Các bất đẳng thức tích phân thuộc loại ostrowski và các áp dụng của nó

(II),tJ.III'wm) III = Iwjm)f(xjlll»)- I (-1)' /~o r~2 r! x[~{(X)'" -0- ~w~""J+~""-0- ~W;"")k.,"(xj"" J] m vui wjnr) (j =0,...,m) lacactn;mgc~uphuongthoa I wjnr)=b- a . )=0 Djnh 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§pxi tichphan ff(x)dx vdi , a mOtsais6duQcbi~uthjtheoIlf(II)IL. l>inhIf 3.1: Clzof: [a,b]~ IR Lientf:lCtren[a,h],saDchof<n-l)La lientf:lCtUYft d(/i (ren [a,bl. Ne'u cac trQngc£1uplU{dngwjm)t/1(3adiiu ki~l1: .[llrfl rMJ~fJIlute lid, jil/(ln loaf f);j{lo,ttJi,. Trang25 (3.2) I X(m) - a <"\' w(m) < X (m) - a I - L... J - 1+1 , ./=0 Vi=O,...,m-l. Khi eMfa co danh gid (3.3) h [1I/(/,/',...,/(I1),611/'WII/)- ff(t)dt a II [(11) II ' [( J I1+1 ( J "+1 ] :::;' if) 'II a+:tw;m) - X~III) + x~::)- a - tw;m) (n + I)! i=O /=0 /=0 :::;IIJ(I1)L 'I1Vlilll)t' (11+1)! i=O 11 [(11) 11 :::;' if) [v(h(III))J1(b-a), (n+I)! trong do [(II) E L [ab] v(h(m)=max{h(m) }h(m) =x(m)-x(m) . '" -' , i=O, m-1 I , I 1+1 I' h D(ic biet, ne'u 11 /(11) 11 <CXJ thE Jim 1 ([,/',...,/(11),6,w ) =fJ(t)dt. '" "(11('0))--+0m . m m a dJ u theocae trong w(m).. J Chun,gminh: I Ta dinhnghTadaycaes6thlfcai(:?=a+Lw;m), )=0 i =0,...,m. Chti Y r~ng m a(m) =a +"\' w(m)= a +b - a =bm+1 L... J . )=0 Do giii thie't(3.2),taco: (m) [ ,(III) (mT]at+1 E X, ,Xi+1 Vi =0,...,m-1. D)nhnghlaat) =a vaHnh, a(m) (m) - (m)I - ao - Wo i i-I ai(:;)-a;m)=0+ Lw;m)-o- Lwt) =w;m),(i=o,,:.,m-l) VO )=0 III Ill-I a(m) - a (111)=0 + '\' w(l11)- a-'\' w(m)= w(m)III+I III L.., J L.., Jill' /=0 /=0 va dod6 m m "\' ( (m) - (1I1))r( (m) )="\' ,(111)/( (m))L.. a,+1 a, X, L.. }I, X, 1=0 1=0 Wlril rlrfJlrjjl"tr f;,.;'-AI,,;)/, 4x.!ifMiOftJi; Trang26 va m I w;m)f(xj"'») )=0 III (-I) r[I m {( (m) I)-I (m) ] r ( (m) t. (m) ] r f (r-I) ( ("'» )1] - - x. -a- W - x -a- W x. I ) S ) s.) r=2 r. )=0 s=o s=o =lm(f,f',...,f(II),6.m'~Vm)' Ap d~Jngba'"td~ngthuc(2.1)tathudu<)cdanhgia (3.3),8 T " h kl ' h" h h ]' d l< E (m) .b- a ,nieJng <)p 11P an O':lC a ell: ;"/11=Xi =a+1---'--, (1=0,...,111) 111 va djnhnghladaycaccongth((cc~uphuongHnhsf): 1",(1,; ,...,f(n) ,6.111'w/11) =fwt) J a+}(b-a))j=O } ~ 111 - I (-Ir [ f {( }(b-a) IW~/11» ) r _ ( }(b-a) - tw.~II1» ) r } fr-I) ( a+}(b--a) ) l r=2 r. j=O 111 s=O 111 s=O m Ij Khi do tathudu<)ch~quasau: He qua3.1: ClIo f: [a,b]~ IR lien t{lCtren [a,b],sao clIofen-I)la lien t~lC tuy~td/;'itren [a,b].Ne'ucac trQngc6.uphl1angw;/II)thoadiJu ki~n: i 1 I' (/II) i+1 . 0 1-<- I <- - -- 11j - , T- ,...,111 , m b - a j=O 111 Khi de)ta C()danhgia: h 1/11(f,f',...,f(II),6./II,w/ll)-ff(t)dt a Il f(II) 11 [( . J "+1 ( . J "+I ] ~. "II twjm)-i ( b-~ ) + (i+l) ( b-a )~twY")(11+I). ;=0 j=O 111 111 )=0 ~Ilf(lI)I\" ( ~ ) II+l, . (11+I)! 111 trong eM fell) E L,,[a,b].. f/JriZrl<fJld/lftr (kit flt4J1 (oqi (iM~o,t}t; Trang 27 h Dgc bi~t,ne'll 11/(11)11",<co,thi J,~~ImC/,f',...,/(II),Wm)= Jf(t)dt d~utheocae It trQng \I,~III). Chungminhh~qua(3.1)suytn!ctie'ptudinh19(3.1).- w :1],;1mfl'flI/If"'~.Jf('f,hf",n kx,; (if.)/;(J(t)/d Trang28 CHUaNG 4 " ? BA.T DANG THUC THUQC LO~I GRUSS BJt d~ngthucthuQclo~iGrUssla bfftd~ngthuctichphanchos\.f lien h~giC'(atkh phancuamOttichhai hamsf{va tichclIacactichphan clla tunghamsf{.Trudche"taco bfftd~ngthucsail: DinhIf 4.1: ClIo h,g ..[a,b]~ IR Ld /wi helmkhd tfch saD cha r/J s hex) s CP va ysg(x) s Tve!i111Qix E [a,b], r/J,</J,y, va r Lacaehangs6:Khi do, tac6 (4.1) ] IT(h,g)1~-«1) - ~Xr - r)4 trong dr5 (4.2) Ih Ih]h T(h,g) =- fh(x)g(x)dx-- fh(x)dx- fg(x)dxb-a b-a b-aa a a vahlings6'~trangbat dcingthac (4.1) Latd(n/1{1'trheanghfarangkh6ng4 thi thaythe'no bangmQts6'khacn/1()/1(}n. Vi~cchungminhbfftd~ngthucnayduQctlmthffytrong[7]. Trongnhi~utai li~u[4,5,6, 11]va cactai li~uthamkhaotrongdo, thl btltd~ngthuc(4.1)g9i la btltc1~ngthucGrUss.T6ngquathdn,btltd~ng thucGrUssc1uQcphatbi~unhusail: Dinh If 4.2: Chofvd g La/wi hamkhd(ich tren[a,b]vays g(x)s r wJinu/i xE[a,h].Khi dr5tacd: (4.3) IT(f,g~r; r (T(f,f))i, .3M, d(f/~r;flute fff'/"I!./'rin !rxr((!jdtt(J(~k( Tran&.J9 (rongde) T(f,f), T(f,g) dur;cxac dinh nhu (4.2). Ta cGngchu 9r~ngT(f,f) =~ [ ff2(X)dx -~ (ff(X)dx J 2 ] ~O. b-a b-aa a Dinh 194.2dfidU<;1cchungminhbdiMalic, Pecaric,va Ujevic [8]vabfft d~ngth((cnaychotamQtdanhgiat6thanb§td~ngthuccuaGruss(4.1). Th~tv~y,gia sa h, g thoacac gia thj.e"t,cuaDinh 194.1.Ap dl;1ng(4.3) voif= g =h,taco: T(h,h)5. <D-~{T(h,h»)'I2,2 hay (4.4) (T(h,h)t25.<D-~.2 Ap d1,lng(4.3) mQtl~nnua cho h, g taco: (4.5) IT(h,g)l5.r - r (T(h,h)YI2,2 va nhlfv~ytaco (4.1)nh(jvao(4.4)va (4.5). Dinh ]9saudayd1javaob§t d~ngthuc(4.3). Dinh Ii 4.3: Choh : a=Xo<Xl < ... <Xk-l<Xk= b la mi)tphepchiacuadogn [a,b], aj (i =0 , ... , k+l) la " k+2" diim saDcho ao=a, aj E[Xi-j,xd (i=1,...,k)vaak+l=b. Gid .'Iiiding f :fa,b] -7 IR lien t~ctuy~td6'i tren [a,b],saDcho &;10ham f(n) :(a,b) --f IR tl1(3am5.f(n)(x) 5.M vdi lriQi X 'E(a,b). Khi db, ta co bitt ddng tIU1C: '$41 clJJI// (Ilfre((eltAf,';n l(Ja; (iJ.)(;(Jf~'i (4.6) Trang30 Jf(t)dt +I (- ~;/ a /=I)' x[~{(/;-°.rf(j-')(X,.,)-(-I)J(~ +o,)f(j~I)(X,)}] _ (fCn-')~)- jCn-I),(a))~(~) 'I+I [ IC;+1( 28; ) r {1 +(-1)n+r} ](b a)(11+1). ;=0 2 r=O h; . M-n [ b- k-l ( h ) 2IHI [ 2n+1 ( 28. ) r ] < 1 a -! Cr ! 1+ -1 r - 2 (211+1)(11!)2t=o2 ~ 2n+1h; { ( )} ( I 1 k-I h n+1 1 r 2 "2 - (n+I)!t=.hc) [~c;.,(~~;J (1+(-l)"H)]]1 d ' h - ' s: XI+I +Xi . 0 k 1trang a i - Xi+/- Xi va Vi =ai+J- , 1= ,..., - .2 Chung minh : Sa dl;}ng(4.2)va (4.3) nhanvaGbdi (b-a)vachQnh(t)=Kn,dt}nhudinh nghTabdi(1.2)vagO)=j(II)(t}, t E[a,b],saGcho: I h h h (4.7) fKn,k(t)j(n)(t)dt-b~a fj(n)(t)dtfKn.k(t)dta a a [ ( J 2 ] .!. M h h 2 ~ ~m (b-a)fK,~.k(t)dt- fKn,k(t)dt , Bay gio tadanhgia h fjcn)(t)dt=jcn-I)(b)- j(I1-I)(a) a va G1 h =fKn,k(t)dt a k-I Xi,l 1 =2:: J,(t-ai+l)"dt 1=0x/ n. 1 k-I= '" J(x - a )n+l ( n+l } (n+1)!to'~ i+1 i+1 - xi-ai+l) 1 k-I = (11+l)IL {cX;+l-ai+I)"+1 +(-1)" (a;+1_X)"+I } . ",0 1 , .'1141r1rf'~11//(h:Ifr/'-/l.!"l" 4m' (J.)6(J((':Jii Trang 31 DungdjnhnghTacuahjva OJ,taco: saocho G1 hi S;:' hi s;: xi+l-ai+I=--u, va ai+l-xi=-+u,2 2 1 k-I {( h ) "+1 ( h' ) ,1+1 } = ,L !- <'5, +(-I)" .-: +8,(11+1)'i~o2 2 = 1 ,I: [ ~C'~+I(-(jir( hi )"+I-r +(-1)"~C~+1«(j,r(~) 11+1-r ](11+1).,~() r=() 2 r=() 2 = (-1)" ,I(hi ) I1+' [ ~c~+, ( 2(ji J r {1+(-l)"H} ] . (n+1).i=() 2 r=() hi Cling v~y G2 h =IK';.k(1)dt a =IX} (t -ai+I)2/1dt ,~oXi (n!)2 1 ~S( )2/1+1 ( )211+1}= (2 1)( 1) 2 ~ ~Xi+1 - ai+1 + ai+1 - Xi 11+ 11. ,~O - 1 k-I {( hi ) 211+1 ( hi . ) 211+1 } - - I --(j + -+(j (2n +1)(11,)2i~O 2' 2' 1 k-I ( / ) 211+1 [ 211+1 ( 2(j_ J r ] = ~ Cr ~ 1+ -1 r (2n+1)(n,)2t=u 2 ~ 2,HI hi { ( )}. Tv d~ngth(fc(1.1),taco th€ vie't: fK/I.k(1)/(11>(t)dt=(-1)" fl(t)dt +(-1)"I (- ~;j a a J~I J. [ k-I ] j (j-I) j (j-I) X t;{(Xi+1-a'+I) / (X'+I)-(Xi -ai+l) I (xJ} va tv ve'tnii cua(4.7),tathoduQc: h 1 h h =IK/I.k (t)/(I1) (t)dt - b - a I/(I1)(1)dtIKI1.k(t)dta a aG] =(-1)"fl(t)dt +(-1)"I (-~r a J~I J. x[~{('; -O.)'fU."(x",>-(-l)J(; +0,)'!U"(X,>}] -( -1)" ( /(I1~I) (b)- 1(11-'),(a)) I (!2)/H' X [~C';+I( 2(ji J r {I +(-I)"H } ](b-a)(I1+I). ,~() 2 r~O hi flJal d;h'fl.Illftr Ifrl!_/!/uiJ/ I(}(,;(!Job(}((ok,. Trang32 sail khi thayv~lOG1. Tli vS phai ClIa (4.7) ta thay v~lOG1 va O2 nhu v~y: h ( h J 2 G~ =(b- a)fK,~,k(t)dt - fKlI,k(t)dt b - a k-] ( h ) 211+1 [ 211+1 ( 25. ) ' ]=(2n+l)(n!)2t;~ ~C;II+I ~ {l+(-lY} [ [ ]J 2 1 k-I h 11+111+1 ' - ,I ( -.!... ) IC~=1( 2/S; J {I+ (-I)"H} 0 (n+l)o;=o 2 ,=0 hi Do d6 M - 111 ~ , IG11s 2 (G4)2, va dinh1y4.3du<;jcchungminh.8 He !1m'!4.1: Choof,h Wlak du:qexaedjnhnhu:trongdjnhly 4.3vahdnnila ta djnhnghia (4.9) )" == - Xi+l+Xi(, ai+1 2 wJi mQi i =0,o..,k-lsaGeho 1(5ils~min{hi:i=l,...,k}..2 Khi doheftdcingtluxesau (4.10) IJf(t)dt+I(-~r II J=I J. (4.8) x[~{(~W(:~8,J fu>',(x,.,)-C~+8,J fU"(xJ}] (f <II-') (b) - f <II-') ( )J k . -III+I ( h ) "+I-' j - a ,,"C' /S' ! {I +(-1) "+' (b-a)(n+l)! ~~ 'HI i 2 . M [ b k-12,HI [ (h ) 211+1-, ] -m -a . . < C' /S'.:...!.- 1+-1' - 2 (2n+l)(n!)2t;~ 211+1 i 2 {() } I . - ( 1 ,IIC~+1(5;,( !2 )"+I-'{1+(-1)"H}J 2 ] 2. (n+l)oi=or=O 2 {!1M,eMJ/!!I/,,!',. I(~/I(;J/ (oa; (!j.)/ifJrtJl; Trang 33 Chung minh dl(Qcsuy tr~(ctie'"ptu (4.6) (j trenb~ngcach thay(4.8) va lam mQtsf) it phep tinh don gian.. H~ml 4.2: Cho /)(1'tkyphiin hog.chh : a =Xo <Xl <...<Xk-l <Xk =bcuadog.n [a,bl, chQ/1 c5;=0 trong(4.8),taco ba'tdcingtluie: (4.11) Iff(t)dt +~;!~(/~r{(-I)' fC;-I)(X;+I)- jC;-I)(x;)} -{I+(-1)"{j(II-I) (b) - j(II-I)(a» ) I (h; ) "+1 Jl (17- a)(n+I)! ;=0 2 1 2 ] - <M-m 2(b-a) k-I ~211+1- {l+(-lr}k-I !i 11+1 2 - 2 [(2"+1)("!)'t=,(2) (n +l)l t=J2) ) . ChungminhduQctrifctie'"psuytu(4.10).. Chu thich4.1: Tnt/J/1glu!pn ie, tasurta (4.11)rang (4.12) Ifj(/)d' +t;!~(~r{r -1)1 ju-n {x,.,):" jU-" (x')1 1 ~ (M -m)~ [ I (!i) 211+1 ] 2. -fin! .J2n+1 ;=02 Tnt(lng lu!p n chcfn, ta suy tll (4.11) rang J /(t)dt+t;!~(~J k-1)1/(1" (x..,)- /(j~" (x,)j -2 ( j(II-1) (b) - j(II-I) (a» ) I (!i) /HI (b- a)(n+I)! ;=0 2 (4.13) He gml4.3: 1 2 ] - M - m 2(b - a) k-I h; 2,,+1 4 k-( h; ,,+1 2 ,; 2 [(2n+I)(n!)'t=oh-) - «n+I)!)'[t=oh-) J . Cho.In) dur;cxaedjnhnlnttrongDjnhiy 4.3vaxetphan hogchd~ueuaGOgHla,b],trongd6 ( b- a )Ek :x; =a+i k' i=O,...,k 'Mal rlrfJ~11/"f("Ilr/' f/'/in Ime (iJdbf}rtJk" Trang34 Khi do, taco b5td~ngthucsau: (4.14) hilI ( b ) J Jf(t)dl +L1 ~ a J=I J. 2k X~{(-IJJ fCH'(a+(i+IJib -aJ) - f(j~"(a+i(b~aJ)} - {I +(-1)" (r(II-I) (b) - /(11-1)(a) J k ( b- a ) /H1 Jl (b- a)(n+I)! 2k k ( b-a ) II+1 ~(M-m) -. n!-J2n+l 2k Chung mink D~tavao cong th(tc(4.11)CJtren vdi chli yding ( b- a ) x= -, hi = Xi+1- i k i =O,...,k. m ._.