Luận văn Các bất đẳng thức tích phân thuộc loại Ostrowski và các áp dụng của nó

f!lJaldrb(? flutefk./'jtluiu h,i (j)~t~()rtOki Trang5 CIIUdNG I CAC DANG THUC TicH PHAN Mt,lcdkh cuachlidngngyla tr'inhbaymQts6cacd~ngthuctkh phanbi~udi~ntheogiatrjhamvacacdC;lOhamcuanotrencackhming tlidngung.C6ngct,lchuye'"ula vi~csadt,lngchungminhquinl,lpva mQt . s6c6ngthuctrongpheptinhvi tkh phan. Tnioc he'"t,ke'"tqua sauday. })jnh IX 1.1. Cho Ik :a =Xo < XI < ",Xk-I < Xk =b lamQtphepplulnho{Zeheuado{Zn ~ [a,b], aj (i=O,...,k+l) La "k +2" .diemsaG eho ao =a,a; E[X;_"X;] (i=l,...,k) vaak+1 =b. Ntu /: [a,b]~ IR c() d{Zoham dtn dip 11-I va /(11-1)lien t1;le tuy~t di;'itrerl [a,b].Khi de)ta'eeldang tlu/e: (1.1) b Jf( t) dt+~(-l)f~f (X -a ) .I f U-]) ( x ) - (x-a )J f U-I)(X )}~ .,~~/"'I 1+1 1+] 11+1. I a .;=1 }. ;=0 b =(-1)" JK,;,k(t)/(II)(t)dt, a trongdonluln PeanodU:(/ehob(Ji: (t-alr, t E [a,x,), 11! (t-a2r, tEfxl,X2)' 11! (1.2) KII,k(t)= (/-ak_,)" , t E [Xk-2,Xk-I)' 11! (f-akr IE[Xk-),b],-, . 11! vcJin,k E IN va f(°>Cx)=lex). Chung minh. Ta chungminhb~ngquin(;lptoaDhQC. Vc3i11p-1, chungtocancht1ngminhdangtl1l1c .~. (1.3) h k-I ff(l)dt - L[(X;+I- a;+1)f(X;+I) - (x; - a;+1)f(xj)] a ;=0 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-')' (I -ak). {E Ixk-I,bl. D~chungminh(1.3),tadungtich phantUngphfinnhtl'sau h k-P", fK1,k (/)( (/)dt==L f K1,k (t).t (I)dl a ;=0x; k-Ix'+1 =L J (t -a;+I)f (t)dl ;=0x; =*i [ (t- al+1 )/(t) Ix;'1 - xJ . :l<I)dl ]1=0 x, x, ~ ~[(x", - a",)[ (x"')- (x,-a",)[(x, ) - '[f(t)dt] k-l k-I x,+, =I [(X;+I-a,+I)/(x;+I)-(x;-a;+I)/(x;)]- I Jf(t)dl ~ ~~ k-I h =L [(X'+I- a,+1)/(X;+I) - (XI - a,+1)f(x;)] - Jf(t)dl ~ a Dodo I> I> k.-1 ff(l)dl +fK1,k(I)f (t)dl=2](X;+1- a;+1)f(x;+I) - (X; - a;+J)f(xJ] a a ;=0 V~yd~ngthuc(1.3)dtl<Jc~!lngminh. (j). 'f,- olZ - ,if "fIJ ,JU,!n {/an ,H/(!(' ');'/.,(Jan .7If)(' 'If'- ::- 'fIJ"., (1),-.J ?U,,/'!JI, .7(,(H( ::.l/((1'fI (11(11dd!1,?11t(f'cIklt .la'trill, UJq;;CMt()l~ti Trang7 Gielsli'r~ng(1.1)dungvdi "n" vatac~nchungminhr~ng(1.1)dungvdi "n+1",tucla, tachungminhd£ngthucsaudaydung: (1.4) " k-I //+.1 '} J/ ' ( )d ",,(-I) f( )1rU-I) ( ) ( )1f U-') ( )} , t t+~ ~ ~~ X;+I-ai+I, X;+I - X; -a;+1 X; a ;=0 }=I J, " = (-1)"+1JKI1+I,k(t)f(n+I)(t)dt, a trongdo nhan Peano Kn+l,k(t) du'<;1ccho bdi: (t - al )"+1 (n+l)! ' tE[a,xl)' (t - a2)"+1 (11+1)1' tE[XpX2)' K n+l,k(t) = (t-a ) "+1 k-l (n+l)! ' tE[Xk-2,Xk-I)' (t -a ) /1+1 k (n+1)!' tElXk-j,bj. Xet tichphan It k-I X,+, JKI1+I,k(t)f"+1\t)dt =I J KI1+I,k(t)f(I1+1)(t)dt a ;=0 Xi =I xJ (t~a;+t)"+1f(n+')(t)dt ;=0 x, (n+l)! sau dodungtichphantUngph~ntadu'<;1c: It J K//+I,k(t)P/HI) (t)dt a k-I [( I ) //+1 Xi>l (t ) // ] =I -a;+I. f(//)(t) IX'" - J -a;+1 f(I1)(t)dt ;=() (11+1)! x, 11!Xi k-l [( ) //+1 ( ) "+1 ] =I X;+t-ai+\ f(I1)(Xi+I)- x;-a;+I, f(I1)(x;);=0 (11+1). (n+l), - IxJ (t -a;+I)" f(I1)(t)dt ;=0Xi n, k-t [( ) '1+1 ( ) "+1 ] =L , Xi+l-ai+" j(I1)(XJ+I)- Xi-ai+l, j(I1)(X,)'=0 (n+1). (n+1). , I i i i " - J K (I) (I1) (t)dt.I1.k '. (1 Ta vie'tl~id~ngthuc ri§y I " 'k..1 [( ),,+1 ( ' ) ,,->1 ] fK (t)j (I1) (t)dt='," Xi+l-tXi+1 j (I1) (x ) - X,-ai+1 j (")(X )l1ok ,L.. ( 1)' 1+1 ( 1)1 ''1=0 n+. n+ . (1.5) 0 I" - JKI1+lok(t)/,,+1)(t)dt. ,0 Theogiathie'tquin~p,taco: " 11 ( 1)1k-I 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 0 a " ( 1)./ k-I ( 1)/1" - " f( )./ j U-I) ( ) ( )1j U-1)( )}+ - .L..~ I L.. ~X'+I-a'+1 Xi+1- Xi -ai+1 X,I ./," .1 'IOn Tir (1.5)va (1.6),taco: (-1)" fj(t)dt +(-lrI (-?/ I (<Xi+1-ai+lf jU-I\Xi+') a 1=1 .J. i=O -(Xi'-ai+I)'/ fU-')(X;)} k-I [( ) /+1 ( ) /+1 ] =2::Xi+l-a'+I, f(I1)(Xi+I)- x,-ai+l, f(/1)(xJ,=0 (n+l). (n+l). " - J K (t)f (I/+') (t)dt11+I.k , a hay " k-I 11 ( 1)1 S r ( )d "" - f( )./ f'U-') ( ) ( )1f 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 (j7J _1 J.2 //, /, / Ii / - / . 41 / / .,7,Ja. a-aJ'fl ",ute "~JI. l/tan waf W,jr,lm!ijh't Trang9 k-I [( ) '1+1 ( ' , ) '1+1 ]=(-1)"2: xi+l-ai+\ f(II)(Xi+I)- x;-a;+I, l")(X,)i=O (n+l). (n+l). h - (-1)/1JK,1f1,k(l)f(II+I)(l)dt a hay h k-I II ( 1);fr()d "" - . f( )1f (;-')( ) ( )J f (;-I) ( )}. t t+ ~~~1Xi+l-ai+1 Xi+1- xi-ai+1 Xi a i=O1=1 J. k-I [( ) '1+1 ( ) '1+1 ] + (_1)"+! L xi+,-a/+1 j(II)(Xi+1)- Xi -ai+1 j(II)(Xi) i=O (n+l)! (n+l)!- h =(-1)"+1 J KII+l.k (I )j(I1+I) (t)dl a hay. (1.7) h k-I II ( 1); Jr( )d "" - . f( )J f (;-I) ( ) ( )1f (;-I) ( )}. 1 1+~~ ~t Xi+1 - ai+l. Xi+1 - Xi - ai+1 Xi a ,=01=1'/' +(-l)"!'~ [(x -a ) "+' f (/I) (x )-(x-a )"+If (II) (X )] ( 1)'£..i Ii" 1...1 1+1 1 1+1 1n+ . i=O h =(_1)11+1J K (I) f (II+I) (I)dlII! l,k. .. a sO'hc:lllgthuhaiva thubacuav~tnii vi~tgoml~ithanhmOtsO'he.mg,do do tathuduQc: (1.8) h k-lll+1 ( 1)1 Jj()d "" - f( )1 f (;-I) ( ) ( )1f (;-I) ( )}t t+ ~~~tXi+l-ai+l. Xi+1- xi-ai+1 Xi a i=O}=1./. h = (-1)"+1f~:11+1,k(1)/(11+1)(t)dl, a nghlala, d~ngthuc(1.1)la dungva Dinhly 1.1duQcchungminh.- H~quasaudaychomOtd~ngthuctichphankhacvdi (1.1)seheru ichtrongcacph§nsau. ryMTdr/,,//-1/,,(('Iff-/'A/'rln f(J(d(il.}t;(J(t)k; Tran~lQ Htimi 1.1. VdiClJnggid thiefcuadjnh(v 1.1,taco : ff({)d{+~(-;n~tx,- a,)J - (x, -a",)J }f(j-I) (X,)](1.9) b =(_1)11JKn,k(l)f(l1) (I)di. a Chung minh. Tli (1.1) taxet s6 hC;lngthil hai va vie'tno thanhtc5ngcua hai s6 hC;lng n ( I)J k-I8 8 -" - "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/)} 1=0< . ¥ k-I /I ( I)'"" - . f( )/ f U-') ( ) ( )J f U-') ( )}+L..L..~~X1+1-ai'l. xi+l-x,-ai+1 X;.i=OJ=2 J. Baygio81du'<Jcvi€t l,;ti .. k-I 81=(a-a,)f(a)+ L(xi -a'+I)f(xi) i=1 k-2 +2)~(Xi+l -ai+,)f(x,+,)}-(b-ak)f(b) i=O k-I =(a-a)f(a)+ L(x; -ai+,)f(xi) ;=1 k-I +L{-(xi -aJf(xi)}-(b-ak)f(b) ;=1 . k-I =(a-a,)f(a)- L(ai+1 -aJf(xi)-(b-ak)f(b). . 1=1 Cling v~yvoi 82du'<Jcvi€t lC;li k-I /I ( 1)' .<; ,,- f( ); [ (/-I) ( ) ( );f U~')( )} L 2 =L..JL..J --=--,tXi+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. =I (-~r{(Xk-ak)/ fU-I)(Xk)}+II (-~r(cXi+1-a;+,»)jU-I)(X'+I)} )=2 J. 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)})=2 J. ,=1)=2 J. - 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)} 1=2 j. /=2 J. . +~[~(-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-akYfU-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),taco k-I 8, +S2=(a-a,).!(a)- I(a,~,-a,).!(x;)-(h-ak).!(h) . ;=1 +I (-~r{(b-ak») fU-')(b) - (a -al») j<)-I)(a)} )=2 J. r ..------- '1:)~.kH.T(JN~IEN THLT\lIEN - OOO~86 J [1M1rlrt/Ift(/",,('(fr/'Ji/uin !.Jai @.)b(J,~t,,: Trang 12 +%[~(ii' kx,-a,J' - (x,-a",dfU-"(x,)] I ~ -{(a, -a)f(a) +~(a", -a,)f(x,) +(b-a,)f(b)} +I (-?l [-(a-a,)} fCj-I) (a) 1=2 J. +I (eXi~a,)} -(Xi -ai+I)}}JU-I)(xJ+(b-akf fU-')(b)] ,=1 Chli Y r~ngXo=a, ao=a,Xk=b Vaak+1 =b taco 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 +I {(x;-ai)} -(Xi -ai+I)}}/U-I)(x;)+(b-ak)} fU-1) (b)] ;=1 k =-I(ai+1 -a;)f(xi) ;=0 +I(-~r [-(xo-a,YfCH)(xo)i=2 J. k-I +I (ex,-aJ/-(x, -aj+,y}rU-')(x;)+(Xk-akYfi-I)(xd] ,=1 k ' = - I(aj+1 -aj)f(x;) ,=0 11( 1)i [ k ]+I~ I {(Xj-aJl -(x; -a;+,f}fu-')(xJ1=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,+82vaoso'h~lOgthU'haicua(1.1)tathoduQcd~ngthU'c(1.9). Bay gio tagia Sltrangcacdi~mchiaXi cuaphanho~chlk laceSdinh, ta tho duQc h<$qua gall. He !loa 1.2. Cho lk :a=xo <XI <",Xk-I<Xk=b La meltphiin ho(}chcua do(}n[a,b]. Ne'u f: [a,b]--+IR gi(fngnhll tron!?dinh Ly1.1.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. Chncac di~ma; U=O,...,k+l)nht!sau: , a+xl Xi-I +X, . ao=a,al=-,a;= , (z=l,...,k),2 2 Xk-I +Xk ' ak = va ak+'=b. 2 D~ngthU'c(1.9)vi€t l~i (1.12) h h ff(t)dt+81 +82 =(-1)" fK",k(t)f(II)(t)dt, a a vdi 8, +82=f (- ?J [ i:kXi - ai)/ - (Xi - ai+I)/ }fU-'J (Xi ) ] . /=1 }! i=O Ta chia so'h~ngthil hai 81+82cua (1.9) thanh3 so'h~ngnhugall: ~Jnlitj{( 'uoAl1I!iJJ&ui.f1JlI}(L(~ . - "",,-," "'-~ 11 (]) ./ [ k k } ] - , , (-I) SI+82=~-j!~ xi-a;)./-,(Xi-ai+t)./fJ (X;) k ' =-2: {(Xi -a;)-(Xi -ai+I)}[(Xi) ;=0 + f (-?} kb-adJ fU-')(b)-(a-aj)} fU-I)(a)} ./=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 =-(a-a,)f(a)+(b-ak)f(h)+ I{(i,-a,)-(x,-a/+,)}f(x,) 1=1 - 1 f k-I }- 2lhof(a) + ~(hi + h,-,)f(x,) + hk-J(b) . S6h~ngthuhai 81+52cua(1.9)vdi i=O,i=k,j~2: (ii) I(-~r (cb-ak)}fU-')(b)-(a-a,)J fU-I)(a)}}=2 .1. =~(_])I {hl- rU-I) (b) - (-I )JhllU-I)(a )}.L. 'I2J k I. O.J=2.1. S6h~ng thil ba 8,+52cua (1.9)vdi ]~i~k-l, 2~.i~n: (iii) ~[~(-;r {(x,-aY - (x,- a,.,)'}Iii"(x,)] =I [ i: , (~!'21)ijh~1-(-I)J h/lrCH) (XI)] ' I~' 1~2.1 f)~tbasO'h~ng(i ), (ii ) va (iii ) vaa 81+52ta 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; Trang15 +i (-?J {(b-adlf(l-I)(b)-(a-al)J fU-') (a)} J=2 ./. k-I [ n ( 1)J { } ] - J J (j-I) +~~fl(Xi-ai) -(xi-ai+i) / (Xi) 1 { k-I , }=-2 ho/(a) +~(hi+hi-I).I (x;)+hk-d(b) +~(-l)J {hJ- fU-I)(b)-(-l)JhJ fU-') (a) }L. '12l k 1 0,J=2./. +I [i (~;:{h;~~1 - (-l)Jh/ lr(j-1)(x/)]1=1l=2 J. =~ [ ~(-l)l {h~ -(-l)J hJ }jU-1)(X) ] L. L. 'I2l /1 1 /1=0 J=I J. =~(-1)J ~ {h~-(-l)JhJ }rU-1)(x). L."2lL.11 I. IJ=I J. 1=0 Cu6i cling ta thudt(Qc(1.11)b~ngcach thay 81+82vao (1.12).. Tn(ong hop talay cacdi~mehiax-eua I cachd~utatlmduoeheqUa. 1 k , .. salt: He qua1.3.Cho (1.14) Ik : Xi =a+:{b~a).i =0,...,k lamQtphfin hOi;lChd~ucua doi;ln[a,b]Va f: [a,b]~ IR sao cho In-I) li@nt\,lctl1yi$td6i tr~n[a,b].Khi d6tac6dAngthd'c: (1.15) J/(t)dt+t(b-a ) .I 0 .I~I 2k x;,[ - Iu "(0)+~I(b)] h = (-1)" J KII,k(t).T(II)(t)dt. 11 . Chu yr~ngsO'hi;lngthli hai cua(1.15)chI chliacacdi;lohamcffpIe ti;li tfftcacacdiemtrongXi' i =1,...,k-1. Chung minh. Sa d\,lng(1.14),tachuy r~ng :YM1d(j,~1(I"f'(' (ir-/' .;'/,(j,J/ !oa; (iM~()fr:Jt; TranR-16 b-a b-a ho=x!-x() =-, hi =xi+!-x/:::::- k k va b-a hi! =Xi -XI-I ""k,(i=t,...,k-l) (1.16) vathe'vao(1.11),taco: " II ( b . ) .1 fI(t)dt +I ~ (1 .1=1 2k . x ;![-f'HI(a)+ ~k-l)' -1)I'HI(x,)+(-I)J fU-"(b)] " =(-1)" fKII,k(t)f(II)(t)dt. a Tinh tacindongian,tUdaytasuytud~ngthuc(1.15).8 Congthucgi6ngTaylorsaildayvoi phfindutichphanclingdung. He gmi 1.4. Cho X :[a,y]~ 1ROJ d~whelmdin clip n sao cho g(lI)lien t~lCtuy~t at)'ltren ju,y]. Khl d6, wYimql 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 (1.18) II (-1).1 [ k ]g(y)=g(a)- ~fl t=o{(Xi- aJ1 - (Xi -ai+l)l }g(J)(xJ y +(-1)" JKII,k(y,t)g(II+)(f)dt. a Chung minh cua(1.17)va(1.18)dlivcsuytn!ctie'ptu(1.1)va(1.9)Hin luvtb~ngeachclwn f =gl, b:::::y..