Đánh giá phép biến hình á bảo giác lên hình vành khăn bị cắt theo các cung tròn đối xứng quay

IiI PBHKABGmi€nD Zenmi§nE cuam(it phdngw, saGehoIzi = R1va Izi = R2zanZur;tuangungvciibientrong C1va bienngoaiC2eilaE. Khi do,vciicaeky hi?unhutrangchuang2, taco ( R2 ) k ( R2 ) k S-(R2, f) >S+(R1,f) R1 +s(R,f) R ' (4.1) trongdo s(R, f) la flingdi?n richngoaieilat(tpdonggiciih(lnbiJi anh caenhatcilt trenduiJngtron Izl =R biJi f. 1 1 Ddngthuexay raq f(z) =alzlK- z+bvciicachiingso'a,b,(a#0). Chungminh.R5rangs(R,f) =S+(R,f) - S-(R, f). Ap d\lngb6d~3.2,taco ( R ) k S-(R, f) >S+(R1,f) Rl 0 Do d6 2 S+(R, f) >S+(Rj,f) (:J K + s(R, f). Clingtheob6 d~3.2,ta Surfa S-(R2, f) 2 > S+(R,f) (i)" > [S+(Rt,f) (~)k+S(R,f)] (i)k. Tli dotaco(4.1)voikhaDangxayfa d~ngthuctheob6d~3.2,nhtt da:lieU. ~ D 4.1 QuaDh~gifia q,m2va M1 Dinhly 4.1. Gill saA lithlnhvimhkhanQ < Izi < 1v6ip nhatcdtdf;mg (2.8),co thi biEnbaagiacblJi h Zenhlnhvimhkhan% < ItI < 1bi cdt rheap nhatcdtdflCcacriabankfnh,saochoIzi =Q, Izi = 1 l&nht(lt tl1(/ngungvdi ItI =%' ItI =l. Khi do,vdigill thiefvaky hi?ulJ chl1(/ng2, doLV(Yicacdc;liZl1(Jngd(ic tntngchoB =f(A),Vf E F, tacocacbatdcingthucdung 1 q <QK , M1_>K m2- qo' (4.2) (4.3) 1 M1 <T(p,QK,q), 1 !L <T(p,QK,q). m2 (4.4) (4.5) 1 1 Deingthactrong(4.2)xayra {:}j(z) =alzlx- z, lal= 1, (4.3)xayra {:}f(z) =alh(z)IK-lh(z), lal= 1, (4.4)xayra {:}f(z) = fo(z)v6ifo trongb6dl 3.4, (4.5)xllyra {:} f(z) =fo(z)vdifotrongb6dl 3.5. Changminh.*Ap d1:1ngb6 d~4.1cho f E F, chil Y S+(Q,f) > 7fq2, S-(1, f) < 7fMi = 7fva s(R, f) =ps, tasur fa ( 1 ) ~ ( 1 ) ~ ( 1 ) ~ 7f >7fq2 Q +ps R >7fq2Q . 18 1Tli doq<QK . 1 1 f)~ngthucKay fa ~ j(z) =alzlK- z +b,va8+(Q,f) =7rq2.Tli do 1 b=0va lal= 1VIM(l, j) = 1.V~yj(z) =alzlK-lz, lal= l. * Vi j 0h-1 la PBHKABG mi~nBo=h(A) lenmi~nB = j(A), lien apd\lngb6 d~3.3,co ( ) K m2 1 M1 K - qo. M1 qo m2 ~ D~ngthucxayfa ~ w =altlK-1tvdit =h(z). Honnuane'uIzi= 1 thlItI = 1va \wl= 1,suyfa lal = 1.V~yj(z) = alh(z)IK-lh(z), lal = 1. *Ap d\lngb6d~3.4taduQc(4.4)clIngvdidi~uki~nxay fa d~ngthuc. *Ap d\lngb6d~3.5vdiR = 1,ta duQc(4.5)clIngvdi di~uki~nxay fa d~ngthuc. 0 4.2 Danhgiaeaedi~nneh DinhIy 4.2. Vai caegiGthilt vaky hi?u iJ chU:l1ng2, \:Ir: Q < r < 1va vj E F, taco caedanhgiadung 2 ( R ) i p8 <82RK - 81 Q , 8(B) >82(1-R*) +81 [(~)k -1] , ( r ) * 2 81 Q <8(r,j) <82rK 1 M6i dangthactrenXGYra ~ j(z) =alzIK-lz+b wJi caehangsffa,b thichh(Jp. Changminh. (4.6) (4.7) (4.8) *Ap d\lngb6d~4.1,taco 2 2 S-(1,f) >S+(Q,f) (~r+psGr. Chli y s+(Q,f) =81,8-(1,f) =82,suyfa 82>81 (~)* +psG) * . 19 1 1 Tli doco (4.6)void~ngthucxayfa {::}j(z) = alzlX- z +bvoicac hAngs6a,b. BaygiGtasexacdinha vab. 1 -1" 1 GQit =IzlK z laPBHBGmienA tenhlnhvanhkhanQK < ItI < 1. Khi do,hamw =at+bbie'nmi~nA tenmi~nB. A \ OQ\ 1 1 "'~ '" / " ~ I / B ~\/ 0 \I b \I ,.:, I 1I I '-' I\ I \ '- I \ ~ I \ I , / " /'- /, '"' '" 1 Tli hlnhtfen,d~dangtinhdu'Qc1+q= lal+ lalQK, Ibl+lal= 1,tuc l+q lal= l«l),lbl=l-lal. l+QK *NhG(4.6),taco 5(B) = 52- 81- p8 2 ( R ) * > 82- 81 - 82RK +81 Q . Tli do co (4.7)voi khanangxayfa d~ngthucnhu'dfflieU. *YOi Q <r < 1d~t Al = An{zllzl<r}, A2 = An {zllzl >r}. Liln 1119tapd\lngb5 d~3.2cho j E F, cacmi~nAI, A2,taco 2 .£. S(r,f) >S+(Q,f) (;r=81 (;y, va ( 1 ) -k 82=8-(1,f) >8(r,f);: . Tli doco(4.8)voikhanangxayfad~ngthucnhu'dfflieU. 20 D H~ qua 4.1. V(ji cac gid thief va ky hitfu(j chlt(jng2, Vr : Q <r <1va "If E F, fa co caedanhgia 2 ( q2 )p8 <7rRK 1 - Q* ' S(B) >J[ [m~+q2 (~)I< - q2 - m~Ri< ] , ( r ) f< 2 7rq2 Q <8(r,f) <7rrK. 1 M6i dang thac tren xdy ra <=}f(z) = a!zIK-1z, lal = 1. Changminh.Ap d\lngdinh194.2vachu9 7rm~7rq2,ta co (4.6a), (4.7a) va (4.8a)voi dAngthuc Kay fa nhu'da lieU voi chung minh tu'dngt\l'nhu'(4.2). D (4.6a) (4.7a) (4.8a) 4.3 C~n!renchom(r,f) va c~ndu'oichoM(r, f) DinhIy 4.3. V(ji cacgidthiefva ky hitfu(j chltdng2, Vr : Q < r < 1va "If E F, taco rs; 1 ( 1 )m(r,f) <V -; rK <rK , M(r,f) > J¥ (~)* (>q(~)*) . (4.9) (4.10) 1 M6i dangthacxdy ra <=}f(z) = alzlK-1z + b, v(ji cachangs6a,b thiGhh(Jp. Hdn mia,v(ji Q < r < R, "If E F, taco m(r,f) < Mj (;)K, (4.9a) vav(ji R <r <1,"If E F, taco M(r, f) >rK. M6i dangthacxdy ra <=}f(z) = alzIK-lz, lal=1. (4.10a) 21 Chang minh. *Ap d\;lllgdinh194.2,congthuc(4.8),voi chu9 1rm2(T,f) <8(T,f)<1rM2(r,f), ta nh~ndu'cjc(4.9)va (4.10)clingvoi di~uki~nxay fa ding thucnhu' chungminh(4.6). *Ne'uQ < r < R, thl mi~nAl khongchuadi~mbiencuaA, do d6 mi~nBl la mi~nnhi lien va bdi PBHBG h, mi~nBl bie'nlen hlnh v~lllhkhan M1 < ItI < p saDchoC1 tu'dnglingvoi ItI = ¥l. Ne'u M1 > m(r,f) thl (4.9a)hi~nnhiendung. Ne'uM1 < m(r,f) thlhlnh vanhkhanM1< Iwl < m(r,f) chuatfongBl va ligancachhaithanh ph~nbiencua Bl lien theotinhddndi~umodulimi~nnhi lien, ta c6 m(r, f) p h<- ay M1 - M1 m(r,f) <p, (4.11) trongd6ding thucxayfa {:}h(w) = aw, laJ= 1. M?t khach0 f laPBHKABGQ <Izi <r lenM1< ItI <p lien ;l«~r, (4.12) tfongd6ding thucxayfa {:}t = blzlK-lz, Ibl= 1. Tli (4.11)va (4.12),ta nh~ndu'cjc(4.9a)clingvoi di~uki~nxay fa ding thuc. Tu'dngt\1',tanh~ndu'cjc(4.10a)clingvoi di~uki~nxayfa ding thuc. D 4.4 C~ndlioi rho m(r,f) va c~n!renrho M(r, f) DinhIf 4.4(xem[14,tr.65]).Vdi caegiGthiefvaky hitlu(j chlt{jng2, "liT: Q <T< 1va"lifE F, taco M(r, f) <U < ...<Uj <Uj-l <... <Ul < 1, m(r,f) >V > ...>Vj>Vj-l >...>VI >q. (4,13) (4.14) trangdo 22 q UI = T(p,r1<,q),VI = [ ( Q ) 1< ] ' T P - ,q, r q , j =2,3,...,Uj = T(p, r1<,Vj-I), Vj = [ ( Q ) 1<,~ ]T P, r Uj-i , (K Q r q)= lirnu.i,U = U ,p, " .i~oo v =V(K,p,Q,r,q) = ~irnVj,.1~oo va T(p,r, s) la hamphl:ldur;cdjnhnghfatrangphdn3.2. Changminh.f)~t AI=An{zllzl <r},BI=f(AI), A2=An{zllzl >r},B2= f(A2), M"(r, f) =rnax{lwllw E 'Y~}, m"(r,f) =rnin{lwll w E 'Y~}, M'(r, f) =rnax{lwllw E 'Y~}, m'(r,f) =rnin{lwllw E 'Y~}, vdi 'Y~,'Y~l~nlu'Qtla thanhph~nbiencuaBI, B2 tu'ongling vdi thanh phfinbien Izi =r cuaAI, A2. R5rangVr :Q <r <1,taco m"(r,f) >m'(r,f) =m(r,f) >q, M'(r,f) <M"(r,f) =M(r,f) <1. Ap d\lngb6d~3.4va3.5chophepbie'nhlnhf E F caerni~nA2,Al vaapd\lngcaetinhcha't(3.7)cuahamT(p,r, s) taco M(r, f) =M"(r, f) <T(p,r1<,m")<T(p,r1<,q)=UI < 1 va q >m(r,f) =m'(r,f) > [ ( Q ) k !L ]T p, ~ ' M' 23 Vi Uj khi j -+ 00 bi ch~ndu'dibdiM (r,f) vadondi~ugiam,trong khi khi j -+ 00 bi ch?ntrenbCiim(r,f) vadondi~utang,dodot5n t~icacgidih~n ~imUj =U, ~imvj = V. .7---+00 .7---+00 Honnlia,coNI(r,f) v. D H~qua4.2.Ket h(lpdjnhly 4.3,djnhly 4.4vacactinhchift(3.7)va (3.21)cuahamT(p,r,s), tacocacdanhgiaddngiansau. Vr :Q <r <1vavf E F, taco 1 M(r, f) <T(p,rK, 0), rn(;,J) <T r' (~)1<,0] , F ( r ) * 1 1 V -; Q <M(r, f) <41'rK , 1 4-iq (;)" <m(r,f)<fj r* (4.15) (4.16) (4.17) (4.18) Chuf 1. Cacc(mtrong(4.15)va(4.16)la totnhifttrongso'cacc(m 1 1 phl;lthuQccungthamso: Cach~so'41',4-1'trong(4.17)va (4.18)la tot nhifttrongso'cach~so'chiphl;lthuQcp. BE chungto di~udo,taxetvi d\l sail. Xet A la hlnh vanhkhanQ < Izl < 1 bi c~tbdi p clingtroll Lj (j =0,...,p- 1)d~ng(2.8)vdiR2=Q. 1 1 PBHKABG Z = h(z) = IzIK- z bie'nA thanhA' la hlnhvanhkhan 1 QK <IZI <1bi c~tbdicacclingtroll { I 1r 7r }Cj= ZIIZI=Rl,et+(2j-1)p<argz<-et+(2j+1)p 11: 2 1 vdi 0 <ex<-, j =0,1,...,p- 1, Ri =RK =QK. P Sando PBHBG 9 bie'nA'len B la hlnhvanhkhanq < Iwl < 1bi c~t bdip do~n OJ ~ {w c<lwl<d, argw=/;} (j=O,...,p-l), 25 q < c < d < 1,saocho IZI = 1tu'dngU'ngIwl = 1va OJ tu'dngU'ngvoi Dj. Do s\fdO'ixU'ngcuaA' quadu'ongtroll IZI =Rl va do tinhduynha't cuag, taco B clingdO'ixU'ngquadu'ongtroll Iwl=Vii,nghlalacd=q va hlnhvanhkhanRl Vii}. V~y,nSud~th =9 0 h(E F) thl theodinhnghlahamph\!T(p,r, s), taco 1 Al(R, fd =d=T(p,RK, Vci) va m(R,h) q q = c=-= 1 d T(p,RK , Vii) q = T [p,(~)k , JQ ]' CO'dinhQ va R, choa -+0 theo[12,tr.l06],taco q -+ O. Nhu' v~y 1 lirnM(R, h) =T(p,RK,0), 00-;0 [ 1 ] . q Q K ll~ m(R,11)= T p,(R) ,0 . Di€u naykSthcjpvoi (3.26)kh~ngdinhchliy1. H~qua4.3.Vi m(lzl,f) < If(z)1<M(lzl, f), Vz E A, nen"If E F, ta co V(K,p, Q, Izl,q)< If(z)1<U(K,p, Q, Izl,q), q 1 [ Q 1< ] < If(z)1<T(p,IzIK,q), T P,(~) ,q q 1 [ Q 1< ] < If(z)1<T(p, IzIK,O), T P, (JZT) ,0 1 4-~q c~'r< If(z)1<4~lzl*. 26 (4.19) (4.20) (4.20a) (4.21) H~qua4.4.Vi m(R,f) <c <d<M(R, f), nentacocacdanhgia q T [P>(~)k >q] q T [p,(~)* ,0] _1 ( R ) -k 1 1 4 1>q Q < C <d < 41'RK , 1 d ~Q K 1-. C q 1 <C <d<T(p,RK,q), (4.22) 1 < C <d<T(p,RK,O), (4.22a) (4.23) (4.24) Tu:angtifchuy 1 co thl thdyrangcacc(intrang(4.20a)va (4.22a) III totnh{[ttrangsr/cacc(inphl;lthuQcLingthamso: Cachf$so'trang (4.21),(4.23)va(4.24)ia totnhdttrangso'cachf$so'chlphl;lthuQcp. 27 ._.