Luận văn Đánh giá cho các phép biến hình K-Á bảo giác những miền chứa trong hình vành khăn

3.Caedanhgia eholapF 3.1Danh gia q{f) D!nh Iy 3.1: Gia saAla hinhvanhkhanQ<Izl<1 bj cat thee p nhat dc;mg(1.1) co thebi~nbaagiacboih IEmhinhvanhkhanqo<ItI<1 bj cat"theop nhat dQc cac tia ban krnhsac cho cac duangtrim Izl=Q,Izl=1tudngung voi It1= q0' It1= 1va diem z = 1 tudngungvoi t = 1. Khi do, voi cac ky hi~uaph~n 1 ta co (3.1) 1 q:;~q(f) ~QK ,Vf E F. Hdn nOa, K IIK-lq(f)=qo f(z)=g[h(z)],ZEA,g(t)=tt , ~ 11 q(f) =Q K fez)= zlzlT ,z E A. Chung minh 1 ~. ~zOQ 11 ~ Hinh 9 Ap dl:lngh~qua2.4cho f E F , taco : 2 2 77:q2~ trQK - S(f{;)K ~. BwOq CJ 1 1 ~ q ~QK . 1 COngthee h~qua 2.4, q=QK fez) = azlzlt-l ,\aJ=1. 21 Nhungvoi f E Fta co f(1) =1nen =>fez) = Z\Zlt-l. Vi f.h-1la PBHK-ABG mi~nBo=h(A)lenmi~nB =f(A) nenap dl;lngbed~2.9taco : ! S ( ~ ) K ~ q "? q(;,vf E F , q qo voi q =q(; ~ f.h-1(t) =atltt-1=ag(t), tUc fez) =ag[h(zj] . Do f(1 )=1nen ta da chungminhxong (3.1). 3.2Danhgia di~ntich: D!nh Iy 3.2: Voi cac kr hi~u O<Q<R<l;O<Q<r<l,VfEF, ta co: da gioi thi~u, \fR thoa (3.3) s(f) ~ s(r,f)'; nrY.[I- ;~J ' S(B)~1llI'[Q~ -I} (3.2) (3.4) 2 ( R ) K 2 TCq2 Q sS(R,f) S TCRK. 1 Dc1ngthuc(3.2),(3.3)va (3.4)xayra~ fez)=zlzlt-lhay q= QK . Chung minh * Ap dl;lngbe d~2.8, ta co : 2 2 ,,~trq2(~r +S(f)(~Y , Suy ra (3.2). 22 D~ngthuGxayra~ fez)=az!zlt-I, al=1. Dof(1)=1nena=1=>fez)=zlzlt-I. * Do(3.2)taco: S(B)=ff(1-q2)-s(f), ",,(I-Q2)-[ nr%(l- ~H. Vi r <1 nenguyra(3.3). Tudngtl!:d~ngthuGxayraa (3.3)~ fez) =zlzlt-I. * Xet Al =A (J {z:IziR . Ap dl)ngbe>de2.7hoc;ic2.8cho f E F I~nluQtlenmienA1va A2'ta co: 2 S(R,f);' nq'(~r' 2 ff ~S(R,f{~)K. => 2 S(R,f) 5, ffRK . TCtdoguyrabatd~ngthUGkep(3.4). I D~ngthuGxayra ~ q=QK. 3.3C~ntrencho m(R,f)vac~nduoicho M(R,f). £)!nhIy 3.3: a)Voi cackrhi~uda neu,V R, Q<R<1,VfEF, taco : (3.5) I m(R,f)5,RK , 23 (3.6) I M(R,f);" q(~r . Dlingthucxayra0 (3.5)ho~c(3.6) ~ f(z) =zlzlt-I. b) HemnO'a,n~u Q <R <r ho~ctudng ung r <R <1 thi ta con co cac danh gia sac hdn : (3.7) m(R,.n"q(~r ho,,"ctllongling (3.8) M(R,f)~RK . Dlingthucxay ra 0 (3.7)ho~c(3.8)~ f(z) =zlzlK-I . Chungminh a)D~t AI =An{z:lzl>R},A2=An{z:lzl<R}, Bj =f(A),j =1,2;feF,Q <R<1, mj =min~rt1:we Ej },Mj=max~rt1:we Ej },j=1,2, Ej la thanhph~nbi€mcua Bj tudngung voi duongtron Izi =R (n~uR:;t:rthi ro rang E1 =E2). Ta co:E(R,f) = EI U E2 da neutrongph~n1 Khido, VfeF,VR:Q<R<1, taco: (3.5a) m(R,f) ~mj,M(R,f) ~Mj,j =1,2 Ap dl:lngh~qua2.5choPBH f e F tren mi~nAI,VR,Q <R <1,ta co : (3.5b) 2 2m - 1> 1--->.. 2 <RK_~-;ml - , RK Tu (3.5a)va (3.5b)suyra(3.5). 24 Ap dl:lngtucmgWcho1E F mi~nA2ta co : (3.6a) 2 ( R ) K 2 2 2> 2- , M2 ';?~=:;.M2 -q Q2 - RK QK TCt(3.5a)va (3.6a)ta chungminhdUc;1c(3.6). D~ngthuGxayra0 (3.5)hoc)c(3.6)~ fez) =azlzlt-l. I Do /(1) =1=>a =1tUGfez) =zlzlK-l. b)N~uQ<R<r thiB2la mi~nnhi lienva 1 PBHBG h bi~nB21enhinh vanh khan B;: q<ItI<r . Dotfnhddndi~ucuamodunnen (3.7a) m2 ~r. , I ' Ap dl:lngbe de 2.6 cho PBHK-ABG h.f ta dUc;1c: (3.7b) ~~ ( R ) K q Q , ~r~q(~r Do (3.5a),(3.7a)va (3.7b)nenguyra(3.7). D~ngthuG0(3.7)xayra~ (3.7a)va(3.7b)xayrad~ngthuG, { hew)=aw,lal=1~ II K-l hf(z)=bzz II K-l b II K-l => af(z)=bzz =>f(z)=-zz . a Do 1 (1)=1 nen E.= 1. a V~yd~ngthuGxayra~ fez) =zlzlK-l. 25 TUC1ng W voi r<R<1ta S8 chungminhdu<;jcdanhgia (3.8)cung k~tlu~nv~truongh<;jpxayrad~ngthuc. 3.4. C~ntren cho M(R,f)VaD(f), c~ndooi cho m(R,f)va danh gia If(z)l. Dinh Ii 3.4: Voi caekrhi~udaneu,'v'R:Q<R<1,'v'fEF,tac6: (3.9) M(R,f) ~u <...<Uj <Uj-l <...<Ul <1I (3.10) m(R,f)~V>...>Vj >Vj-l >",>Vl >q. Voi: ~ q u,~T(p,R',q),v,=+Ht',q]' ui =T(p,R!ic,vi-')'Vi=+.(~)!ic, U~-,r~2,3,..., U =U(K,p,Q,R,q)= limuj'J~'" V =V(K,p,Q,R,q) =limvj"J~'" Chungminh Xet AI =A!1~zlR},BI=f(AI),Bz=j(Az). GQi r~ va r~ la bien cua 81va 82tUC1ngung voi bien Izi = R cua A1 va A21 M'(R,f) =max~wl:wE r~},m'(R,f) =min~~:wE r~}, MIt(R,f)=max~~:wEr~},mlt(R,f)=min~wl:w E r~}. Khi d6 ta c6 : m(R,f) >q,M(R,f) <1,M'(R,f) ~M(R,f), mlt(R,f)~m(R,f),MIt(R,f) =M(R,f),m'(R,f) =m(R,f). 26 XetPBH f mi~nA21emB2vatheobed~2.10,taco: M"(R,f):$;T(P,RYK,mil). DoUnhddndi~u3 cuahamT va m" >q nen: (3.9a) M(R,f) =M"(R,f) <T(p,RYK,m")<T(p,RYK,q)=Ul' " XetPBHf mi~nA1lenB1va theobed~2.11,taco : q .m'(R,f)? [ ( Q ) YK ~ ]T p, R 'M DoUnhddndi~ucuahamT va ~ >q nen M (3.10a) q q =v,' m(R,f)~m'(R,f)~ TH~t',~r+Ht',q] Gia sa (3.9a) va (3.10a) dung cho UjVa Vj Wc M(R,f)<uj va m(R,f) >Vj' ta S8 chung minhchungcOngdungcho Uj+l vaVj+l' Th~tv~ydoUnhddndi~ucuahamT va m"(R,f)?m(R,f»Vj taco " f) YK ") T( RYK ) ( R ~ )M(R,f)=M (R, :$;T(p,R ,m < p, ,m <T p, ,Vj =Uj+l' Tvdng tv, cOng doUnhddndieucua va L>!L>!L taco . . M' M u. ] q q > q =Vj+1"m(R,f)=m'(R,f)? [ ( Q ) YK L ] > [ ( Q ) YK,!L ] T [ P,(Q)YK,!L]T p, - , , T p, R M R UjR M V~yta da chung minh: M(R,f) Vj'vi. 27 Tu tfnhddndi~ucua hamT, voi UI q lam khoi diem, ta de dangthg.yUj la daygiamva bj ch~nduoiboi M(R,f), vi la daytangva bj ch~ntrEmboi m(R,f). Dodot6nt9icacgioih9n U = limuj'V = limvj' j-->oo ]-->00 Dodo taco cacdanhgia(3.9),(3.10). H~qua3.1: \:IR, Q<R<1,\:IfEF, ta co: (3.11) (3.12) M(R, f) <T(p,RJiK,0) q m(R,f) > [ 1 ]( Q ) JiK T p, R ,0 ChtJy: Sau nay(xemh~qua3.7)ta S8chi racac c~ntrongh~qua naylat6tnhg.trongs6 cacc~nphl;Jthu(>cungcacd9iluc;mg. H~qua3.2 : Voi cacgiathi~tda neu,taco : M(R,f) <4x'RJiK, h ( R ) JiK m(R,f) >4 Pq Q . (3.13) (3.14) Cac h~s6 0v~phailat6tnhg.trongs6 cac h~s6 chiphl;Jthu(>cp. Chungminh Do (3.11)va tfnhchg.t6 cuahamT (xem2.1.2c) taco (3.13). TudngWtU(3.12)vatfnhchg.t6 cuahamT suyra (3.14). Do tfnhchg.t7 cua hamT nen limT(~/,O)=1va do chuycuah~ r-->O 4 P r qua3.1tathg.ycach~s6tronghaidanhgianaylat6tnhg.t. . 28 H~qua3.3: Vaicackyhi~udaneuaphfIn1,theeb~td~ngthuctamgiacta co: D(f):::;2M(r,f),vf EF. D~ngthuccothexayravaip= 2 (xemchungminhdinhIy3.7). TCldo,do (3.9),taco: D(f) <2T(p,rI/K,q), vf E F ho~cnho(3.11)va(3.13),taco: (3.15) (3.15a) D(f) <2T(p,rI/K,0), D(f) <2.41!prl!K,Vf E F. Vaip=2h~56trongdanhgiacu6iclInglat6tnh~t. H~qua3.4: Vi m(R,f):::;If(z)1:::;M(R,f)vaiR =IzI,zE A, NentheedinhIy3.4taco: q T ( P, ( Q J l!K J <If(z)! <T(p,lzI1!K,q), Izl ,q Ho~ctheeh~qua3.2taco (3.16) 4:y'q(l~t <I/(z)1<4Y.olzlY.c. Cach~56trongdanhgianaylat6tnh~t.. 29 3.5.BAtphudngtrinh chinh Trangml;jc naytaS8apdl:mgIythuy~tdQdaiclJc tridagiaithi~utrang ph~n2.1.3vabed~2.12dethi~tI~pcacb~tphudngtrinhchfnhrangbuQc h~uh~tcacd9iIu(,jngd~ctrungchomi~ng6cvami~nanh. D!nh Iy 3.5: Vai cackyhi~uvagiathi~trangph~n1, vf E F ta co: ff- ap 1 ff- ap c ap 1 1-In2 -+ In2-+-In2 -::; Kffln-. In! d In~ q Inl q q r Q Q (3.17) ff - ap 2 ff - ap 2 ap 2 Kff 2 ~ (1- d) + 1 (c- q) +1 (1- q) ::;2 (1- q ).In- In- In- r Q Q (3.18) Chung minh ""'" Sj .'. ...~......... A . (8) """"'" J "" S ". ...::::>.::::::: J z , """""""" ,,"""""~"""' , H. 1 \. 'J i 30 0 In; c:J I R". IInIn; {;-a J {: -aJ 2a Hinh10 Goi: + S} latUgiacGongnamtrongA gioihc;lnboi Izi=1,Izl=r, . 2TC .2TC argz =a +(j -1)-,argz =-a+ J-. p P + S') la tUgiacGongnamtrongA gioih(~mboi Izi=r,lzr=Q, argz=a +(j -1)( 2; Jargz ~-a +je;)j =1,2, ,p. +S"}latUgiacGOngnamtrongA gioih(~mboi Izi=1,Izi=Q, . ( 2TC J . ( 2TC Jargz = -a +(j - 1) P ,argz =a +(j - 1) P . PBH Vf E F bi~nmi~nA lenmi~nB, khido : +S}trothanhH} co mQtc9nh namtren I~=1va mQtc9nh namtrenQ'j. +S~tro thanhH'}co mQtc9nh namtren Iwl=qva mQtc9nh nam trenQ'j . +S;tro thanh H}" co mQtc9nhnamtrenI~=1va mQtc9nh namtrenI~=q . T6n t9i cac PHBHBG trongdo cac dinh CU8tUgiac Gongthanh dinh hinhchOnh<;1t + h,bi~ns)en hinhchi]nhi;ltR c(lnh{; -a)vaIn!: sac choc<;Inhnamtren[zl ~ 1lingVO;C<;Inh{; -a l 31 +h, bi~nS;len hinhchu nh~tR' C<lnhz(; -a )vaIn ~ sac choe<;InhnamtrenIzl =Q(Jngvo;c(lnh2(;-a). +h3 bi~n Sj':len hinhchOnhatR" canh 2a va In~sao cho .. {2 c~mhnam tren Izi = 1ung voi cc;mh2a. Dodo: + PBH g1tU Hj len R co the xem la hQpcua PBHK-ABG r1 va PBHBG h1.Vi v~y,gl = h1.f-1 la PBH K-ABG . +g2 = h2.f-1 : PBHK-ABGtUH'jlenR'. + g3=h3.f-1: PBHK-ABG tU H"j len R". Viv~y,apdl;Jngbod~2.12voi P(W)=I~IVOiWEHj taco: Sp(Hj) "2(;-~ J [ fP(W)ld~J ' =2(;-~J ( fldWIJ 2. KIn- r KIn- r I~ r r Vi w=tehp nen Idwl=leiq>dt+iteiq>dtl=leiq>lldt+itdtl=ldt+itdtl=~(dt)2+(tdq»2 ~.Idtl. ~ I~I;, I~I , Suyra:Sp(H)? 2(~-~] ( fdtJ ' ~ {~-at~. KIn- d t KIn.!. d r r 32 ( ff J 2 --a Tuong tlJ: Sp(H'j);::: P r In2,:" Sp(H"j);::: 2a1 In2!. KIn- q KIn- q Q Q M~tkhac: , " 1 pSp(H) +pSp(H) +pSp(Hj):::;2ffln-. . q Dodo taco batcitIngthuc(3.17). N~ufez) =zlzlK-l thic = d =rKvaq=QKnencitIngthucxayra. Lylu~ntuongtlJtrennhungvaipew)=1,WedQdo Euclide,tachung minhdU9c(3.18). 3.6Dimhgiac(f)vad(f). Vi d(f):::;M(r,f)va e(f);:::m(r,f)vai f EFnen c~ntrencuad(f) va c~nduaicuac(f)cothelaytCtdinhIy3.4vacaeh~quacuano.Trangph~n naytachixaydl!ngc~nduaicuad(f)vac~ntrencuac(f). H~qua3.5: Vaicaekyhi~uvagiathi~ttrangph~n1,Vf E F, taco: (3.19) ~ (kff ap Jdel) >qVff-ap Inq-lnQ , (3.20) e(f) V 1:~§J( Kff-~ J q(f) <q ap Inq lnQ . Chungminh a) Tv (3.17)suyra: 1 1 1 ff -ap In2- + ap In2- <Kffin-, In! d Inl q q r Q 33 ~1 I 2 1In-; I Kff - ap In -, 21.-< 1 1 q In d 7r-ap In- In Qq ~ ~ ( KTC ap ) d >qVTC-ap Inq-InQ GOngtU(3.17)suyra: ff-ap 2C ap 21 1-In -+-In -<Kffin-, In~ q Inl q q Q Q ~ r In- In2~< Q I Kff - ap I In2l, q ff-ap Inl Inl q q Q In~ C -.iL (KTC - ap ) - <qV TC-ap Inq InQ q ~ Ta thflyk~tquathuduc;1Ca(3.19)va (3.20)c6nhuc;1cdiemlacacc~n cuad va ~ deuphl,JthuQcvaoq lad<;liluc;1ngchuabi~t.BaygiGta c6the q khacphl,Jcdieud6nhobode2.4boi: £)!nhIy 3.6: Voicacgiathi~tvakyhi~unhua ph~n1,Vf E F, tac6: (3.21) Kp1l: del) >r1l:-ap, (3.22) Kp1l: :~ «~t~. 34 Chungminh ". S .... ~ "" A zO° B Oq ~-------- Hinh11 GQi Sj 113tU giac Gong gioi h9n bai 27r .27r Izl=1,lzl =r,argz =a+(j -l)-,argz =-a+ J-. p p 1 S" t O J > \J"- . S J1 PBHK-ABGVf E F biE§nSj IEmS'j 113tU giac Gong co m(>tc9nh nam tren lui=1va m(>tc9nh namtrenQ'j. PBHBG h biE§nmi~nnhjliengioih9nbai lui =1 va Q'j lenhinhvanh khans <ItI<1 sao cho lui=1 tuongung voi ItI=1.Tu giac Gong S'j tra thanh S"j 113daibangco haic9nhnamtren ItI=1 va It\=s. Nhuv~yt6nt9iPBHK-ABG h.f biE§nmi~nSjlenS"j. TheadinhnghiaPBHK-ABGtaco: 1 " -m(S.):::; m(S). K } }(3.21a) Do bed~2.4taco: (3.21b) m(S):::; 27r1 . In- s M<1itkhac 35 (3.21c) 2 ( ff a J m(Sj) = P 1 ' In- r Ket h<;1p(3.21a),(3.21b)va(3.21c)taco: ff --a P ff-<- I - l' Kln- In- r s ~ In! ~ Kpff In!. S ff-ap r Mc;itkhacdotinhddndi~ucuamodunmi~nnhilientaco 1 1 -<-, d s Tv dosuyra lnl < Kpff In!, d ff-ap r V~ytaco (3.21). Lylu~ntudngtl!taco (3.22). H~qua 3.6: Neu a =canst, dc;itC = Kpff , cho 1-r= & ~ 0 thi thee (3.21)taco ff-ap (3.23) c c 1-d ::JC& khi & ~ 0 Tudngtu neua =canst,datC = Kpff vacho! -1=& ~ 0 . . ff-ap Q Tv(3.22) taco: (3.24) : -l«~r -1=(1+£)c-1~C& khi &40 Cac danhgia (3.23)va (3.24)lat6iuuv~b~cIanh~s6. 36 Th~tv~y,n~ulay p=1,ex~ 0 thi haidanh gia naytro thanh 1-d<K(1-r)+o(1-r) khi r~l, c ( r J [ r J . r q -1 <K Q -1 +0 Q -1 khl Q ~ 1. Trangkhidon~uxetPBH w= zlzIK-l(E F) thide dang thay c=d=rK, q=QK. TCtdo 1-d=1-rK =l-[l-(l-r)r ~K(l-r) khi r~l, ~-l=(;r-1~[1+(;-1)r-l~K(;-l) khi ;-+1. . De chiratfnht6iuucuacacdanhgiachaboidinhIy 3.4 va cach~ quacuanotaS8chungminhbod~sau 86 d~3.1: Ton t9i [, E F va r E (Q,l)sac chavaicackyhi~uquenthuQctaco: (3.25) d(r,fo)=r(p,rl!K,Ji) , (3.26) c(r,fo) =r(p,r~K,-fi) . .........--.............:A ..' ... (OQ)r Ii '\" ,."" ~ ChQn r =.JQ E (Q,l). Chung minh HInf1 2 :1- 5 1 ....... i 0')/1 "'1// 1 , Xet PBHK-ABG t=h(z)=zlzlTl bien A thanh A' la hinh vanh khan Q~ <ItI<1 vai p nhatcat d9ng 37 ii' - { ' 11 - h. ( '- 1)21r <- ,21r '- 12 }~l.- t,t-r ,a+} 5:argt- a+} ,}-"...,p.1 P P T6n t<;liduy nh~tPBHBG W=g(t)bi~nA' thanh B la hinh vanh khan q e ban kfnh { 21r }Lj = w:c5:lwl5:d;argw=--f;j=1,2,..,p;q<c<d<1 sac eho Itl=l tuong ung I~=1 va Q'j tuongung Li" Vi r=& nen ~* r =l.Q* . , 1 V~yA' W d6i xung quaduongtrim ItI=rK , Vi B eOng W d6i xung qua duong trim I~=..rq nen cd =(..rqY=q. Theo nguyenIy d6i xung va tfnheh~teua PBH hinh vanh khan rh <It\<1 phai duQe9 bi~nbaa giae len hinh vanh khan ..rq<I~<1 v6i p nhatcatdQCbankinh Lj ~ {w;,{qos;H,; d,argw ~ 2:oj ~ I,...P}. N~udc%lt[, =g.h thi theedinh nghia hamT(p,r.s) ta co (3.25), Do c =!I tanhElndude(3.26). d ' , H~qua 3.7: Cae e~neua cae danh gia (3.11),(3.12) va (3.15) v6i p=2 la t6t nh~t trongs6 cae e~nphl,lthuQeeungthams6. Th~tv~y,tru6eh~tta ehOyrangtheechungminhtrenanheuaeung C(a)={z/z=rei<fJ,-as;qJ5:aboi w=fo(z) laeung C'(a) ~ {WIw ~ [ie",- ; <;<p<;; } 38 Vi v~yn~uc6djnhQ= r2 vachoa ~ 0,WccungC(a) co v~diemz =r thi anh C'(a) cua no cOngphai thuv~mQtdiem,Wc q ~ O.Tu do n~uchQn 1=10vaR =r =&=consthitheo(3.11)va(3.25)taco r(p,rl/K,Jq)= d(r,fo) =M(r,fo) <r(p,rl/K,O). => limM(r,j~) =r(p,rl/K,O).a~O TudngW,theo(3.12)va (3.26)taco q [ ( Q ) 1/K ] <m(r,fo) =c(r,fo) =T[ q T p, -;: ,0 T p,rVK,~]. => r m(r,fo)- 1 al.To q -r[p,rl/K,O] 39