các bất đẳng thức tích phân 2

trình bày về các bất đẳng thức tích phân 2

JlUJl W ha1 iu&uJ Yule Trang 13 @JuLdnfJ 2: @ae ha1 ~ tJuk

CHUaNG II

"-CAC BAT DANG THUC TICH PHAN

Trong chuang nay, chung t6i mu6n nghien cUu cac ba't dAng thlic tich phan bi6u di~n theo gia tri ham va cac d~o ham cua no tren cac khmlng tuang ling K€t qua trong phffn nay cho phep tim l~i cac ba't dAng thlic thuQc lm;li Ostrowski va cac ba't dAng thlic lien quan khac

Djnh Iy 2.1.

dol tren [a,b] va f(n) E L'"([a,b D Khi d6 ta c6 beit dang thac

(2.1)

b ff(t)dt- I (b-X)k+l + (-l)k(x-a)k+l (k)

~ IIf(n)1100 [(x-ay+l +(b-xy+l]

(n + 1)!

~ Ilf(n)t (b - aY+\ \Ix E [a,b],

(n + 1)!

trong d6

Cac b(Jt dang thac nay la sitc va hang so' 1 la tot nhttt.

Chung minh.

Dung dAngthlic (1.1), ta du<;1c

(2.2)

b

J~t Jb 1J/fL(lJIUJ tJule Trang 14 ~ 2: @De hat ilLinLJ t1uLe

b

=IfKn (x,t)f(n) (t)dt

a

b

:s;suplf(n) (t)1 flKn (X, t)ldt

aSISb a

~ Ilf'.' II.[t ~~)" dt+ f(b :,1)" dl]

= Ilf(n)!L[(x-ay+l + (b-Xy+l].

(n + I)!

Bfft d~ng thuc thu hai cua (2.1) du'<;1csuy fa tu bfft d~ng thuc sail

(2.3)

Bay gio ta d~ C?P d6n tinh s~c cua bfft d~ng thuc (2.1)

n

Ta co

( t - a + b J

n-k

va

(2.6)

(

ff(t)dt = f~ t - a + b

)

ndt

= 1+ (-IY

(

b - a

)n+l.

(n + I)! 2 Khi do, tu (2.1), ta co

JJ~t to' bat ttdrUJ tJum Trang 15 ~ 2: &i£ bat ilkuJ tJum

l+(-lr

(b-a )n+l n-'(b-x)k+'+(-l)k(x-a)k+' 1

(

a+b

)

n-k

(n + I)!

Thay x = a; b vao (2.7), ta du<;1c

(

b-a

J

n+' ~ 2C

(

b-a

J

n+l.

(n + I)! 2 (n + I)! 2

chung minh hoan tfft

Ta cling chti y rang ham s6

hn :[a,b]~IR, hJx)=(x-ar+1 +(b-xr+',

co tinh chfft

(

a+b )

(b-a )n+l

xE[a,b] n n - 2 = 2n'

D d' b'" d? h' '" h'" hA

d " (21)kh ' 1'" a+b

0 0 at ang t tic tot n at n (;In u<;1cta 1 ta ay x = 2'

Lffy x = a ;b trong (2.1) Khi do, ta thu du<;1ch~ qua sau

H~ qua 2.2.

Gid sa rling ham f nhu trang djnh ly 2,1, ta co bat dang thac

(2,10)

b ff(t)dt- I 1+(-1)' (b-a)'" f Ck)

(

a+b

)

~ Ilfcn)t

n+l

MQt ke't qua khac t6ng quat bfft d~ng thuc hinh thang 1a h~ qua sau

J/UJl M IffllluLruJ tJule Trang 16 @1uLdrl{J 2: @ae hat ilLi.nq tJuLe

H~ qua 2.3.

V6i cac giG thiet nhu trang dinh ly 2.1, ta co bat dang thac

(2.11 )

b ff(t)dt-I (b-a)k+l f(kJ(a)+(-l)kf(kJ(b)

{

1 n = 2r,

< 1 (b-aJ"'llf'"'II.x 2"','-1 n:2r+l

Chung minh.

Dung d~ng thuc (1.14), ta du<;$c

(2.12)

b

b

a

b

::; Ilf(n) IL ~Tn (t)ldt.

a

* N€u n = 2r, khi d6

= ~ !

[

(b - a)2r+l+ (b - a)2r+l

]

- (b-a)2r+l

(2r + I)!

- (b-ay+l

(n + I)!

* N€u n=2r+1 dAt h (t) = ( b-t )2r+l_

J~t W Iffli iu1ruJ tluIR- Trang 17 ~ 2: @Liehat ilfing tluIR-

Chli Y ding

hzr+1(t)

= 0, khi t = a + b

2

Khi d6

(2.14 )

a+b

flhzr+1(t)ldt = f[(b - t)zr+1- (t - a )Zr+l]dt

b

+ f[(t-a)Zr+1 -(b-t)Zr+1]dt

a+b

Z

- 2(b-a)Zr+z

2r+2

4(b;af'

2r+2

= zr~z[Z(b-a)2n2 - (b-a)2n222r ]

( 2 -~ J

Dod6

a 2r+1 (2r+l)!a22r+1

2r+ 22r+1

JIiL}l ro 1Jt11ilJ"uJ lJum Trang 18 ~ 2: @ae 1Jt11~ lJuI£

Ba't dAngthuc (2.11) duQcsuy ra tu (2.12), (2.13) va (2.15).

V?y h~ qua 2.3 duQcchung minh

Ba't dAng thuc sail day theo chuftn 11.1100 cho khai tri€n gi6ng Taylor (1.19) cling dung.

H~ qua 2.4.

Gia sa riing ham g nhu trong h~ qua 1.4 Khi d6 ta c6 bilt dang thac

(2.16) g(y) - g(a) - ~L.J (y - X)k+1 + (-I)k (x - a)k+1 (hi)(

II

(n+l)

II

~ g 00 [(y-xy+l +(x-ay+l]

(n + I)!

II

(n+l)

II

~ g 00 (y-ay+l, 'v'xE[a,y].

(n + 1)!

Chung minh.

Cho x E [a,y], tu cac dAngthuc (1.19), (1.20), ta co

y ,

a

~ Ilg(n+I)IL~Kn(x,t)ldt a

=l/g(n+l)t

[ iCt-ay dt+f(y-tYa n! x n! dt ]

J~t yj' lull ilLirl{J iJuU! Trang 19 ~2: @LieWil~iJuU!

II (n+l) II

= g 00 [(x-ay+l +(Y_Xy+l]

(n + I)!

II

(n+l)

II

S g 00 (y-aY+\

(n + I)!

trong do, bit d~ng thuc sail cling cua (2.16) du<jcchung minh nho (2.3)

Chu y 2.1.

if Trong (2.16), liy x = a, ta du<jc

k

II

(n+l)

II

Ta cling bi€t rang (2.18) cho mQt danh gia tITcong thuc khai tri€n Taylor c6

(2.18)

di€n xung quanh di€m x =a ma ai cling bi€t.

iif Trong (2.16), liy x = a ~ y , ta du<jc

(

a+y )

II

(n+l)

II

Bit d~ng thuc (2.19) chung to rang voi g E Coo([a,b]) thl chu6i

(

a+ Y J

(2.19) chI chua nhung dt,loham cip Ie cua g

JJltll Jij' luLl ilJ.ruJ iJule Trang 20 ~ 2: @Li£hif1 ilJ.ruJ iJule

Ch6 Y 2.2

if Trong bat dAngthuc (2.1), lay n = 1, ta c6

(2.21)

!1(t)dt-(b-a)/(x) ~ (x-a)' ;(b-X)' 11/'11., \fXE[a,b].

Tinh to an don gian ta thu du<:jc

(2.22) -[(x-a)1 2 +( -x) ]=-( b 2 1 b -a) 2 + x ( a + b )2.

Khi d6, ta thu du<:jcbat dAng thuc Ostrowski

2

a+b

(2.23)I/(x) fl(t)dt::; -+ 2 l(b-a)ll/l", \ixE[a,b].

iif Trong bat dAng thuc (2.10), lay n = 1 ta du<:jcbat dAng thuc trung di€m

(2.24 )

!f(IJdl-(b-aJf( a;b)l:;; ~(b-aJ21If'II

iiif Trong bat dAng thuc (2.11), lay n = 1, ta du<:jcbat dAng thuc hlnh thang

(2.25)

!f(t)dt-(b-a/(a); f(b)[ ~ ~ (b-a)'lIft.

ivf Trong bat dAng thuc (2.16), lay n = 1, ta thu du<:jcbat dAng thuc

2

a+y

\ix E [a,y].

Ch6 Y 2.3.

if Trong bat dAng thuc (2.1), lay n = 2, khi d6 ta du<:jc

Jlfi}l M Iffli ili1uJ 1JttI£ Trang 21 @/w'dmJ 2: @LieMl ~ 1JttI£

(2.27)

b

(

a+b

)

fl(t)dt-(b-a)/(x)+(b-a) X-2 II (X)

1

Bay gio, ta chli y rang

(2.28)

khi do, ta Hm l(;liduc;5cbat d~ng thlic trong [2]

(2.29)

b

(

a+b

)

fl(t)dt - (b - a)/(x) + (b - a) x -2 II (x)

2

~ 1-+

I(b )3

11

II

II

24 2 (b - a)2 - a I ",' Vx E [a,b].

ii/ Trong bat d~ng thlic (2.10), lay n = 2, ta thu duc;5cbat d~ng thlic trung

di~m c6 di~n

(2.30)

(2.31)

b fl(t)dt - (b - a) lea) + I(b) (b - a)2 II (a) - II (b)

< (b - a) 3111IIt

iv/ Cu6i cling, trong (2.16), lay n = 2, ta thu duc;5cbat d~ng thlic

(2.32)

g(y)- g(a) -(y-a)gl (x) + (y-a{x- a~ Y )gll (x)

J~t .uf IJiiL~ tJui'R Trang 22 ~ 2: @ae IJiiL~ tJui'R

(

X - a + Y

J

2

~ 1-+

I( )3

11

///

11

242 (y-a)2 y-a g oo,VxE[a,y].