Luận văn thạc sĩ toán học -chuyên ngành Toán Giải Tích-Chuyên đề :Đánh giá lớp phép biến hình Á bảo giác lên vành khăn bị cắt theo các cung tròn đối xứng quay
28 Lu(m van Th(lc sj Toan h(Jc - Tntclng Thu(1n Chuang cA C DANH GIA LOP HAM G Trang chu'ang nay, chung toi danh gia cae d£;lilu'cjngd~c tru'ng cha mi~n chuffn cling nhu' moduli cua cae lOp ham G Vi~c danh gia ban kfnh Q (g), E G, dong m9t vai tro quail trQng trang vi~c danh gia cae d£;li lu'cjngkhac, VI the' chung Wi b~t dftu voi danh gia 5.1 Danh gia ban kinh Q(g) Dinh Iy 5.1 VcJicae giG thief va ky hi£lu iJ ehLtdng 2, V9 E G, ta e6 K S~~) (1+ r' (5.1) < Q(g) < /i-*, d6 dang thue trai xdy B = Bo va g(w) = alwIK-1w,w E 1 B, lal = va dang thue phdi xdy B = Bo va g(w) = blwlK-W,w E B, Ibl= Chung minh Ap d\lng b6 d~ 4.1 cha PBHKABG ngu'cjcf ta co 32 > 81 > 81 (Q~g)r+ps (R~9)r (Q~g))K +ps C2.~ psrIf E G, (doR(g) < 1) Tli day, suy fa c~n du'oi cua Q(g) Q(g) > = g-l, = (1+ S~~)rIf (5.2) Khi B = Bo va g(w) = alwIK-1w, la! = thl d~ng thuc Kayfa M~H khac, ne'u f.1 > 1, ap d\lng [13,dinh 19 1] cho PBHKABG E G ta nh~n dllc;1c 7r12 > 7rQ2 (g) {l K Suy fa Q(g) < {l-K Tli day, nhd [14,c6ng thuc 2.5], ta co d~ng thuc Kay fa B -k-1 = Bo ~ va g(w) = blwl w, Ibl= 1, wEB Ne'u {l = thl daub gia c~n tfen Ia hi~n nhien va d~ng thuc kh6ng th~ Kay fa H~ qua 5.1 VI 8(B) < 7r - 81 - p8, 81 (5.1) dzt(fc vief dztcJi d(lng P8 ( Q(g) > qK - -:; > 7rq2, cg,n dztcJicua Q(g) -if ) (g E G), (5.3) suy Q(g) > Deing thac (5.3) ho(ic (5.4) xdy {:} (5.4) qK B = Bo va g(w) = alwIK-1w, w E B, lal = 5.2 Danh gia c~n dtioi ban kinh R(g) K Vi R(g) > Q(g), \/g E G, tli dinh 19 5.1, ta co R(g) > (1 + S~~))-2 M~t khac, ta co dinh 19 sail: Dinh Iy 5.2 V6i caegid thief va ky hi?u iJ chztdng 2, \/9 E G va > 0, ta co cae danh gia: 1f R(g) > R(g) > ( 1- 8(B) ) 82 ( 82 - p8 29 81{l KI ) (5.5) j If (5.6) Cluing minh Thea (5.2), voi > 82 a 1< + p8 ( R(g) ) > 81 ( Q(g)) -* 1< >R (5.7) (81+ p8) Suy fa R k 81 (1'*)K 1< + p8 ( R(g) ) D Tli day suy fa (5.6) 5.3 Cae danh gia khae eho Q(g), R(g) va Ig(w)1 Blob Iy 5.3 VJi cae gia thitt va ky hi£1u(J chZlcJng2, 'l/g E G, ta co -x Q(g) < J-L RK(p, Iwl,q) < Ig(w)1< RK (p, 1~I,q) - RK (p, -K 1;I,q Q (g) < f-l , (5.8) , (5.9) ) (qK ; v6it' =T [p, C~~i I) k , q] Tli tu'dng t1/ nhu' tren, \:/gE G va \:/w E B, ta co -k Q(g) q I ( Ig(w)1 ) = R(p, t, q) > R ( p,~, q) , Ke't hcjp voi (5.1), suy cac c~n tren cua Ig(w) i (5.8) Tu'ong t1/, nho ba't d~ng thlic (4.22), ta co th~ chi cac ba't d~ng thlic (5.9) f)anh gia c~n du'oi (5.10) d6i voi Q(g) duQc suy tr1/c tie'p t11(5.9) D H~ qua 5.2 Tit (3.24), ta nh(m du(lc cac danh gia ddn gilln v fii cac gill thief va ky hi~u iJ chuang 2, \:/gE G, ta co 4-*lwIK < Ig(w)1 < 4{fQ(g) C~I) K < 4*p-k K K K (~) 4-P dK < R(g) < 4P Q(g) 4-~ (c ) _2f,' CJ K K < C~I) K, K 4P jL-K ( ~) ' < Q(g) < R(g) (5.11) (5.12) (5.13) , (q:r< Q(g) (5.14) Tit (3.12) va (3.16), ta thfly rling cac h~ slf chi phl:l thuQc vaa K va p trang (5.11)-(5.14) la tot nh{{t Chzl y Truong hcjp cac ph~n bien o-jthoai hoa p di~m roi r(;lc,hi~n nhien danh gia (5.11) vftn dung ne'u ta thay cac da'u < bdi < Nhu' v~y, bAng cach thac tri~n lien t\lC ham z 31 = (w) t(;lip di~m bien dfi nell, ta thffy (5.11) v~n dung cho PBHKABG d6i xung quay p lfin z = g(w) mi~n nhi lien B nQi tie'p tfong hlnh vanh khan q < Iwl < leD hlnh vanh khan Q < Izi < Chu j (5.14) co th~ s~c hon (5.4) q -+ 0, C -+ vdi di~u ki~n d = canst va" -q = canst c H~ qua 5.3 TruiJng hC;pC1, C2 va cac CJjlan ll1c;tla cac dl1(Jngtran Iwl = Qt, Iwl = va cac nhat cdt tren duiJngtran Iwl = R', ta co cac danh gia sau ~ ,1 Q'K < Q < Q K, (5.15) ,1 RK (5.16) R" Q') (p,R', Q') < R < RK (p, Q ~, Trang (5.15) dling thac traixay {::}B = Bo, g(w) = alwIK-1w, w E 1 B, lal = va dling thac phai xay {::}B = Bo, g(w) = blwlK- w, w E B, Ibl = Chang minh Th~t v~y, q = M1 = Q', C = d = R', J.t = J" S(B) = 7r(1 - Q'2), 81 = 7rQ'2, = O Do do, ap d1;lngdinh 19 5.1, ta nh~n du'Qc(5.15), cling vdi di~u ki~n xay fa ding thuc D Tu (5.9) va (5.15), nh~n du'Qc(5.16) H~ qua 5.4 Ktt hC;p(5.2) vai (5.9), ta tim l(Ii c(ln dual cila Q(g) co thi sdc h(Jn (5.1) nhu sau 82 > 81 81 ( Q(g) ) 1< + p8 ( R(g) ) ( Q(g) ) > 1< 1< K q -k -k p,~, q J.t ( ( + p8 R ) ) Suy -~ 82 - p8.R2 (p,~, q) J.t~ Q(g) > ( 81 32 ) H~ qua 5.5 Tit (5.9) va (5.10), ta nh(m dLt(le,nhiJ (3.12) va (3.17), cae danh gia sau day ddi vai Ide d{j h{ji I¥ ctla R(g), ~i~i va Q(g) lrong eae tntiJng h(lp gifJi h(ln K1f2 1- R(g) < 1- RK(p,d,O) ~ K[l- R(p,d,O)] ~ 2p n p(l-d) (5.17) d -+ 1, tlle la R(g) -+ d -+ CJ - Q(9) < - RK R(g) P, c' ( ~ ~ -; I, tlic fa ~i: i -; !{ [ 1- ~ R (p, ~,O )] ~ C ~ ~ (5.18) -; K [(1- R(p, d, 0)) + (1 - R (p,~, 0))] !{ 1f2 K 1f2 2p In p(l-d) + (5.19) 2p In p(l-~) ~ -+ e FJanh giG (5.19) n6i r2ingQ(g) ddn tdi ntu d 5.4 K1f28 2p In p(l-~) < 1-RK(p,d,0)RK(p,~,0) 1-Q(g) d -+ va ) -+ va :l -+ c Danh gia g6c md j3(g) R5 rang ta luan co < {3(g) < 21f, E G, nhien ta mu6n co danh gia p t6t hdn nhfi'ng tru'ong hejp nao Mu6n v~y ta dung phu'dng phap dQ d~li-di~n tich hay gQi Ia dQ dai Qtc tri Ahlfors va Beurling [1] d~ xu'ong nam 1950, giup giiH quye't nhi~u bai toan t6i u'u PBHBG Md rQng phu'dng phap cho PBHKABG, ta co b6 d~ sau: B6 d~ 5.1 Trong m(lt phdng z eho hlnh ehil nh~t D = {z = x + iyl < x < a, < y < b} 33 Gia sa ham so' W = j(z) th1!c hi~n mQt PBHKABG hlnh chTl nh(lt D ZenmQt ta giac Gong H cila m(it phdng W saD cho cac dlnh 0, a, a + ib va ib cila D ztm Zufft tu(jng ang V(ji cac dlnh WI, W2, W3 va W4 cila H GQi r ZahQ cac cung r H noi cc;mhWIW2wJi c(;mhW3W4cila H GiGsa co ham dQdo p = p(w) < Ip(r) = >0 lien tf:lC H saD cho lp,dW,< 00, V"{ E r va < SetH) = JJ HP2dudv < 00, W = u +iv lJ(it l p = inf l p ( r fEr ) Khi do, ta co Sp(H) > a K blpo (5.20) Ddng thac (j (5.20) co thl xay W4 W3 a + ib ib WI D ,Dx x a W2 Hinh 5.1: PBHKABG hlnh chu nh?t D Jen tu giac cong H Chang minh *Tru'dng h Denco l Do do, d€ p21f'(zWldvl y 1x E f, > ~ ta co a SetH) (l plf'(zJ[[dvl) a i J0 (l plf'(z)lIdYI) dx= i J0 (1 > pldwl) dx a 12 J > y;lp dx = a2 y;lp *Tru'ong hQp K > Xet T/= h(w) la PBHBG tu giac H leD hlnh chii'nh~t D' = {'TJ= S + it < s < a', < t < b'} I cho cac dlnh WI, W2, W3 va W4 cua H l~n hiQt tu'ong umg vdi cac dlnh 0, a', a' + ib' va ib' cua D' Ap dvng chung minh tren cho anh x~ ngu'Qch-I, ta co a' Sp(H) > bll~ M~t khac, anh x~ h f la PBHKABG hlnh chii'nh~t D leD hlnh chii' nh~t D' Den co a' 1a ->-b' - K b IBilt dAng thuc co d~ng !