

2

Ap d1,lng (4.3) mQt l~n nua cho h, g ta co:

(4.5)

IT(h,g)l5. r - r (T(h,h)YI2,
2

va nhlfv~y ta co (4.1) nh(j vao (4.4) va (4.5).
Dinh ]9 sau day d1ja vao b§t d~ng thuc (4.3).

Dinh Ii 4.3:
Cho h : a= Xo < Xl < ... < Xk-l < Xk = b la mi)t phep chia cua dogn
[a,b], aj

(i

=0

, ... , k+l) la " k+2" diim saD cho ao

= a,

aj E[Xi-j,xd

(i= 1,...,k) va ak+l = b.
Gid .'Iiiding f :fa,b] -7 IR lien t~c tuy~t d6'i tren [a,b], saD cho
&;10 ham f(n) :(a,b) --f IR tl1(3a m 5. f(n)(x) 5. M vdi lriQi X 'E( a,b). Khi db, ta
co bitt ddng tIU1C:


'$41

clJJI// (Ilfre ((elt Af,';n

(4.6)
Jf(t)dt
a

+

I

/=I)'

Trang 30

l(Ja; (iJ.)(;(Jf~'i

(- ~;/

x[~{(/;-°.rf(j-')(X,.,)-(-I)J(~

~~

_

+o,) f(j~I)(X,)}]
'I+I

IC;+1

- jCn-I),(a)
( fCn-')~) a)(11+1). ) ;=0 ( 2 )
(b

<
-

M-n

1

2

b-

a

2IHI

h

k-l

[ r=O

2n+1

-!

(

h;

!

1+

2n+1h; )

(

[~

)

r {1

+ (-1)n+r }
. ]

r

28.

Cr

+
t=o
[ (211 1)(11!)2 ( 2 )

28;

{

-1 r

( )} ]
I

1

k-I

h

n+1

- ( (n+I)!t=.hc)

1

r

[~c;.,(~~;J

' s:
trang d a h i - Xi+/ - Xi va Vi = ai+J'

XI+I

2

(1+(-l)"H)]]

+ Xi
.
, 1= 0 ,...,
2

"2

1

k- 1.

Chung minh :
Sa dl;}ng (4.2) va (4.3) nhan vaG bdi (b-a) va chQn h(t) = Kn,dt}nhu dinh

nghTabdi (1.2) va gO) =j(II)(t} , t E[a,b], saG cho:
h

h

h

I

(4.7)

fKn,k(t)j(n)(t)dta

b~a a
fj(n) (t)dt a
fKn.k(t)dt
2

M

h

h

2

( fKn,k(t)dt J ] ,

~m [ (b-a)fK,~.k(t)dt-

~

.!.

Bay gio ta danh gia
h

fjcn)(t)dt = jcn-I)(b) - j(I1-I)(a)

a

va

h

G1

=

fKn,k (t)dt

a

k-I Xi,l 1

=2:: J,(t-ai+l)"dt
1=0 x/ n.

=

1

k-I
'" J( x

(n+1)!to'~

1

- a
i+1

)
i+1

n+l

n+l

(
-

}

xi-ai+l)

k-I

.
= (11+ l)IL",0{cX;+l-ai+I)"+1

1
+ (-1)" (a;+1 _X)"+I

}
,


.'1141 r1rf'~11//(h: Ifr/'-/l.!"l"

Trang 31

4m' (J.)6(J((':Jii

Dung djnh nghTa cua hj va OJ, ta co:
xi+l-ai+I=--u,

sao cho

hi

S;:'

1 k-I h
= (11+1)'i~o 2
,L !-{(

G1

hi

va

2

s;:

ai+l-xi=-+u,
2

"+1

,1+ 1

h'
<'5,

+ (-I)"

)

hi

= (11+1).,~() ~ C'~+I r ( 2 )
1 ,I: r=() (-(ji
[

( .-: + 8, )
2

}

~
"+I-r + (-1)" ~
C~+1 «(j, r

11+1-r

(2)

r=()

]

I1+'

,I(

hi

= (-1)"

(n + 1). i=()

2)

h

Cling v~y

2(ji

~c~+,
[

.

r {1+(-l)"H}

(

r=()

J

hi

]

I

= a K';.k (1)dt

G2

= IX} (t

-ai+I)2/1 dt

,~o Xi

=

(n!)2

1
1 2

~S~
~ (

(2 11+ 1)(11.) ,~O

2/1+1

Xi+1

-

)

ai+1

211+1

(

+ ai+1 - Xi

-

=

hi

I {( --(j
2'

(

~

Cr

(2n + 1)(n,)2 t=u( 2 )

)

2'

211+1 211+1

/

.

+ -+(j

)

(2n + 1)(11,)2 i~O

k-I

211+1

hi

k-I

1

}
-

211+1

1

-

)

2(j_

1+ -1

~

2,HI hi J
(

[~

}

r

{

r

( )}.]

Tv d~ng th(fc (1.1), ta co th€ vie't:

fK/I.k(1)/(11>(t)dt = (-1)" fl(t)dt
a

a

+ (-1)"

I

J~I

(- ~;j

J.

k-I
j
X

[ t;{(Xi+1 -a'+I)

(j-I)

/

j

(X'+I)-(Xi -ai+l)

(j-I)

I

(xJ} ]

va tv ve' tnii cua (4.7), ta tho duQc:

1

h

G]

=

I

a K/I.k (t)/(I1)

(t)dt

- b -

= (-1)" fl(t)dt + (-1)"
a

x[ ~{(';
-( -1)"

(

h

h

aa
I/(I1)(1)dtI KI1.kt)dt

(
a

I (-~r
J.
J~I

-O.)'fU."(x",>-(-l)J(;
(b) - 1(11-'),(a)
(b-a)(I1+I).

/(I1~I)

I ( !2)

) ,~()

2

+0,)' !U"(X,>}]
/H'

X

~ C';+I( 2(ji J

[ r~O

hi

r {I

+ (-I)"H

}

]


Trang 32

flJal d;h'fl. Illftr Ifrl!_/!/uiJ/ I(}(,; (!Job(}((ok,.

sail khi thay v~lOG1.
Tli vS phai ClIa (4.7) ta thay v~lO G1 va O2 nhu v~y:
h

G~

= (b -

f

2

h

f

a) K,~,k(t)dt - ( KlI,k (t)dt J

b- a

= (2n+l)(n!)2

211+1

h

k-]

t; ( ~)

25.

211+1

(

[ ~C;II+I

'

~)

{l+(-lY} ]
2

1
-

k-I

,I (

h
-.!...
2

M-

Do d6

[

IC~=1

) [ ,=0

(n+l)o;=o

2/S;

11+1 11+1

(

J

{I + (-I)"H}

0

~

111

hi

'

]J

IG11s 2 (G4)2,
va dinh 1y 4.3 du<;jcchung minh.8
He !1m'!4.1:
,

Cho of,h Wl ak du:qexae djnh nhu:trong djnh ly 4.3 va hdn nila ta
djnh nghia
(4.8)

)"
(, ==ai+1 - Xi+l + Xi

2

wJi mQi i = 0, o..,k-l saG eho
(4.9)

1(5ils~min{hi:i=l,...,k}..

2

Khi do heft dcing tluxe sau
(4.10)

IJf(t)dt+I(-~r J.
J=I
II

x[~{(~W(: ~8,J fu>',(x,.,)-C~ +8,J fU"(xJ}]

-

(f

<
-

(b ) - f


(b-a)(n+l)!

M -m
2

k -III+I

(a

b -a
[ (2n+l)(n!)2

,,"C'

)J

~~
.

/S'
'HI

i

h

"+I-'

!

( 2)

I + (-1 )"+'
.
j

{

211+1-,

k-12,HI

t;~ [ C'

. .
/S'.:...!.-

211+1

i

1+-1'

2
(h )

{()
I

2

!2

.

-

2.
1 ,IIC~+1(5;, ( 2 ) "+I-'{1+(-1)"H}
J ]

( (n+l)oi=or=O

}
]


{!1M, eMJ/!!I/,,!',.

I(~/I(;J/

Trang 33

(oa; (!j.)/ifJrtJl;

Chung minh dl(Qc suy tr~(c tie'"ptu (4.6) (j tren b~ng cach thay (4.8)
va lam mQt sf) it phep tinh don gian..

H~ml

4.2:
Cho /)(1'tky phiin hog.ch h : a

[a,bl,

(4.11)

chQ/1 c5;

= 0 trong

Iff(t)dt

+

= Xo < Xl

< ... <

Xk-l

<

Xk

= b cua

dog.n

(4.8), ta co ba't dcing tluie:

~ ;!~(/~r

-{I + (-1)"

{(-I)'

{ j(II-I) (b) Jl

fC;-I)(X;+I)-

jC;-I)(x;)}
"+1

I

j(II-I)(a»

h;

) ;=0 ( 2 )

(17 a)(n + I)!
-

-1

2


-

2(b-a)

2

~

k-I

211+1-

[ (2"+1)("!)'t=,(2)

{l+(-lr}k-I

!i

11+1

2

(n +l)l t=J 2) ) ] .

Chung minh duQc trifc tie'"psuy tu (4.10)..
Chu thich 4.1: Tnt/J/1glu!p n ie, ta sur ta (4.11) rang
(4.12)

Ifj(/)d' +

t ;!~(

~

r

{r

-1)1 ju-n

I

~ (M -m)~

!i

{x,.,):" jU-" (x')1
1
211+1

2.

(
-fin! .J2n + 1 [ ;=0 2 )

Tnt(lng

(4.13)

]

lu!p n chcfn, ta suy tll (4.11)

J/(t)dt +

t;!~( J

~

k -1)1 /(1"

-2 j(II-1) (b) - j(II-I) (a»

(

(b - a)(n + I)!

(x..,)-

I ( !i )
)

rang

/(j~" (x,)j

/HI

2

;=0

2

M

- m

2(b - a)

k-I

,;

h;

2,,+1

4

k-(

h;

,,+1

1
-

2

2 [(2n+I)(n!)'t=oh-) - «n +I)!)' [t=oh- ) J ] .
He gml 4.3: Cho .In) dur;c xae djnh nlnt trong Djnh iy 4.3 va xet phan
hogch d~u eua GOgHla,b], trong d6
b- a
Ek

:x; =a+i ( k'

) i=O,...,k


'Mal rlrfJ~11/"f(" Ilr/' f/'/in

Trang 34

Ime (iJdbf}rtJk"

Khi do, ta co b5t d~ng thuc sau:
(4.14)

J

b

hilI

Jf(t)dl

+

a

X

-

L 1 ( ~2k
J.

)

J=I

~ {(-IJJ
{I

fCH'( + (i + IJib -aJ)
a

+ (-1)" (r(II-I) (b) - /(11-1)(a) k
(b - a)(n + I)! J (

Jl

k
~(M-m)

Chung mink

- f(j~"(a + i(b~ aJ)}

b-a

n!-J2n+l

(

-.

b- a

/H1

2k )

II+1

2k )

D~ta vao cong th(tc (4.11) CJtren vdi chli
hi = Xi+1
-

x=
i

(

b- a
-,
k

)

m

i = O,...,k. .

y ding