Một số bất đẳng thức thuộc loại ostrowski và các ứng dụng

t- I (b-X)k+l + (-l)k(x-a)k+l (k) a k=O (k +1)! f (x) ~IIf(n)1100[(x-ay+l +(b-xy+l] (n+1)! ~Ilf(n)t (b- aY+\ \Ix E [a,b], (n+1)! trongd6 Ilf(n)1100= suplf(n) (t)1< +00.a5,/:;;b Cacb(Jtdangthacnayla sitcvahangso'1 la totnhttt. Chungminh. DungdAngthlic(1.1),tadu<;1c (2.2) b ff(t)dt- I (b-X)k+l +(-l)k(x-a)k+l a k=O (k +1)! f(k) (x) J~t Jb 1J/fL(lJIUJ tJule... Trang 14 ~ 2: @Dehat ilLinLJ t1uLe... b =I fKn(x,t)f(n)(t)dt a b :s;suplf(n) (t)1flKn (X, t)ldt aSISb a ~ Ilf'.' II.[t~~)" dt+ f(b :,1)" dl] =Ilf(n)!L[(x-ay+l+(b-Xy+l]. (n+I)! V?y bfftd~ngthucthunhfftcua(2.1)du'<;1cchungminh. Bfftd~ngthucthuhaicua(2.1)du'<;1csuyfa tubfftd~ngthucsail (x-ay+l +(b-x)n+l:S;(b-ay+l, \fxE[a,b].(2.3) Bay gio tad~C?Pd6ntinhs~ccuabfftd~ngthuc(2.1). X6thamf: [a,b]~ IR nhu'sau (2.4) Jet) =~ ( t- a+b ) n n! 2. Taco (2.5) f(k) (t) = 1 ( t - a+b J n-k (n- k)! 2 ' Ilf(n)L, =suplf(n)(01=1,as/sb va (2.6) b b (ff(t)dt =f~t - a+b) ndt a an! 2 =1+(-IY ( b - a ) n+l. (n+I)! 2 Khi do,tu(2.1),taco JJ~t .to'bat ttdrUJ tJum... Trang15 ~ 2: &i£ batilkuJ tJum... l+(-lr ( b-a )n+l n-'(b-x)k+'+(-l)k(x-a)k+' 1 ( a+b ) n-k (2.7)1 - -2: x-- (n+I)! 2 k=O (k +I)! (n- k)! 2 ~ C [(x-a)n+1+(b-xr+1]. (n+I)! Thayx =a; b vao(2.7),tadu<;1c (2.8) l+(-lr ( b-a J n+' ~ 2C ( b-a J n+l. (n+I)! 2 (n+I)! 2 NhuV?y C ~1vi a<b va C =1 1ahangs6 t6tnhfft.Do do, dinh1ydU<;1c chungminhhoantfft. Ta clingchtiyranghams6 hn:[a,b]~IR, hJx)=(x-ar+1 +(b-xr+', cotinhchfft (2.9) inf h (x) =h ( 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 atn (;In u<;1cta. 1ta ay x =2' Lffyx=a;b trong(2.1).Khi do,tathudu<;1ch~quasau. H~qua2.2. Gid sa rlinghamf nhutrangdjnhly 2,1,taco batdangthac (2,10) b ff(t)dt- I 1+(-1)' (b-a)'" f Ck)( a+b )a k=O (k+I)! 2k+1 2 ~ Ilfcn)t 2n( (b- a) n+l n+1)! . MQtke'tquakhact6ngquatbfftd~ngthuchinhthang1ah~quasau. J/UJl M IffllluLruJ tJule... Trang16 @1uLdrl{J2: @ae hat ilLi.nq tJuLe... H~qua2.3. V6i cacgiGthietnhutrangdinhly 2.1,tacobatdangthac (2.11) b ff(t)dt-I (b-a)k+lf(kJ(a)+(-l)kf(kJ(b) a k:O (k+1)! 2 { 1 n=2r, < 1 (b-aJ"'llf'"'II.x 2"','-1 n:2r+l - (n +1)! 22r+1' Chung minh. Dungd~ngthuc(1.14),tadu<;$c (2.12) b ff(t)dt-I (b-a)k+1f(k)(a)+(-l)kf(k)(b) a k=O (k+I)! 2 b =I II: (t)f(n)(t)dt a b ::;Ilf(n)IL ~Tn(t)ldt.a * N€u n =2r, khid6 (2.13) flT2r(t)ldt=~f(b - t)2r+(t- a)2rdta (2r).a 2 =~ ! [ (b - a)2r+l+ (b - a)2r+l ](2r)! 2 2r+1 2r+1 - (b-a)2r+l (2r +I)! - (b-ay+l (n +I)! * N€u n=2r+1 dAth (t) =(b-t )2r+l_ (t-a )2r+l tE [a b], . 2r+l , ,. J~t W Iffli iu1ruJtluIR-... Trang17 ~ 2: @Liehat ilfing tluIR-... Chli Y ding hzr+1(t) =0, khi t=a+b 2 ' >0, khi tE[a a+b), , 2 <0, khi tE (a+b b] 2 ' . Khi d6 (2.14) a+b b ""2 flhzr+1(t)ldt= f[(b - t)zr+1- (t - a)Zr+l]dt a a 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)2n2]22r = 1 (b-a)2r+2 ( 2-~ J2r+2 22r - 1- 2 (b-a ) 2r+2 22r+1_ 1 r+2 x 22r . Dod6 (2.15) bib 1 fiT (t)ldt = f- I h (t)ldt a 2r+1 (2r+l)!a22r+1 - 1- ( (b ) 2r+ 22r+1 2r+2)! -a 'x -I22r+1 JIiL}l ro 1Jt11ilJ"uJ lJum... Trang18 ~ 2: @ae1Jt11~ lJuI£... 1 22r+1-1 = (b-a)n+1x . (n +I)! 22r+1 Ba'tdAngthuc(2.11)duQcsuyratu(2.12),(2.13)va(2.15). V?y h~qua2.3duQc hungminh.. Ba'tdAngthucsail day theochuftn11.1100 chokhaitri€n gi6ngTaylor(1.19) clingdung. H~qua2.4. Gia sa riinghamg nhutrongh~qua1.4.Khi d6tac6biltdangthac (2.16) g(y) - g(a) - ~(y - X)k+1+(-I)k (x - a)k+1L.J (hi)( k=O (k +I)! g x) 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)! Chungminh. Chox E [a,y],tucacdAngthuc(1.19),(1.20),taco (2.17) g(y) - g(a) - ~(y - X)k+1+(-I)k (x - a)k+1L.J (hi)( ~ ~+1)! g ~ y , =I fKn(x,t)g(n+l)(t)dt a ~Ilg(n+I)IL~Kn(x,t)ldta =l/g(n+l)t[iCt-ay dt+f(y-tYdt]a n! x n! J~t yj' lull ilLirl{JiJuU!... 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)! trongdo,bit d~ngthucsailclingcua(2.16)du<jchungminhnho(2.3).. Chuy2.1. if Trong(2.16),liy x=a, tadu<jc k II (n+l) II g(y) - I (y - a) g(k)(a)S g 00 (y - ay+l,Vy ~a. k=O k! (n+I)! Ta clingbi€t rang(2.18)chomQtdanhgiatITcongthuckhai tri€n Taylorc6 (2.18) di€n xungquanhdi€m x =a maai clingbi€t. iif Trong(2.16),liy x =a~y , tadu<jc (2.19) g(y)-g(a)-I 1-(-I)k (y-a)k (k) ( a+y )k=l k! 2k g 2 II (n+l) IIs g 00 (y - a)n+l Vx E [a y]. 2n(n+l)! ' , Bit d~ngthuc(2.19)chungtorangvoi g ECoo([a,b])thlchu6i (2.20) g(a)+f 1-(-I)k (y-a)k (k)( a+Y Jk=l k! 2k g 2 hQit\1nhanhv~g(y) nhanhhonchu6ithongthuongf (y - ~)k g(k)(a), mak=O k. chu6inaychlnhla chu6iTaylorcuag. HonmIa,taclingchuyrangtrong (2.19)chIchuanhungdt,lohamcip Ie cua g. JJltll Jij' luLl ilJ.ruJ iJule... Trang20 ~ 2: @Li£hif1ilJ.ruJ iJule... Ch6 Y 2.2. if TrongbatdAngthuc(2.1),lay n=1,tac6 (2.21) !1(t)dt-(b-a)/(x) ~ (x-a)' ;(b-X)' 11/'11.,\fXE[a,b]. Tinh toandongiantathudu<:jc (2.22) 1 2 b 2 1 b 2 ( a+b ) 2 -[(x-a) +( -x) ]=-( -a) + x-- . 2 4 2 Khi d6,tathudu<:jcbatdAngthucOstrowski ( ) 2 a+b x-- I b 1 2 (2.23)I/(x)--fl(t)dt::; -+ 2 l(b-a)ll/l", \ixE[a,b].b-a a 4 (b-a) iif TrongbatdAngthuc(2.10),lay n = 1tadu<:jcbatdAngthuctrungdi€m (2.24) !f(IJdl-(b-aJf( a;b)l:;; ~(b-aJ21If'II.. iiif TrongbatdAngthuc(2.11),lay n=1,tadu<:jcbatdAngthuchlnhthang (2.25) !f(t)dt-(b-a/(a); f(b)[ ~~(b-a)'lIft. ivf TrongbatdAngthuc(2.16),lay n=1,tathudu<:jcbatdAngthuc ( ) 2 a+y (2.26) Ig(y)-g(a)-(y-a)g/(x)I:o:I.!.+ x- 2, l(y-a)'llgllll,4 (y- a) 00 \ix E [a,y]. Ch6 Y 2.3. if Trong bat dAngthuc (2.1), lay n =2, khi d6 ta du<:jc Jlfi}l M Iffli ili1uJ 1JttI£... Trang21 @/w'dmJ2: @LieMl ~ 1JttI£... (2.27) b ( a+b )fl(t)dt-(b-a)/(x)+(b-a) X-2 II (X) 1 ~6[(x-a)3 +(b-x)3]lIlll", VxE[a,b]. Baygio,tachliy rang (2.28) (x -a)' +(b-x)' =(b-a{(b;a)'+3(x- a;bn khi do,taHml(;liduc;5cbatd~ngthlictrong[2] (2.29) b ( a+b )fl(t)dt- (b- a)/(x) +(b- a) x -2 II (x) ( x- a+b J 2 1 1 2 ~1-+ I( b ) 3 11 II II24 2 (b- a)2 - a I ",' Vx E [a,b]. ii/ Trongbatd~ngthlic(2.10),lay n=2, tathuduc;5cbat d~ngthlictrung di~mc6di~n (2.30) !f(l)dl-Cb-a)f( a~b)l;;; 2~Cb-a)'llf't. iii/ Trongbatd~ngthlic(2.11),lay n=2, tathuduc;5cbatd~ngthlic (2.31) b fl(t)dt - (b- a)lea) +I(b) (b- a)2 II (a)- II (b) a 2 2 2 < (b - a) 3111IIt .- 6 iv/Cu6icling,trong(2.16),lay n=2, tathuduc;5cbatd~ngthlic (2.32) g(y)- g(a)-(y-a)gl (x)+(y-a{x- a~Y)gll(x) J~t .uf IJiiL ~ tJui'R... Trang22 ~ 2: @ae IJiiL ~ tJui'R... ( X - a+Y J 2 1 1 2 ~1-+ I( ) 3 11 /// 11 242 (y-a)2 y-a g oo,VxE[a,y]. ._.