VỀ LÝ THUYẾT NEVANLINNA CHO HÌNH VÀNH KHUYÊN VÀ VẤN ĐỀ DUY NHẤT

["45 2.2. Hai \u0111\u1ecbnh l\u00fd duy nh\u1ea5t cho \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh Trong ph\u1ea7n n\u00e0y ch\u00fang t\u00f4i ph\u00e1t bi\u1ec3u v\u00e0 ch\u1ee9ng minh hai k\u1ebft qu\u1ea3 v\u1ec1 c\u00e1c \u0111i\u1ec1u ki\u1ec7n \u0111\u1ea1i s\u1ed1 li\u00ean quan \u0111\u1ebfn c\u00e1c si\u00eau m\u1eb7t \u1edf v\u1ecb tr\u00ed t\u1ed5ng qu\u00e1t \u0111\u1ed1i v\u1edbi ph\u00e9p nh\u00fang Veronese \u0111\u1ec3 hai \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh tr\u00ean h\u00ecnh v\u00e0nh khuy\u00ean l\u00e0 tr\u00f9ng nhau. 2.2.1. Tr\u01b0\u1eddng h\u1ee3p kh\u00f4ng x\u00e9t ngh\u1ecbch \u1ea3nh c\u1ee7a t\u1eebng si\u00eau m\u1eb7t G\u1ecdi D l\u00e0 m\u1ed9t si\u00eau m\u1eb7t b\u1eadc d trong Pn(C) v\u00e0 Q l\u00e0 \u0111a th\u1ee9c thu\u1ea7n nh\u1ea5t b\u1eadc d, v\u1edbi c\u00e1c h\u1ec7 s\u1ed1 l\u1ea5y trong C, x\u00e1c \u0111\u1ecbnh D. Khi \u0111\u00f3 Q bi\u1ec3u di\u1ec5n \u0111\u01b0\u1ee3c d\u01b0\u1edbi d\u1ea1ng: nd Q(z0, . . . , zn) = akz0ik0 . . . znikn, k=0 trong \u0111\u00f3 nd = n+d \u2212 1 v\u00e0 ik0 + \u00b7 \u00b7 \u00b7 + ikn = d, ak \u2208 C v\u1edbi k = 1, . . . , nd. n Ta g\u1ecdi a = (a0, . . . , and) l\u00e0 vect\u01a1 li\u00ean k\u1ebft v\u1edbi si\u00eau m\u1eb7t D (ho\u1eb7c Q). Ti\u1ebfp theo ta nh\u1eafc l\u1ea1i kh\u00e1i ni\u1ec7m h\u1ecd c\u00e1c si\u00eau m\u1eb7t \u1edf v\u1ecb tr\u00ed t\u1ed5ng qu\u00e1t \u0111\u1ed1i v\u1edbi ph\u00e9p nh\u00fang Veronese. Cho D = {D1, . . . , Dq} l\u00e0 m\u1ed9t h\u1ecd c\u00e1c si\u00eau m\u1eb7t c\u1ed1 \u0111\u1ecbnh, trong \u0111\u00f3 Dj \u0111\u01b0\u1ee3c x\u00e1c \u0111\u1ecbnh b\u1edfi \u0111a th\u1ee9c thu\u1ea7n nh\u1ea5t Qj trong C[z0, . . . , zn] b\u1eadc dj, v\u1edbi m\u1ed7i j = 1, . . . , q. G\u1ecdi mD l\u00e0 b\u1ed9i chung nh\u1ecf nh\u1ea5t c\u1ee7a c\u00e1c b\u1eadc d1, . . . dq, k\u00ed hi\u1ec7u nD = n + mD \u2212 1. n V\u1edbi m\u1ed7i j = 1, . . . , q, ta \u0111\u1eb7t Q\u2217j = Qmj D\/dj v\u00e0 g\u1ecdi a\u2217j l\u00e0 vect\u01a1 li\u00ean k\u1ebft v\u1edbi \u0111a th\u1ee9c Q\u2217j. H\u1ecd D c\u00e1c si\u00eau m\u1eb7t c\u1ed1 \u0111\u1ecbnh \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 \u1edf v\u1ecb tr\u00ed t\u1ed5ng qu\u00e1t \u0111\u1ed1i v\u1edbi ph\u00e9p nh\u00fang Veronese n\u1ebfu q > nD v\u00e0 v\u1edbi m\u1ed7i h\u1ecd t\u00f9y \u00fd th\u00ec c\u00e1c ch\u1ec9 s\u1ed1 i1, . . . , inD+1 \u2208 {1, . . . , q}, c\u00e1c vect\u01a1 ai\u22171, . . . , ai\u2217nD+1 l\u00e0 \u0111\u1ed9c l\u1eadp tuy\u1ebfn t\u00ednh. Cho \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh f : \u2206 \u2192 Pn(C), v\u1edbi m\u1ed9t si\u00eau m\u1eb7t D b\u1eadc d trong Pn(C) x\u00e1c \u0111\u1ecbnh b\u1edfi \u0111a th\u1ee9c thu\u1ea7n nh\u1ea5t Q, ta k\u00ed hi\u1ec7u","46 Ef (D) := {z \u2208 \u2206 | Q \u25e6 f (z) = 0 kh\u00f4ng k\u1ec3 b\u1ed9i}; Ef (D) := {(z, m) \u2208 \u2206 \u00d7 N | Q \u25e6 f (z) = 0 v\u00e0 \u03bdQ(f)(z) = m \u2a7e 1}. V\u1edbi m\u1ed9t h\u1ecd c\u00e1c si\u00eau m\u1eb7t D = {D1, . . . , Dq} trong Pn(C), ta \u0111\u1ecbnh ngh\u0129a Ef (D) := Ef (Dj) v\u00e0 Ef (D) := Ef (Dj). Dj \u2208D Dj \u2208D V\u1edbi c\u00e1c kh\u00e1i ni\u1ec7m v\u00e0 k\u00ed hi\u1ec7u nh\u01b0 tr\u00ean, n\u0103m 2021 ch\u00fang t\u00f4i \u0111\u00e3 ch\u1ee9ng minh m\u1ed9t d\u1ea1ng \u0111\u1ecbnh l\u00fd duy nh\u1ea5t cho \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh tr\u00ean h\u00ecnh v\u00e0nh khuy\u00ean v\u1edbi m\u1ee5c ti\u00eau l\u00e0 c\u00e1c si\u00eau m\u1eb7t nh\u01b0 sau: \u0110\u1ecbnh l\u00fd 2.2.1 ([39]). Cho f v\u00e0 g l\u00e0 hai \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh kh\u00f4ng suy bi\u1ebfn \u0111\u1ea1i s\u1ed1 t\u1eeb \u2206 v\u00e0o Pn(C) sao cho Of (r) = o(Tf (r)) v\u00e0 Og(r) = o(Tg(r)). Cho D = {D1, . . . , Dq} l\u00e0 m\u1ed9t h\u1ecd g\u1ed3m q > nD + 1 + 2nD2 \/mD c\u00e1c si\u00eau m\u1eb7t \u1edf v\u1ecb tr\u00ed t\u1ed5ng qu\u00e1t \u0111\u1ed1i v\u1edbi ph\u00e9p nh\u00fang Veronese trong Pn(C). Gi\u1ea3 s\u1eed f (z) = g(z) v\u1edbi m\u1ecdi z \u2208 Ef (D) \u222a Eg(D). Khi \u0111\u00f3 f \u2261 g. Ch\u1ee9ng minh. Ta ch\u1ee9ng minh b\u1eb1ng ph\u1ea3n ch\u1ee9ng. Gi\u1ea3 s\u1eed f \u0338\u2261 g. G\u1ecdi k l\u00e0 s\u1ed1 nguy\u00ean d\u01b0\u01a1ng \u0111\u1ee7 l\u1edbn, ta s\u1ebd ch\u1ecdn sau, tr\u01b0\u1edbc h\u1ebft ta ch\u1ee9ng minh v\u1edbi m\u1ed7i r, 1 < r < R : (q(k + 1 \u2212 nD)\u2212(nD + 1)(k + 1))Tf (r) \u2a7d nD2 k (Tf (r) + Tg(r)) + (k + 1)Of (r), (2.1) mD trong \u0111\u00f3 Of (r) x\u00e1c \u0111\u1ecbnh nh\u01b0 trong \u0110\u1ecbnh l\u00fd 2.1.3. Th\u1eadt v\u1eady, gi\u1ea3 s\u1eed f = (f0, . . . , fn) l\u00e0 bi\u1ec3u di\u1ec5n t\u1ed1i gi\u1ea3n c\u1ee7a f . G\u1ecdi Qj l\u00e0 \u0111a th\u1ee9c thu\u1ea7n nh\u1ea5t b\u1eadc dj trong C[z0, . . . , zn] x\u00e1c \u0111\u1ecbnh Dj v\u1edbi m\u1ed7i j = 1, . . . , q. V\u1edbi m\u1ed7i j = 1, . . . , q, \u0111\u1eb7t Qj\u2217 = QjmD\/dj , khi \u0111\u00f3 c\u00e1c \u0111a th\u1ee9c Q1\u2217, . . . , Q\u2217q c\u00f3 c\u00f9ng b\u1eadc mD v\u00ec mD l\u00e0 b\u1ed9i s\u1ed1 chung nh\u1ecf nh\u1ea5t c\u1ee7a d1, . . . , dq.","47 K\u00ed hi\u1ec7u (z0 : \u00b7 \u00b7 \u00b7 : zn) l\u00e0 h\u1ec7 t\u1ecda \u0111\u1ed9 thu\u1ea7n nh\u1ea5t trong Pn(C) v\u00e0 {I0, . . . , InD} l\u00e0 t\u1eadp h\u1ee3p t\u1ea5t c\u1ea3 c\u00e1c (n + 1)\u2212b\u1ed9 s\u1ed1 t\u1ef1 nhi\u00ean sao cho \u03c3(Ij) = mD, j = 0, . . . , nD \u0111\u00e3 \u0111\u01b0\u1ee3c s\u1eafp x\u1ebfp theo tr\u1eadt t\u1ef1 t\u1eeb \u0111i\u1ec3n, t\u1ee9c l\u00e0 Ii < Ij v\u1edbi m\u1ecdi i < j \u2208 {0, . . . nD}. V\u1edbi m\u1ed7i z = (z0 : \u00b7 \u00b7 \u00b7 : zn) \u2208 Pn(C), ta k\u00ed hi\u1ec7u zI = z0i0 . . . , znin, trong \u0111\u00f3 I = (i0, . . . , in) \u2208 {I0, . . . , InD}. K\u00ed hi\u1ec7u \u03f1mD : Pn(C) \u2192 PnD(C) l\u00e0 ph\u00e9p nh\u00fang Veronese b\u1eadc mD, t\u1ee9c l\u00e0 n\u1ebfu k\u00ed hi\u1ec7u (w0 : \u00b7 \u00b7 \u00b7 : wnD) l\u00e0 h\u1ec7 t\u1ecda \u0111\u1ed9 thu\u1ea7n nh\u1ea5t trong PnD(C) th\u00ec \u03f1mD \u0111\u01b0\u1ee3c x\u00e1c \u0111\u1ecbnh nh\u01b0 sau \u03f1mD(z) = (w0(z) : \u00b7 \u00b7 \u00b7 : wmD(z)), trong \u0111\u00f3 wj(z) = zIj , Ij \u2208 {I0, . . . , InD} v\u1edbi m\u1ed7i j = 0, . . . , nD. Ta \u0111\u1eb7t F = (F0 : \u00b7 \u00b7 \u00b7 : FnD) = \u03f1mD \u25e6 f, d\u1ec5 th\u1ea5y Fj = f Ij , j = 0, . . . , nD. Khi \u0111\u00f3 F l\u00e0 \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh t\u1eeb \u2206 v\u00e0o PnD(C) v\u00e0 F = (F0, . . . , FnD) l\u00e0 m\u1ed9t bi\u1ec3u di\u1ec5n t\u1ed1i gi\u1ea3n c\u1ee7a F . T\u1eeb gi\u1ea3 thi\u1ebft r\u1eb1ng f l\u00e0 kh\u00f4ng suy bi\u1ebfn \u0111\u1ea1i s\u1ed1 ta suy ra F kh\u00f4ng suy bi\u1ebfn tuy\u1ebfn t\u00ednh. V\u1edbi m\u1ed7i \u0111a th\u1ee9c Dj \u2208 {D1, . . . , Dq}, k\u00ed hi\u1ec7u aj = (aj0, . . . , ajnD) l\u00e0 vect\u01a1 li\u00ean k\u1ebft v\u1edbi Qj\u2217. V\u1edbi m\u1ed7i j = 1, . . . , q, k\u00ed hi\u1ec7u Lj = aj0w0 + \u00b7 \u00b7 \u00b7 + ajnDwnD, khi \u0111\u00f3 Lj l\u00e0 m\u1ed9t d\u1ea1ng tuy\u1ebfn t\u00ednh trong PnD(C). G\u1ecdi Hj l\u00e0 si\u00eau ph\u1eb3ng trong PnD(C) x\u00e1c \u0111\u1ecbnh b\u1edfi d\u1ea1ng tuy\u1ebfn t\u00ednh Lj v\u00e0 ta n\u00f3i r\u1eb1ng si\u00eau ph\u1eb3ng Hj li\u00ean k\u1ebft v\u1edbi si\u00eau m\u1eb7t Dj. Theo gi\u1ea3 thi\u1ebft h\u1ecd {D1, . . . , Dq} \u1edf v\u1ecb tr\u00ed t\u1ed5ng qu\u00e1t \u0111\u1ed1i v\u1edbi ph\u00e9p nh\u00fang Veronese trong Pn(C), ta suy ra h\u1ecd c\u00e1c si\u00eau ph\u1eb3ng {H1, . . . , Hq} \u1edf v\u1ecb tr\u00ed t\u1ed5ng qu\u00e1t trong PnD(C). \u00c1p d\u1ee5ng M\u1ec7nh \u0111\u1ec1 2.1.3 cho \u00e1nh x\u1ea1 ch\u1ec9nh h\u00ecnh F : \u2206 \u2192 PnD(C) v\u00e0 h\u1ecd c\u00e1c si\u00eau ph\u1eb3ng Hj \u1edf v\u1ecb tr\u00ed t\u1ed5ng qu\u00e1t ta c\u00f3","48 q (2.2) \u2225 (q \u2212 nD \u2212 1)TF (r) \u2a7d NFnD(r, Hj) + OF (r). j=1 B\u00e2y gi\u1edd ta \u01b0\u1edbc l\u01b0\u1ee3ng B\u1ea5t \u0111\u1eb3ng th\u1ee9c (2.2). Theo \u0111\u1ecbnh ngh\u0129a \u0111\u01b0\u1eddng cong F , v\u1edbi m\u1ed7i j = 1, . . . , q, nD Hj \u25e6 F = aj.F := ajk.Fk = Qj\u2217 \u25e6 f. k=0 Suy ra Nf (r, Qj\u2217) = NF (r, Hj); NfnD(r, Qj\u2217) = NFnD(r, Hj). (2.3) H\u01a1n n\u1eefa, t\u1eeb \u0111\u1ecbnh ngh\u0129a \u0111\u01b0\u1eddng cong F , ta c\u00f3 mf (r, Qj\u2217) 1 2\u03c0 \u2225f (rei\u03b8)\u2225mD 1 2\u03c0 \u2225f (r\u22121ei\u03b8)\u2225mD = 2\u03c0 0 log |Qj\u2217 \u25e6 f (rei\u03b8)|d\u03b8 + 2\u03c0 0 log |Q\u2217j \u25e6 f (r\u22121ei\u03b8)|d\u03b8 1 2\u03c0 \u2225F (rei\u03b8)\u2225 = log |Hj \u25e6 F (rei\u03b8)|d\u03b8 (2.4) 2\u03c0 0 1 2\u03c0 \u2225F (r\u22121ei\u03b8)\u2225 + log |Hj \u25e6 F (r\u22121ei\u03b8)|d\u03b8 + O(1) 2\u03c0 0 = mF (r, Hj) + O(1). (2.5) K\u1ebft h\u1ee3p (2.3) v\u00e0 \u0110\u1ecbnh l\u00fd 2.1.2 v\u00e0 (2.5) ta c\u00f3 TF (r) = NF (r, Hj) + mF (r, Hj) + O(1) (2.6) = Nf (r, Q\u2217j ) + mf (r, Q\u2217j ) + O(1). T\u1eeb M\u1ec7nh \u0111\u1ec1 1.1.8, ta c\u00f3 Nf (r, Qj\u2217) + mf (r, Qj\u2217) 1 2\u03c0 \u2225f (rei\u03b8)\u2225mD 1 2\u03c0 \u2225f (r\u22121ei\u03b8)\u2225mD = log |Q\u2217j \u25e6 f (rei\u03b8)|d\u03b8 + 2\u03c0 0 log |Q\u2217j \u25e6 f (r\u22121ei\u03b8)|d\u03b8 2\u03c0 0 1 2\u03c0 + log |Qj\u2217 \u25e6 f (rei\u03b8)|d\u03b8 2\u03c0 0 1 2\u03c0 + log |Q\u2217j \u25e6 f (r\u22121ei\u03b8)|d\u03b8 + O(1) 2\u03c0 0","49 = mD 1 2\u03c0 1 2\u03c0 + O(1) 2\u03c0 (2.7) log \u2225f (rei\u03b8)\u2225d\u03b8 + log \u2225f (r\u22121ei\u03b8)\u2225d\u03b8 0 2\u03c0 0 = mDTf (r) + O(1). Nh\u01b0 v\u1eady, t\u1eeb (2.6) ta c\u00f3 TF (r) = mDTf (r) + O(1). (2.8) \u0110i\u1ec1u n\u00e0y k\u00e9o theo OF (r) = Of (r). (2.9) K\u1ebft h\u1ee3p (2.2), (2.3), (2.8) v\u00e0 (2.9), ta c\u00f3 \u2225 (q \u2212 nD \u2212 1)Tf (r) \u2a7d 1 q NfnD (r, Q\u2217j ) + Of (r). (2.10) mD j=1 B\u00e2y gi\u1edd ta \u01b0\u1edbc l\u01b0\u1ee3ng v\u1ebf ph\u1ea3i c\u1ee7a (2.10). V\u1edbi m\u1ed7i j \u2208 {1, . . . , q}, t\u1eeb M\u1ec7nh \u0111\u1ec1 2.1.1 v\u00e0 (2.7) ta c\u00f3 NfnD(r, Qj\u2217) = NfnD(r, Q\u2217j \u2a7d k) + NfnD(r, Q\u2217j , > k) = k k 1NfnD(r, Q\u2217j , \u2a7d k) + k k 1NfnD(r, Q\u2217j , \u2a7d k) + + + NfnD(r, Qj\u2217, > k) \u2a7d k k 1NfnD(r, Q\u2217j , \u2a7d k) + nD 1 Nf1(r, Qj\u2217, \u2a7d k) + k+ + nDNf1(r, Qj\u2217, > k) \u2a7d k k 1NfnD(r, Q\u2217j , \u2a7d k) + nD 1 Nf (r, Q\u2217j , \u2a7d k) + k+ nD + k+ 1 Nf (r, Q\u2217j , > k) \u2a7d k k 1NfnD(r, Qj\u2217, \u2a7d k) + nD 1 Nf (r, Qj\u2217) + k+ \u2a7d k k 1NfnD(r, Qj\u2217, \u2a7d k) + nDmD Tf (r) + O(1), + k+1 L\u1ea5y t\u1ed5ng tr\u00ean t\u1eadp c\u00e1c ch\u1ec9 s\u1ed1 j = 1, 2, . . . , q, ta c\u00f3","50 1 q NfnD (r, Qj\u2217) \u2a7d (k k q NfnD(r, Q\u2217j , \u2a7d k) mD j=1 + 1)mD j=1 + qnD Tf (r) + O(1). (2.11) k+1 \u00c1p d\u1ee5ng (2.11) v\u00e0o (2.10), ta c\u00f3 (q \u2212 nD \u2212 1)Tf (r) \u2a7d k 1) q NfnD(r, Qj\u2217, \u2a7d k) mD(k + j=1 + qnD Tf (r) + Of (r). k+1 T\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi q \u2212 qnD \u2212 nD \u2212 1 Tf (r) \u2a7d k 1) q NfnD(r, Q\u2217j , \u2a7d k) + Of (r). k+1 mD(k + j=1 Suy ra (q(k + 1 \u2212 nD)\u2212(nD + 1)(k + 1))Tf (r) \u2a7d k q NfnD(r, Qj\u2217, \u2a7d k) + (k + 1)Of (r) mD j=1 \u2a7d nDk q Nf1(r, Q\u2217j , \u2a7d k) + (k + 1)Of (r). (2.12) mD j=1 Do f \u2261\u0338 g n\u00ean t\u1ed3n t\u1ea1i hai ch\u1ec9 s\u1ed1 \u03b1, \u03b2 \u2208 {0, . . . , n}, \u03b1 =\u0338 \u03b2 sao cho f\u03b1g\u03b2 \u0338\u2261 f\u03b2g\u03b1. Gi\u1ea3 s\u1eed z0 \u2208 \u2206 l\u00e0 kh\u00f4ng \u0111i\u1ec3m c\u1ee7a Qj\u2217(f ) v\u1edbi b\u1ed9i nh\u1ecf h\u01a1n hay b\u1eb1ng k, khi \u0111\u00f3 z0 l\u00e0 kh\u00f4ng \u0111i\u1ec3m c\u1ee7a Qj(f ) v\u00ec Q\u2217j = QjmD\/dj , suy ra z0 \u2208 Ef (D) \u222a Eg(D). T\u1eeb gi\u1ea3 thi\u1ebft ta c\u00f3 g(z0) = f (z0), k\u00e9o theo f\u03b1(z0) = g\u03b1(z0). f\u03b2(z0) g\u03b2(z0) Do \u0111\u00f3 f\u03b1(z0)g\u03b2(z0) = f\u03b2(z0)g\u03b1(z0) v\u00ec f\u03b1, g\u03b1, f\u03b2, g\u03b2 l\u00e0 c\u00e1c h\u00e0m ch\u1ec9nh h\u00ecnh. \u0110i\u1ec1u n\u00e0y k\u00e9o theo z0 l\u00e0 kh\u00f4ng \u0111i\u1ec3m c\u1ee7a h\u00e0m f\u03b1g\u03b2 \u2212f\u03b2g\u03b1. Ch\u00fa \u00fd r\u1eb1ng h\u1ecd D \u1edf v\u1ecb tr\u00ed t\u1ed5ng qu\u00e1t \u0111\u1ed1i v\u1edbi ph\u00e9p nh\u00fang","51 Veronese n\u00ean t\u1ed3n t\u1ea1i kh\u00f4ng qu\u00e1 nD si\u00eau m\u1eb7t Dj trong h\u1ecd D sao cho Dj \u25e6 f (z0) = Q(f )(z0) = 0. \u0110i\u1ec1u n\u00e0y k\u00e9o theo q 1 . (2.13) r, f\u03b1g\u03b2 \u2212 f\u03b2g\u03b1 Nf1(r, Q\u2217j , \u2a7d k) \u2a7d nDN0 j=1 \u0110\u1eb7t H = f\u03b1g\u03b2 \u2212 f\u03b2g\u03b1, khi \u0111\u00f3 H l\u00e0 h\u00e0m ch\u1ec9nh h\u00ecnh, t\u1eeb M\u1ec7nh \u0111\u1ec1 1.1.8 ta c\u00f3 11 2\u03c0 1 2\u03c0 N0(r, H ) = 2\u03c0 log |H(rei\u03b8)|d\u03b8 + log |H(r\u22121ei\u03b8)|d\u03b8 + O(1). 0 2\u03c0 0 Ngo\u00e0i ra, v\u1edbi m\u1ed7i z \u2208 \u2206 ta c\u00f3 log |H(z)| = log |(f\u03b1g\u03b2 \u2212 f\u03b2g\u03b1)(z)| \u2a7d log max{|f\u03b1(z)g\u03b2(z)|, |f\u03b2(z)g\u03b1(z)|} + log 2 = max{log |f\u03b1(z)g\u03b2(z)|, log |f\u03b2(z)g\u03b1(z)|} + log 2 = max{log |f\u03b1(z)| + log |g\u03b2(z)|, log |f\u03b2(z)| + log |g\u03b1(z)|} + log 2 \u2a7d max{log |f\u03b1(z)|, log |f\u03b2(z)|} + max{log |g\u03b1(z)|, log |g\u03b2(z)|} + log 2 = log max{|f\u03b1(z)|, |f\u03b2(z)|} + log max{|g\u03b1(z), |g\u03b2(z)|} + log 2 \u2a7d log \u2225f (z)\u2225 + log \u2225g(z)\u2225 + log 2. Do \u0111\u00f3 1 2\u03c0 1 2\u03c0 2\u03c0 0 log |H(rei\u03b8)|d\u03b8 + 2\u03c0 0 log |H(r\u22121ei\u03b8)|d\u03b8 1 2\u03c0 1 2\u03c0 \u2a7d 2\u03c0 0 log \u2225f (rei\u03b8)\u2225d\u03b8 + 2\u03c0 0 log \u2225f (r\u22121ei\u03b8)\u2225d\u03b8 1 2\u03c0 1 2\u03c0 + 2\u03c0 0 log \u2225g(rei\u03b8)\u2225d\u03b8 + 2\u03c0 0 log \u2225g(r\u22121ei\u03b8)\u2225d\u03b8 + O(1). = Tf (r) + Tg(r) + O(1).","52 T\u1eeb \u0111\u00f3, (2.13) tr\u1edf th\u00e0nh q Nf1(r, Q\u2217j , \u2a7d k) \u2a7d nD(Tf (r) + Tg(r)) + O(1). j=1 Do \u0111\u00f3 (2.12) tr\u1edf th\u00e0nh (q(k + 1 \u2212 nD)\u2212(nD + 1)(k + 1))Tf (r) \u2a7d n2Dk (Tf (r) + Tg(r)) + (k + 1)Of (r). mD Nh\u01b0 v\u1eady (2.1) \u0111\u01b0\u1ee3c ch\u1ee9ng minh. T\u01b0\u01a1ng t\u1ef1 cho \u00e1nh x\u1ea1 g ta c\u00f3 (q(k + 1 \u2212 nD)\u2212(nD + 1)(k + 1))Tg(r) \u2a7d n2Dk (Tf (r) + Tg(r)) + (k + 1)Og(r). (2.14) mD K\u1ebft h\u1ee3p (2.1) v\u00e0 (2.14), ta c\u00f3 (q(k + 1 \u2212 nD)\u2212(nD + 1)(k + 1))(Tf (r) + Tg(r)) \u2a7d 2n2Dk (Tf (r) + Tg(r)) + (k + 1)(Of (r) + Og(r)). mD \u0110i\u1ec1u n\u00e0y k\u00e9o theo q(k + 1 \u2212 nD) \u2212 (nD + 1)(k + 1) \u2212 2n2Dk mD \u2a7d Of (r) + Og(r)(k + 1) (2.15) Tf (r) + Tg(r) \u0111\u00fang v\u1edbi m\u1ecdi s\u1ed1 1 < r < R. T\u1eeb gi\u1ea3 thi\u1ebft Of (r) v\u00e0 Og(r), ta c\u00f3 lim sup Of (r) + Og(r) = 0. r \u2192 R Tf (r) + Tg(r) Cho r \u2192 R trong (2.15) ta \u0111\u01b0\u1ee3c q(k + 1 \u2212 nD) \u2212 (nD + 1)(k + 1) \u2212 2nD2 k \u2a7d 0. mD \u0110i\u1ec1u n\u00e0y t\u01b0\u01a1ng \u0111\u01b0\u01a1ng v\u1edbi k(qmD \u2212 (nD + 1)mD \u2212 2n2D) + (q \u2212 qnD \u2212 (nD + 1))mD \u2a7d 0.","53 N\u1ebfu ta ch\u1ecdn k > (qnD \u2212 q + nD + 1)mD , qmD \u2212 (nD + 1)mD \u2212 2nD2 th\u00ec t\u1eeb gi\u1ea3 thi\u1ebft q > nD + 1 + 2nD2 ta c\u00f3 m\u1eabu thu\u1eabn. Nh\u01b0 v\u1eady figj \u2261 fj gi v\u1edbi mD m\u1ed7i i \u0338= j \u2208 {0, . . . , n}, t\u1ee9c l\u00e0 f \u2261 g. \u0110i\u1ec1u n\u00e0y k\u00e9o theo k\u1ebft lu\u1eadn c\u1ee7a \u0110\u1ecbnh l\u00fd 2.2.1. 2.2.2. Tr\u01b0\u1eddng h\u1ee3p c\u00f3 xem x\u00e9t \u0111i\u1ec1u ki\u1ec7n ngh\u1ecbch \u1ea3nh c\u1ee7a t\u1eebng si\u00eau m\u1eb7t \u0110\u1ecbnh l\u00fd 2.2.2 ([39]). Cho f v\u00e0 g l\u00e0 hai \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh kh\u00f4ng suy bi\u1ebfn \u0111\u1ea1i s\u1ed1 t\u1eeb \u2206 v\u00e0o Pn(C) sao cho Of (r) = o(Tf (r)) v\u00e0 Og(r) = o(Tg(r)). Cho D = {D1, . . . , Dq} l\u00e0 m\u1ed9t h\u1ecd g\u1ed3m q > nD + 1 + 2nD\/mD c\u00e1c si\u00eau m\u1eb7t \u1edf v\u1ecb tr\u00ed t\u1ed5ng qu\u00e1t \u0111\u1ed1i v\u1edbi ph\u00e9p nh\u00fang Veronese trong Pn(C). Gi\u1ea3 s\u1eed (a) f (z) = g(z) v\u1edbi m\u1ecdi z \u2208 Ef (D) \u222a Eg(D), (b) Ef (Di) \u2229 Ef (Dj) = \u2205 v\u00e0 Eg(Di) \u2229 Eg(Dj) = \u2205 v\u1edbi m\u1ecdi i \u0338= j \u2208 {1, . . . , q}. Khi \u0111\u00f3 f \u2261 g. Ch\u1ee9ng minh. Ta c\u0169ng ch\u1ee9ng minh \u0110\u1ecbnh l\u00fd 2.2.2 b\u1eb1ng ph\u1ea3n ch\u1ee9ng. Gi\u1ea3 s\u1eed f \u2261\u0338 g. G\u1ecdi k l\u00e0 m\u1ed9t s\u1ed1 nguy\u00ean d\u01b0\u01a1ng \u0111\u1ee7 l\u1edbn ta s\u1ebd ch\u1ecdn sau. V\u1edbi c\u00e1c gi\u1ea3 thi\u1ebft trong \u0110\u1ecbnh l\u00fd 2.2.2 v\u00e0 ch\u1ee9ng minh t\u01b0\u01a1ng t\u1ef1 nh\u01b0 \u0110\u1ecbnh l\u00fd 2.2.1, v\u1edbi m\u1ed7i s\u1ed1 th\u1ef1c r : 1 < r < R ta c\u00f3 (q(k + 1 \u2212 nD)\u2212(nD + 1)(k + 1))Tf (r) \u2a7d nDk q Nf1(r, Qj\u2217, \u2a7d k) + (k + 1)Of (r), (2.16) mD j=1 \u0111\u00fang v\u1edbi m\u1ed7i j = 1, 2, . . . , q, trong \u0111\u00f3 Q\u2217j = QjmD\/dj . V\u00ec f \u0338\u2261 g n\u00ean t\u1ed3n t\u1ea1i hai s\u1ed1 \u03b1, \u03b2 \u2208 {0, . . . , n}, \u03b1 \u0338= \u03b2 sao cho f\u03b1g\u03b2 \u0338\u2261 f\u03b2g\u03b1.","54 Ta bi\u1ebft r\u1eb1ng, n\u1ebfu z0 \u2208 \u2206 l\u00e0 kh\u00f4ng \u0111i\u1ec3m c\u1ee7a Qj\u2217(f ) v\u1edbi b\u1ed9i nh\u1ecf h\u01a1n ho\u1eb7c b\u1eb1ng k, th\u00ec z0 l\u00e0 kh\u00f4ng \u0111i\u1ec3m c\u1ee7a h\u00e0m f\u03b1g\u03b2 \u2212 f\u03b2g\u03b1. T\u1eeb gi\u1ea3 thi\u1ebft Ef (Di) \u2229 Ef (Dj) = \u2205 v\u1edbi m\u1ed7i c\u1eb7p i \u0338= j \u2208 {1, . . . , q}, ta suy ra n\u1ebfu z0 l\u00e0 kh\u00f4ng \u0111i\u1ec3m c\u1ee7a Qj\u2217(f ) th\u00ec z0 s\u1ebd kh\u00f4ng l\u00e0 kh\u00f4ng \u0111i\u1ec3m c\u1ee7a Q\u2217i (f ) v\u1edbi m\u1ecdi i \u2208 {1, . . . , q}, i \u0338= j. Do \u0111\u00f3 q 1 \u2a7d Tf (r) + Tg(r) + O(1). r, f\u03b1g\u03b2 \u2212 f\u03b2g\u03b1 Nf1(r, Q\u2217j , \u2a7d k) \u2a7d N j=1 Nh\u01b0 v\u1eady B\u1ea5t \u0111\u1eb3ng th\u1ee9c (2.16) tr\u1edf th\u00e0nh (q(k + 1 \u2212 nD)\u2212(nD + 1)(k + 1))Tf (r) \u2a7d nDk (Tf (r) + Tg(r)) + (k + 1)Of (r). (2.17) mD T\u01b0\u01a1ng t\u1ef1 v\u1edbi \u00e1nh x\u1ea1 g ta c\u00f3 (q(k + 1 \u2212 nD)\u2212(nD + 1)(k + 1))Tg(r) \u2a7d nDk (Tf (r) + Tg(r)) + (k + 1)Og(r). (2.18) mD K\u1ebft h\u1ee3p (2.17) v\u00e0 (2.18), ta c\u00f3 (q(k + 1 \u2212 nD)\u2212(nD + 1)(k + 1))(Tf (r) + Tg(r)) \u2a7d 2nDk (Tf (r) + Tg(r)) + (k + 1)(Of (r) + Og(r)). mD K\u00e9o theo qmD(k + 1 \u2212 nD) \u2212 mD(nD + 1)(k + 1) \u2212 2nDk \u2a7d Of (r) + Og (r) (k + 1)mD Tf (r) + Tg(r) \u0111\u00fang v\u1edbi m\u1ecdi s\u1ed1 th\u1ef1c 1 < r < R. Cho r \u2192 R, ta c\u00f3 k(qmD \u2212 (nD + 1)mD \u2212 2nD) + (q \u2212 qnD \u2212 (nD + 1))mD \u2a7d 0. N\u1ebfu ta ch\u1ecdn k > (qnD \u2212 q + nD + 1)mD , qmD \u2212 (nD + 1)mD \u2212 2nD","55 th\u00ec t\u1eeb gi\u1ea3 thi\u1ebft q > nD + 1 + 2nD ta c\u00f3 m\u1eabu thu\u1eabn. Nh\u01b0 v\u1eady figj \u2261 fj gi v\u1edbi mD m\u1ecdi i \u0338= j \u2208 {0, . . . , n}, t\u1ee9c l\u00e0 f \u2261 g. \u0110\u1ecbnh l\u00fd 2.2.2 \u0111\u01b0\u1ee3c ch\u1ee9ng minh. Nh\u1eadn x\u00e9t. 1. Trong \u0110\u1ecbnh l\u00fd 2.2.2, s\u1ed1 si\u00eau m\u1eb7t t\u1ed1i thi\u1ec3u th\u1ecfa m\u00e3n gi\u1ea3 thi\u1ebft l\u00e0 nD + 1 + 2nD\/mD. Ch\u00fa \u00fd r\u1eb1ng khi h\u1ecd c\u00e1c si\u00eau m\u1eb7t l\u00e0 c\u00e1c si\u00eau ph\u1eb3ng th\u00ec v\u1ecb tr\u00ed t\u1ed5ng qu\u00e1t \u0111\u1ed1i v\u1edbi ph\u00e9p nh\u00fang Veronese ch\u00ednh l\u00e0 \u1edf v\u1ecb tr\u00ed t\u1ed5ng qu\u00e1t th\u00f4ng th\u01b0\u1eddng trong Pn(C). Trong tr\u01b0\u1eddng h\u1ee3p n\u00e0y nD = n v\u00e0 mD = 1 n\u00ean q = 3n + 2, tr\u00f9ng v\u1edbi s\u1ed1 si\u00eau ph\u1eb3ng c\u1ea7n thi\u1ebft trong k\u1ebft qu\u1ea3 c\u1ee7a Fujimoto. 2. Ta bi\u1ebft r\u1eb1ng, m\u1ed9t h\u00e0m ph\u00e2n h\u00ecnh h tr\u00ean h\u00ecnh v\u00e0nh khuy\u00ean l\u00e0 si\u00eau vi\u1ec7t khi lim sup T0(r, h) = \u221e khi R = \u221e v\u00e0 lim sup \u2212 T0(r, h) r) = \u221e khi r \u2192 \u221e log r log(R \u2212 r\u2192R R < \u221e. Do \u0111\u00f3, \u0111\u1ed1i v\u1edbi m\u1ed9t \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh f = (f0 : \u00b7 \u00b7 \u00b7 : fn) th\u00ec ch\u1ec9 c\u1ea7n m\u1ed9t h\u00e0m fj l\u00e0 si\u00eau vi\u1ec7t th\u00ec Of (r) = o(Tf (r)).","56 K\u1ebft lu\u1eadn Ch\u01b0\u01a1ng 2 Trong Ch\u01b0\u01a1ng 2, lu\u1eadn \u00e1n \u0111\u00e3 thu \u0111\u01b0\u1ee3c c\u00e1c k\u1ebft qu\u1ea3 ch\u00ednh sau : - Gi\u1edbi thi\u1ec7u m\u1ed9t s\u1ed1 kh\u00e1i ni\u1ec7m c\u01a1 b\u1ea3n c\u1ea7n thi\u1ebft s\u1eed d\u1ee5ng \u0111\u1ebfn trong ch\u1ee9ng minh v\u1ea5n \u0111\u1ec1 duy nh\u1ea5t c\u1ee7a \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh tr\u00ean h\u00ecnh v\u00e0nh khuy\u00ean: h\u00e0m \u0111\u1ebfm b\u1ed5 sung, c\u00e1c \u0111\u1ecbnh l\u00fd c\u01a1 b\u1ea3n cho \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh tr\u00ean h\u00ecnh v\u00e0nh khuy\u00ean v\u1edbi m\u1ee5c ti\u00eau l\u00e0 c\u00e1c si\u00eau ph\u1eb3ng. - Ph\u00e1t bi\u1ec3u v\u00e0 ch\u1ee9ng minh hai \u0111\u1ecbnh l\u00fd: \u0110\u1ecbnh l\u00fd 2.2.1 v\u00e0 \u0110\u1ecbnh l\u00fd 2.2.2 v\u1ec1 v\u1ea5n \u0111\u1ec1 duy nh\u1ea5t cho \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh tr\u00ean h\u00ecnh v\u00e0nh khuy\u00ean trong c\u00e1c tr\u01b0\u1eddng h\u1ee3p si\u00eau m\u1eb7t \u1edf v\u1ecb tr\u00ed t\u1ed5ng qu\u00e1t \u0111\u1ed1i v\u1edbi ph\u00e9p nh\u00fang Veronese. Hai k\u1ebft qu\u1ea3 n\u00e0y cho ch\u00fang ta c\u00e1c \u0111i\u1ec1u ki\u1ec7n \u0111\u1ee7 \u0111\u1ec3 hai \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh kh\u00f4ng suy bi\u1ebfn \u0111\u1ea1i s\u1ed1 tr\u00ean m\u1ed9t h\u00ecnh v\u00e0nh khuy\u00ean l\u00e0 \u0111\u1ed3ng nh\u1ea5t.","57 Ch\u01b0\u01a1ng 3 V\u1ea5n \u0111\u1ec1 duy nh\u1ea5t cho h\u00e0m nguy\u00ean li\u00ean quan \u0111\u1ebfn gi\u1ea3 thuy\u1ebft Bru\u00a8ck 3.1. Ki\u1ebfn th\u1ee9c b\u1ed5 tr\u1ee3 Trong ph\u1ea7n n\u00e0y ch\u00fang t\u00f4i nh\u1eafc l\u1ea1i m\u1ed9t s\u1ed1 kh\u00e1i ni\u1ec7m v\u00e0 k\u00ed hi\u1ec7u trong L\u00fd thuy\u1ebft Nevanlinna cho \u0111a th\u1ee9c vi ph\u00e2n v\u00e0 h\u1ecd chu\u1ea9n t\u1eafc c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh tr\u00ean C, c\u1ea7n thi\u1ebft trong c\u00e1c ch\u1ee9ng minh k\u1ebft qu\u1ea3 c\u1ee7a ch\u00fang t\u00f4i. 3.1.1. Ph\u00e2n b\u1ed1 gi\u00e1 tr\u1ecb cho \u0111a th\u1ee9c vi ph\u00e2n Cho h\u00e0m ph\u00e2n h\u00ecnh kh\u00e1c h\u1eb1ng g(z) tr\u00ean m\u1eb7t ph\u1eb3ng ph\u1ee9c C v\u00e0 p \u0111\u1ea1o h\u00e0m \u0111\u1ea7u ti\u00ean c\u1ee7a n\u00f3. M\u1ed9t \u0111a th\u1ee9c vi ph\u00e2n P c\u1ee7a g \u0111\u01b0\u1ee3c \u0111\u1ecbnh ngh\u0129a b\u1edfi np P (z) := \u03b1i(z) (g(j)(z))Sij , i=1 j=0 trong \u0111\u00f3 Sij, 0 \u2a7d i, j \u2a7d n, l\u00e0 c\u00e1c s\u1ed1 nguy\u00ean kh\u00f4ng \u00e2m v\u00e0 \u03b1i(z), 1 \u2a7d i \u2a7d n l\u00e0 c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh nh\u1ecf \u0111\u1ed1i v\u1edbi g. \u0110\u1eb7t pp d(P ) := min Sij v\u00e0 \u03b8(P ) := max jSij. 1\u2a7di\u2a7dn 1\u2a7di\u2a7dn j=0 j=0 N\u0103m 2002, J. Hinchliffe ([21]) \u0111\u00e3 ch\u1ee9ng minh k\u1ebft qu\u1ea3 sau, cho m\u1ed9t \u0111\u00e1nh gi\u00e1 gi\u1eefa c\u00e1c h\u00e0m Nevanlinna c\u1ee7a h\u00e0m ph\u00e2n h\u00ecnh v\u00e0 h\u00e0m \u0111\u1ebfm c\u1ee7a \u0111a th\u1ee9c vi ph\u00e2n.","58 M\u1ec7nh \u0111\u1ec1 3.1.1 ([21]). Cho g l\u00e0 m\u1ed9t h\u00e0m ph\u00e2n h\u00ecnh si\u00eau vi\u1ec7t v\u00e0 a =\u0338 0 l\u00e0 m\u1ed9t h\u1eb1ng s\u1ed1 ph\u1ee9c, g\u1ecdi P l\u00e0 m\u1ed9t \u0111a th\u1ee9c vi ph\u00e2n kh\u00e1c h\u1eb1ng c\u1ee7a g v\u1edbi d(P ) \u2a7e 2. Khi \u0111\u00f3 \u03b8(P ) + 1 1 + 1 1 ) T (r, g) \u2a7d N (r, ) N (r, a d(P ) \u2212 1 g d(P ) \u2212 1 P \u2212 + o(T (r, g)), \u0111\u1ed1i v\u1edbi m\u1ecdi r \u2208 [1, +\u221e) n\u1eb1m ngo\u00e0i m\u1ed9t t\u1eadp c\u00f3 \u0111\u1ed9 \u0111o Lebesgues h\u1eefu h\u1ea1n. Khi f l\u00e0 m\u1ed9t h\u00e0m nguy\u00ean si\u00eau vi\u1ec7t, b\u1ea5t \u0111\u1eb3ng th\u1ee9c tr\u00ean tr\u1edf th\u00e0nh T (r, g) \u2a7d \u03b8(P ) + 1 1 1 1 ) + o(T (r, N (r, ) + N (r, a g)), d(P ) g d(P ) P \u2212 \u0111\u1ed1i v\u1edbi m\u1ecdi r \u2208 [1, +\u221e) n\u1eb1m ngo\u00e0i m\u1ed9t t\u1eadp c\u00f3 \u0111\u1ed9 \u0111o Lebesgues h\u1eefu h\u1ea1n. Cho f l\u00e0 m\u1ed9t h\u00e0m ph\u00e2n h\u00ecnh tr\u00ean m\u1eb7t ph\u1eb3ng ph\u1ee9c C, ta nh\u1eafc l\u1ea1i b\u1eadc \u03c3(f ) c\u1ee7a h\u00e0m ph\u00e2n h\u00ecnh f \u0111\u1ecbnh ngh\u0129a b\u1edfi \u03c3(f ) = lim sup log T (r, f ) . r\u2192\u221e log r V\u00e0 si\u00eau b\u1eadc c\u1ee7a f \u0111\u01b0\u1ee3c \u0111\u1ecbnh ngh\u0129a b\u1edfi \u03c32(f ) = lim sup log log T (r, f ) . r\u2192\u221e log r Tr\u01b0\u1eddng h\u1ee3p \u0111\u1eb7c bi\u1ec7t, n\u1ebfu f l\u00e0 m\u1ed9t h\u00e0m nguy\u00ean, bi\u1ec3u di\u1ec5n \u0111\u01b0\u1ee3c d\u01b0\u1edbi d\u1ea1ng chu\u1ed7i l\u0169y th\u1eeba \u221e th\u00ec ta k\u00ed hi\u1ec7u f (z) = anzn, n=0 \u00b5(r, f ) = max {|anzn|}, n\u2208N,|z|=r \u03bd(r, f ) = sup{n : |an|rn = \u00b5(r, f )}, M (r, f ) = max |f (z)|. |z|=r Trong tr\u01b0\u1eddng h\u1ee3p n\u00e0y b\u1eadc c\u1ee7a f c\u00f3 th\u1ec3 bi\u1ec3u di\u1ec5n \u0111\u01b0\u1ee3c d\u01b0\u1edbi d\u1ea1ng \u03c3(f ) = lim sup log log(M (r, f )) . r\u2192\u221e log r","59 M\u1ec7nh \u0111\u1ec1 3.1.2 ([29]). N\u1ebfu f l\u00e0 m\u1ed9t h\u00e0m nguy\u00ean v\u1edbi b\u1eadc \u03c3(f ), khi \u0111\u00f3 \u03c3(f ) = lim sup log \u03bd(r, f ) . r\u2192\u221e log r M\u1ec7nh \u0111\u1ec1 3.1.3 ([29]). Cho f l\u00e0 m\u1ed9t h\u00e0m nguy\u00ean si\u00eau vi\u1ec7t, \u03b4 l\u00e0 m\u1ed9t s\u1ed1 th\u1ef1c th\u1ecfa m\u00e3n 0 < \u03b4 < 1 Cho z l\u00e0 m\u1ed9t s\u1ed1 ph\u1ee9c th\u1ecfa m\u00e3n |z| = r v\u00e0 . 4 1 \u2212 +\u03b4 |f (z)| > M (r, f )\u03bd(r, f ) 4 . Khi \u0111\u00f3 t\u1ed3n t\u1ea1i m\u1ed9t t\u1eadp F \u2282 R+ c\u00f3 \u0111\u1ed9 \u0111o logarit h\u1eefu h\u1ea1n, t\u1ee9c l\u00e0 dt < +\u221e, t F sao cho f (m)(z) \u03bd(r, f ) m = (1 + o(1)) f (z) z \u0111\u00fang v\u1edbi m\u1ecdi m \u2a7e 1 v\u00e0 r \u2208\u0338 F. L\u1ea5y E0(z) = 1 \u2212 z, Em(z) = (1 \u2212 z)ez+z2\/2+\u00b7\u00b7\u00b7+zm\/m, m \u2208 Z+, khi \u0111\u00f3 ta c\u00f3 k\u1ebft qu\u1ea3 sau \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 \u0111\u1ecbnh l\u00fd bi\u1ec3u di\u1ec5n Weierstrass. M\u1ec7nh \u0111\u1ec1 3.1.4 ([29]). Cho f l\u00e0 m\u1ed9t h\u00e0m nguy\u00ean, v\u1edbi b\u1ed9i kh\u00f4ng \u0111i\u1ec3m t\u1ea1i z = 0 l\u00e0 m \u2a7e 0. Ta g\u1ecdi c\u00e1c kh\u00f4ng \u0111i\u1ec3m kh\u00e1c c\u1ee7a f l\u00e0 a1, a2, . . . , m\u1ed7i kh\u00f4ng \u0111i\u1ec3m \u0111\u01b0\u1ee3c l\u1eb7p l\u1ea1i s\u1ed1 l\u1ea7n b\u1eb1ng b\u1ed9i c\u1ee7a n\u00f3. Khi \u0111\u00f3 f c\u00f3 bi\u1ec3u di\u1ec5n \u221e z , an f (z) = eg(z)zm Emn n=1 v\u1edbi g l\u00e0 m\u1ed9t h\u00e0m nguy\u00ean v\u00e0 mn l\u00e0 c\u00e1c s\u1ed1 t\u1ef1 nhi\u00ean. H\u01a1n n\u1eefa, n\u1ebfu f c\u00f3 b\u1eadc \u03c1 h\u1eefu h\u1ea1n th\u00ec g l\u00e0 m\u1ed9t \u0111a th\u1ee9c v\u1edbi b\u1eadc kh\u00f4ng v\u01b0\u1ee3t qu\u00e1 \u03c1. 3.1.2. H\u1ecd chu\u1ea9n t\u1eafc c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh K\u00ed hi\u1ec7u S l\u00e0 m\u1eb7t c\u1ea7u Riemann v\u00e0 \u03c0 : C = C \u222a{\u221e} \u2192 S l\u00e0 ph\u00e9p chi\u1ebfu c\u1ea7u. \u0110\u1ecbnh ngh\u0129a 3.1.5. Cho z1, z2 \u2208 C, k\u00ed hi\u1ec7u v\u00e0 M1 = \u03c0(z1), M2 = \u03c0(z2) l\u00e0 hai \u0111i\u1ec3m tr\u00ean m\u1eb7t c\u1ea7u S t\u01b0\u01a1ng \u1ee9ng l\u1ea7n l\u01b0\u1ee3t v\u1edbi c\u00e1c \u0111i\u1ec3m z1, z2. \u0110\u1ed9 d\u00e0i c\u1ee7a","60 \u0111o\u1ea1n th\u1eb3ng M1M2 \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 kho\u1ea3ng c\u00e1ch c\u1ea7u gi\u1eefa hai \u0111i\u1ec3m z1, z2 v\u00e0 k\u00ed hi\u1ec7u l\u00e0 \u03c1S(z1, z2). Hi\u1ec3n nhi\u00ean, n\u1ebfu z1 \u2261 z2 th\u00ec \u03c1S(z1, z2) = 0. H\u01a1n n\u1eefa ta d\u1ec5 d\u00e0ng t\u00ednh to\u00e1n \u0111\u01b0\u1ee3c kho\u1ea3ng c\u00e1ch c\u1ea7u gi\u1eefa c\u00e1c \u0111i\u1ec3m z1 =\u0338 z2 trong m\u1eb7t ph\u1eb3ng ph\u1ee9c m\u1edf r\u1ed9ng nh\u01b0 sau: \u2022 N\u1ebfu z1, z2 \u2208 C th\u00ec \u03c1S(z1, z2) = |z1 \u2212 z2| ; (1 + |z1|2) 1 (1 + |z2|2) 1 2 2 \u2022 N\u1ebfu z1 \u2208 C, z2 = \u221e th\u00ec \u03c1S(z1, z2) = 1 . (1 + |z1 |2 ) 1 2 \u0110\u1ecbnh ngh\u0129a 3.1.6. M\u1ed9t d\u00e3y c\u00e1c \u0111i\u1ec3m {zn, n = 1, 2, . . . } c\u1ee7a C \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 h\u1ed9i t\u1ee5 \u0111\u1ed1i v\u1edbi kho\u1ea3ng c\u00e1ch c\u1ea7u (hay c\u00f2n g\u1ecdi l\u00e0 h\u1ed9i t\u1ee5 c\u1ea7u) n\u1ebfu v\u1edbi m\u1ed7i s\u1ed1 \u03b5 > 0, t\u1ed3n t\u1ea1i m\u1ed9t s\u1ed1 nguy\u00ean d\u01b0\u01a1ng N sao cho, v\u1edbi m\u1ecdi n \u2265 N, m \u2265 N , ta c\u00f3: \u03c1S(zn, zm) < \u03b5. (3.1) B\u1ed5 \u0111\u1ec1 3.1.7. N\u1ebfu m\u1ed9t d\u00e3y c\u00e1c \u0111i\u1ec3m {zn, n = 1, 2, . . . } c\u1ee7a C h\u1ed9i t\u1ee5 \u0111\u1ed1i v\u1edbi kho\u1ea3ng c\u00e1ch c\u1ea7u th\u00ec t\u1ed3n t\u1ea1i m\u1ed9t \u0111i\u1ec3m duy nh\u1ea5t z\u2217 \u2208 C sao cho: lim \u03c1S (zn, z\u2217) = 0. (3.2) r\u2192\u221e Ph\u1ea7n t\u1eed z\u2217 trong b\u1ed5 \u0111\u1ec1 tr\u00ean \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 gi\u1edbi h\u1ea1n c\u1ee7a d\u00e3y {zn} \u0111\u1ed1i v\u1edbi kho\u1ea3ng c\u00e1ch c\u1ea7u hay c\u00f2n g\u1ecdi l\u00e0 gi\u1edbi h\u1ea1n c\u1ea7u c\u1ee7a d\u00e3y {zn}. \u0110\u1ecbnh ngh\u0129a 3.1.8. Cho S = {fn(z), n = 1, 2, . . . } l\u00e0 m\u1ed9t d\u00e3y c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh x\u00e1c \u0111\u1ecbnh trong mi\u1ec1n D v\u00e0 E m\u1ed9t t\u1eadp con c\u1ee7a D. D\u00e3y S \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 h\u1ed9i t\u1ee5 \u0111\u1ec1u tr\u00ean E \u0111\u1ed1i v\u1edbi kho\u1ea3ng c\u00e1ch c\u1ea7u (hay c\u00f2n g\u1ecdi l\u00e0 h\u1ed9i t\u1ee5 c\u1ea7u \u0111\u1ec1u), n\u1ebfu v\u1edbi m\u1ed7i s\u1ed1 d\u01b0\u01a1ng \u03b5, t\u1ed3n t\u1ea1i m\u1ed9t s\u1ed1 nguy\u00ean d\u01b0\u01a1ng N sao cho v\u1edbi m\u1ecdi n \u2265 N, m \u2265 N ta c\u00f3: \u03c1S(fn(z), fm(z)) < \u03b5 (3.3) v\u1edbi m\u1ecdi z \u2208 E.","61 Gi\u1ea3 s\u1eed {fn(z), n = 1, 2, . . . } h\u1ed9i t\u1ee5 c\u1ea7u \u0111\u1ec1u tr\u00ean E. Khi \u0111\u00f3 v\u1edbi m\u1ed7i \u0111i\u1ec3m z0 \u2208 E, d\u00e3y {fn(z0), n = 1, 2, . . . } l\u00e0 h\u1ed9i t\u1ee5 c\u1ea7u v\u00e0 theo B\u1ed5 \u0111\u1ec1 3.1.7 n\u00f3 c\u00f3 duy nh\u1ea5t m\u1ed9t gi\u1edbi h\u1ea1n c\u1ea7u trong E, ta k\u00ed hi\u1ec7u l\u00e0 w0. Do \u0111\u00f3 ta thi\u1ebft l\u1eadp \u0111\u01b0\u1ee3c m\u1ed9t h\u00e0m f x\u00e1c \u0111\u1ecbnh tr\u00ean E: f (z0) = w0 v\u1edbi m\u1ed7i z0 \u2208 E. H\u00e0m f x\u00e1c \u0111\u1ecbnh nh\u01b0 v\u1eady \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 h\u00e0m gi\u1edbi h\u1ea1n c\u1ee7a d\u00e3y h\u00e0m {fn(z), n = 1, 2, . . . } \u0111\u1ed1i v\u1edbi kho\u1ea3ng c\u00e1ch c\u1ea7u. Gi\u1ea3 s\u1eed {fn(z), n = 1, 2, . . . } h\u1ed9i t\u1ee5 c\u1ea7u \u0111\u1ec1u tr\u00ean E v\u00e0 f l\u00e0 h\u00e0m gi\u1edbi h\u1ea1n c\u1ee7a d\u00e3y h\u00e0m {fn(z), n = 1, 2, . . . } \u0111\u1ed1i v\u1edbi kho\u1ea3ng c\u00e1ch c\u1ea7u. V\u1edbi m\u1ed9t s\u1ed1 d\u01b0\u01a1ng \u03b5, t\u1eeb gi\u1ea3 thi\u1ebft {fn(z), n = 1, 2, . . . } h\u1ed9i t\u1ee5 c\u1ea7u \u0111\u1ec1u tr\u00ean E suy ra t\u1ed3n t\u1ea1i m\u1ed9t s\u1ed1 nguy\u00ean d\u01b0\u01a1ng N sao cho khi n \u2265 N, m \u2265 N ta c\u00f3 \u03b5 \u03c1S (fn(z ), fm(z)) < , 2 v\u1edbi m\u1ecdi z \u2208 E. Do \u0111\u00f3 khi n \u2265 N, m \u2265 N v\u00e0 z \u2208 E, ta c\u00f3: \u03c1S(fn(z), f (z)) \u2a7d \u03c1S(fn(z), fm(z)) + \u03c1S(fm(z), f (z)) \u03b5 < 2 + \u03c1S(fm(z), f (z)). Cho m \u2192 \u221e, ta nh\u1eadn \u0111\u01b0\u1ee3c \u03b5 \u03c1S(fn(z), f (z)) \u2a7d 2 < \u03b5, v\u1edbi m\u1ecdi z \u2208 E, v\u1edbi m\u1ecdi n \u2a7e N . Khi \u0111\u00f3 ta n\u00f3i r\u1eb1ng d\u00e3y h\u00e0m fn(z) h\u1ed9i t\u1ee5 \u0111\u1ec1u \u0111\u1ed1i v\u1edbi kho\u1ea3ng c\u00e1ch c\u1ea7u (hay h\u1ed9i t\u1ee5 c\u1ea7u \u0111\u1ec1u) \u0111\u1ebfn h\u00e0m f (z) tr\u00ean E khi n \u2192 +\u221e. B\u1ed5 \u0111\u1ec1 3.1.9. N\u1ebfu f (z) l\u00e0 m\u1ed9t h\u00e0m ph\u00e2n h\u00ecnh trong mi\u1ec1n D th\u00ec f (z) li\u00ean t\u1ee5c trong D \u0111\u1ed1i v\u1edbi kho\u1ea3ng c\u00e1ch c\u1ea7u. T\u1ee9c l\u00e0, v\u1edbi m\u1ed7i \u0111i\u1ec3m z0 \u2208 D ta lu\u00f4n c\u00f3: lim \u03c1S (f (z), f (z0)) = 0. (3.4) z \u2192 z0 \u0110\u1ecbnh ngh\u0129a 3.1.10. Cho S = {fn(z), n = 1, 2, . . . } l\u00e0 m\u1ed9t d\u00e3y c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh x\u00e1c \u0111\u1ecbnh trong m\u1ed9t mi\u1ec1n D. M\u1ed9t \u0111i\u1ec3m z0 \u2208 D \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 C0- \u0111i\u1ec3m c\u1ee7a d\u00e3y S n\u1ebfu t\u1ed3n t\u1ea1i m\u1ed9t h\u00ecnh tr\u00f2n U = {|z \u2212 z0| < r} \u2282 D sao cho","62 d\u00e3y S l\u00e0 h\u1ed9i t\u1ee5 \u0111\u1ec1u trong U \u0111\u1ed1i v\u1edbi kho\u1ea3ng c\u00e1ch c\u1ea7u. D\u00e3y S \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 C0-d\u00e3y trong D, n\u1ebfu m\u1ed7i \u0111i\u1ec3m c\u1ee7a D \u0111\u1ec1u l\u00e0 m\u1ed9t C0-\u0111i\u1ec3m c\u1ee7a S. Gi\u1ea3 s\u1eed d\u00e3y S h\u1ed9i t\u1ee5 c\u1ea7u \u0111\u1ec1u trong h\u00ecnh tr\u00f2n U = {|z \u2212 z0| < r}, khi \u0111\u00f3 d\u00e3y S c\u00f3 h\u00e0m gi\u1edbi h\u1ea1n f (z) x\u00e1c \u0111\u1ecbnh trong U \u0111\u1ed1i v\u1edbi kho\u1ea3ng c\u00e1ch c\u1ea7u v\u00e0 khi n \u2192 +\u221e th\u00ec fn(z) h\u1ed9i t\u1ee5 \u0111\u1ec1u \u0111\u1ebfn f (z) trong U \u0111\u1ed1i v\u1edbi kho\u1ea3ng c\u00e1ch c\u1ea7u. M\u1ec7nh \u0111\u1ec1 3.1.11. Cho S = {fn(z), n = 1, 2, . . . } l\u00e0 C0-d\u00e3y c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh trong m\u1ed9t mi\u1ec1n D. Khi \u0111\u00f3 h\u00e0m gi\u1edbi h\u1ea1n f (z) c\u1ee7a d\u00e3y S \u0111\u1ed1i v\u1edbi kho\u1ea3ng c\u00e1ch c\u1ea7u l\u00e0 m\u1ed9t h\u00e0m ph\u00e2n h\u00ecnh trong D ho\u1eb7c \u221e. \u0110\u1ecbnh ngh\u0129a 3.1.12. Cho D \u2282 C l\u00e0 m\u1ed9t mi\u1ec1n v\u00e0 F l\u00e0 m\u1ed9t h\u1ecd c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh tr\u00ean D. H\u1ecd F \u0111\u01b0\u1ee3c g\u1ecdi l\u00e0 h\u1ecd chu\u1ea9n t\u1eafc tr\u00ean D n\u1ebfu m\u1ecdi d\u00e3y {fn} \u2282 F lu\u00f4n t\u1ed3n t\u1ea1i m\u1ed9t d\u00e3y con c\u1ee7a {fn} h\u1ed9i t\u1ee5 c\u1ea7u \u0111\u1ec1u tr\u00ean m\u1ecdi t\u1eadp con compact c\u1ee7a D. {z \u2208 |z| 1}. n V\u00ed d\u1ee5. K\u00ed hi\u1ec7u U = : < Cho fn(z) = ,n = 1, 2, 3, ..., tr\u00ean C z U . Khi \u0111\u00f3 fn l\u00e0 h\u00e0m ph\u00e2n h\u00ecnh v\u00e0 {fn} h\u1ed9i t\u1ee5 c\u1ea7u \u0111\u1ec1u \u0111\u1ecba ph\u01b0\u01a1ng t\u1edbi \u221e trong U. \u0110\u1ecbnh ngh\u0129a 3.1.13. Cho F l\u00e0 h\u1ecd c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh trong m\u1ed9t mi\u1ec1n D v\u00e0 z0 l\u00e0 m\u1ed9t \u0111i\u1ec3m c\u1ee7a D. Ta n\u00f3i r\u1eb1ng h\u1ecd F l\u00e0 chu\u1ea9n t\u1eafc t\u1ea1i z0, n\u1ebfu t\u1ed3n t\u1ea1i m\u1ed9t h\u00ecnh tr\u00f2n U = {|z \u2212 z0| < r} \u2282 D sao cho h\u1ecd F l\u00e0 chu\u1ea9n t\u1eafc tr\u00ean h\u00ecnh tr\u00f2n U . T\u1eeb \u0111\u1ecbnh ngh\u0129a ta d\u1ec5 d\u00e0ng suy ra n\u1ebfu F chu\u1ea9n t\u1eafc tr\u00ean D th\u00ec F chu\u1ea9n t\u1eafc m\u1ed7i \u0111i\u1ec3m c\u1ee7a D. Chi\u1ec1u ng\u01b0\u1ee3c l\u1ea1i ta c\u00f3 k\u1ebft qu\u1ea3 sau: M\u1ec7nh \u0111\u1ec1 3.1.14. Cho F l\u00e0 h\u1ecd c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh trong m\u1ed9t mi\u1ec1n D. N\u1ebfu h\u1ecd F chu\u1ea9n t\u1eafc t\u1ea1i m\u1ed7i \u0111i\u1ec3m c\u1ee7a D th\u00ec F chu\u1ea9n t\u1eafc tr\u00ean D. \u0110\u1ecbnh ngh\u0129a 3.1.15. Cho F l\u00e0 h\u1ecd c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh trong m\u1ed9t mi\u1ec1n D. Ta n\u00f3i r\u1eb1ng F l\u00e0 b\u1ecb ch\u1eb7n \u0111\u1ec1u \u0111\u1ecba ph\u01b0\u01a1ng trong D, n\u1ebfu cho m\u1ed7i z0 c\u1ee7a D,","63 t\u1ed3n t\u1ea1i m\u1ed9t h\u00ecnh tr\u00f2n U = {|z \u2212 z0| < r} n\u1eb1m trong D v\u00e0 m\u1ed9t s\u1ed1 d\u01b0\u01a1ng M sao cho v\u1edbi m\u1ed7i f \u2208 F |f (z)| \u2264 M, (3.5) \u0111\u00fang v\u1edbi m\u1ecdi z \u2208 U . M\u1ec7nh \u0111\u1ec1 3.1.16. Cho F l\u00e0 h\u1ecd c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh trong m\u1ed9t mi\u1ec1n D. N\u1ebfu F b\u1ecb ch\u1eb7n \u0111\u1ec1u \u0111\u1ecba ph\u01b0\u01a1ng trong D, khi \u0111\u00f3 F chu\u1ea9n t\u1eafc trong D. Cho F l\u00e0 m\u1ed9t h\u1ecd c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh tr\u00ean mi\u1ec1n D \u2282 C, v\u1edbi m\u1ed7i f \u2208 F , \u0111\u1ea1o h\u00e0m c\u1ea7u c\u1ee7a h\u00e0m f \u0111\u01b0\u1ee3c \u0111\u1ecbnh ngh\u0129a b\u1edfi: f #(z) = 1 |f \u2032(z)| + |f (z)|2 . \u0110\u1ecbnh l\u00fd 3.1.17 (\u0110\u1ecbnh l\u00fd Marty). M\u1ed9t h\u1ecd F c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh tr\u00ean m\u1ed9t mi\u1ec1n D \u2282 C l\u00e0 chu\u1ea9n t\u1eafc tr\u00ean mi\u1ec1n D khi v\u00e0 ch\u1ec9 khi t\u1eadp c\u00e1c \u0111\u1ea1o h\u00e0m c\u1ea7u {f #(z), f \u2208 F } b\u1ecb ch\u1eb7n \u0111\u1ec1u tr\u00ean m\u1ed7i t\u1eadp con compact c\u1ee7a D. M\u1ec7nh \u0111\u1ec1 3.1.18 (B\u1ed5 \u0111\u1ec1 Zalcman, [53]). Cho F l\u00e0 m\u1ed9t h\u1ecd c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh tr\u00ean \u0111\u0129a m\u1edf \u25b3 = {z \u2208 C : |z| < 1}. Khi \u0111\u00f3 n\u1ebfu F kh\u00f4ng chu\u1ea9n t\u1eafc t\u1ea1i m\u1ed9t \u0111i\u1ec3m z0 \u2208 \u25b3, th\u00ec v\u1edbi m\u1ed7i s\u1ed1 th\u1ef1c \u03b1 th\u1ecfa m\u00e3n \u22121 < \u03b1 < 1, t\u1ed3n t\u1ea1i 1) m\u1ed9t s\u1ed1 th\u1ef1c r, 0 < r < 1 v\u00e0 m\u1ed9t d\u00e3y \u0111i\u1ec3m zn, |zn| < r, zn \u2192 z0, 2) d\u00e3y c\u00e1c s\u1ed1 d\u01b0\u01a1ng \u03c1n, \u03c1n \u2192 0+, 3) d\u00e3y c\u00e1c h\u00e0m fn, fn \u2208 F th\u1ecfa m\u00e3n d\u00e3y h\u00e0m gn(\u03be) = fn(zn + \u03c1n\u03be) h\u1ed9i \u03c1\u03b1n t\u1ee5 c\u1ea7u \u0111\u1ec1u v\u1ec1 h\u00e0m g(\u03be) tr\u00ean c\u00e1c t\u1eadp con compact c\u1ee7a C, trong \u0111\u00f3 g(\u03be) l\u00e0 m\u1ed9t h\u00e0m ph\u00e2n h\u00ecnh kh\u00e1c h\u1eb1ng v\u00e0 g#(\u03be) \u2a7d g#(0) = 1. H\u01a1n n\u1eefa, b\u1eadc c\u1ee7a g kh\u00f4ng l\u1edbn h\u01a1n 2. M\u1ec7nh \u0111\u1ec1 3.1.19 ([9]). Cho g l\u00e0 m\u1ed9t h\u00e0m nguy\u00ean v\u00e0 M l\u00e0 m\u1ed9t h\u1eb1ng s\u1ed1 d\u01b0\u01a1ng. N\u1ebfu g#(\u03be) \u2a7d M \u0111\u1ed1i v\u1edbi m\u1ecdi \u03be \u2208 C, th\u00ec g c\u00f3 b\u1eadc cao nh\u1ea5t l\u00e0 1.","64 Ch\u00fa \u00fd. Trong M\u1ec7nh \u0111\u1ec1 3.1.18, n\u1ebfu F l\u00e0 m\u1ed9t h\u1ecd c\u00e1c h\u00e0m ch\u1ec9nh h\u00ecnh, th\u00ec g l\u00e0 m\u1ed9t h\u00e0m ch\u1ec9nh h\u00ecnh d\u1ef1a tr\u00ean \u0111\u1ecbnh l\u00fd Hurwitz. Do \u0111\u00f3, b\u1eadc c\u1ee7a g kh\u00f4ng l\u1edbn h\u01a1n 1 theo M\u1ec7nh \u0111\u1ec1 3.1.19. 3.2. V\u1ea5n \u0111\u1ec1 duy nh\u1ea5t Trong ph\u1ea7n n\u00e0y ch\u00fang t\u00f4i s\u1ebd ph\u00e1t bi\u1ec3u v\u00e0 ch\u1ee9ng minh m\u1ed9t \u0111\u1ecbnh l\u00fd v\u1ec1 v\u1ea5n \u0111\u1ec1 duy nh\u1ea5t cho c\u00e1c h\u00e0m nguy\u00ean li\u00ean quan \u0111\u1ebfn gi\u1ea3 thuy\u1ebft Bru\u00a8ck. K\u1ef9 thu\u1eadt ch\u1ee9ng minh k\u1ebft qu\u1ea3 n\u00e0y d\u1ef1a tr\u00ean m\u1ed9t k\u1ebft qu\u1ea3 v\u1ec1 ti\u00eau chu\u1ea9n chu\u1ea9n t\u1eafc c\u1ee7a h\u1ecd c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh ph\u1ee9c \u0111\u01b0\u1ee3c ch\u00fang t\u00f4i ph\u00e1t bi\u1ec3u v\u00e0 ch\u1ee9ng minh trong ph\u1ea7n \u0111\u1ea7u ti\u00ean c\u1ee7a m\u1ee5c n\u00e0y. 3.2.1. Ti\u00eau chu\u1ea9n chu\u1ea9n t\u1eafc c\u1ee7a h\u1ecd c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh \u0110\u1ec3 ch\u1ee9ng minh k\u1ebft qu\u1ea3 c\u1ee7a ch\u00fang t\u00f4i v\u1ec1 ti\u00eau chu\u1ea9n chu\u1ea9n t\u1eafc c\u1ee7a h\u1ecd c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh ph\u1ee9c, tr\u01b0\u1edbc h\u1ebft ch\u00fang t\u00f4i ch\u1ee9ng minh m\u1ed9t s\u1ed1 b\u1ed5 \u0111\u1ec1 c\u1ea7n thi\u1ebft sau \u0111\u00e2y: M\u1ec7nh \u0111\u1ec1 3.2.1 ([47]). Cho f l\u00e0 m\u1ed9t h\u00e0m ph\u00e2n h\u00ecnh si\u00eau vi\u1ec7t v\u00e0 a l\u00e0 m\u1ed9t h\u1eb1ng s\u1ed1 ph\u1ee9c. G\u1ecdi n \u2208 N, k, nj, tj \u2208 N\u2217, j = 1, . . . , k th\u1ecfa m\u00e3n kk n + nj \u2a7e tj + 3. j=1 j=1 Khi \u0111\u00f3 ph\u01b0\u01a1ng tr\u00ecnh f n(f n1)(t1) . . . (f nk )(tk) = a c\u00f3 v\u00f4 s\u1ed1 nghi\u1ec7m. H\u01a1n n\u1eefa, n\u1ebfu f l\u00e0 m\u1ed9t h\u00e0m nguy\u00ean si\u00eau vi\u1ec7t, kh\u1eb3ng \u0111\u1ecbnh kk \u0111\u00fang khi n + nj \u2a7e tj + 2. j=1 j=1 Ch\u1ee9ng minh. \u0110\u1eb7t P (f ) = f n(f n1)(t1) . . . (f nk)(tk). Ta th\u1ea5y (f nj )(tj) = cm0,m1,...,mtj f m0(f \u2032)m1 . . . (f )(tj) mtj ,","65 trong \u0111\u00f3 cm0,m1,...,mtj l\u00e0 c\u00e1c h\u1eb1ng s\u1ed1, m0, m1, . . . , mtj l\u00e0 c\u00e1c h\u1eb1ng s\u1ed1 kh\u00f4ng \u00e2m th\u1ecfa m\u00e3n m0 + \u00b7 \u00b7 \u00b7 + mtj = nj v\u00e0 tj jmj = tj . Do \u0111\u00f3 ta d\u1ec5 d\u00e0ng t\u00ednh j=1 to\u00e1n \u0111\u01b0\u1ee3c kk d(P ) = n + nj v\u00e0 \u03b8(P ) = tj. j=1 j=1 S\u1eed d\u1ee5ng M\u1ec7nh \u0111\u1ec1 3.1.1 v\u1edbi f v\u00e0 P (f ), ta c\u00f3 k T (r, f ) \u2a7d tj + 1 1 1 N (r, 1 ) N (r, ) + a j=1 k P \u2212 f k n + nj \u2212 1 n + nj \u2212 1 j=1 j=1 + o(T (r, f )). kk V\u00ec n + nj \u2a7e tj + 3, ta thu \u0111\u01b0\u1ee3c ph\u01b0\u01a1ng tr\u00ecnh j=1 j=1 f n(f n1)(t1) . . . (f nk )(tk) = a c\u00f3 v\u00f4 s\u1ed1 nghi\u1ec7m. H\u01a1n n\u1eefa, n\u1ebfu f l\u00e0 m\u1ed9t h\u00e0m nguy\u00ean si\u00eau vi\u1ec7t, ta c\u00f3 k T (r, f ) \u2a7d j=1 tj + 1 N (r, 1 ) + 11 kf N (r, ) k P \u2212a n + nj n + nj j=1 j=1 + o(T (r, f )). kk Do \u0111\u00f3 \u0111i\u1ec1u ki\u1ec7n n + nj \u2a7e tj + 2 k\u00e9o theo j=1 j=1 f n(f n1)(t1) . . . (f nk )(tk) = a c\u00f3 v\u00f4 s\u1ed1 nghi\u1ec7m. M\u1ec7nh \u0111\u1ec1 3.2.2 ([47]). Cho f l\u00e0 m\u1ed9t h\u00e0m h\u1eefu t\u1ef7 kh\u00e1c h\u1eb1ng v\u00e0 a l\u00e0 m\u1ed9t h\u1eb1ng s\u1ed1 ph\u1ee9c kh\u00e1c kh\u00f4ng. Cho n \u2208 N, k, nj, tj \u2208 N\u2217, j = 1, . . . , k th\u1ecfa m\u00e3n kk nj \u2a7e tj, n + nj \u2a7e tj + 2, j = 1, . . . , k. j=1 j=1","66 Khi \u0111\u00f3 ph\u01b0\u01a1ng tr\u00ecnh f n(f n1)(t1) . . . (f nk )(tk) = a c\u00f3 \u00edt nh\u1ea5t hai kh\u00f4ng \u0111i\u1ec3m ph\u00e2n bi\u1ec7t. Ch\u1ee9ng minh. Ta x\u00e9t hai tr\u01b0\u1eddng h\u1ee3p sau: Tr\u01b0\u1eddng h\u1ee3p 1. f l\u00e0 m\u1ed9t \u0111a th\u1ee9c. Gi\u1ea3 s\u1eed f l\u00e0 m\u1ed9t \u0111a th\u1ee9c b\u1eadc m \u2a7e 1, khi \u0111\u00f3 f n(f n1)(t1) . . . (f nk )(tk) c\u0169ng l\u00e0 m\u1ed9t \u0111a th\u1ee9c c\u00f3 b\u1eadc l\u00e0 kk kk m n + nj \u2212 tj \u2a7e n + nj \u2212 tj \u2a7e 2. j=1 j=1 j=1 j=1 Do \u0111\u00f3, n\u1ebfu f n(f n1)(t1) . . . (f nk )(tk) \u2212 a c\u00f3 m\u1ed9t kh\u00f4ng \u0111i\u1ec3m duy nh\u1ea5t l\u00e0 z0, th\u00ec f n(f n1)(t1) . . . (f nk)(tk) \u2212 a = A(z \u2212 z0)l, trong \u0111\u00f3 l \u2a7e 2 v\u00e0 A l\u00e0 m\u1ed9t h\u1eb1ng s\u1ed1 kh\u00e1c kh\u00f4ng, suy ra (f n(f n1)(t1) . . . (f nk)(tk))\u2032 = Al(z \u2212 z0)l\u22121. \u0110i\u1ec1u n\u00e0y k\u00e9o theo z0 l\u00e0 kh\u00f4ng \u0111i\u1ec3m duy nh\u1ea5t c\u1ee7a (f n(f n1)(t1) . . . (f nk )(tk))\u2032. V\u00ec f l\u00e0 \u0111a th\u1ee9c n\u00ean f c\u00f3 kh\u00f4ng \u0111i\u1ec3m. Gi\u1ea3 s\u1eed z\u2217 l\u00e0 m\u1ed9t kh\u00f4ng \u0111i\u1ec3m b\u1ed9i m \u2a7e 1 c\u1ee7a f , khi \u0111\u00f3 z\u2217 l\u00e0 kh\u00f4ng \u0111i\u1ec3m b\u1ed9i kk m n + nj \u2212 tj \u2a7e 2 j=1 j=1 c\u1ee7a f n(f n1)(t1) . . . (f nk)(tk) v\u00e0 do \u0111\u00f3 n\u00f3 l\u00e0 m\u1ed9t kh\u00f4ng \u0111i\u1ec3m c\u1ee7a (f n(f n1)(t1) . . . (f nk )(tk))\u2032.","67 Nh\u01b0 \u0111\u00e3 n\u00f3i \u1edf tr\u00ean, (f n(f n1)(t1) . . . (f nk)(tk))\u2032 c\u00f3 kh\u00f4ng \u0111i\u1ec3m duy nh\u1ea5t l\u00e0 z0 n\u00ean z0 ph\u1ea3i l\u00e0 kh\u00f4ng \u0111i\u1ec3m c\u1ee7a f . Ta th\u1ea5y r\u1eb1ng 0 = f n(z0)(f n1)(t1)(z0) . . . (f nk)(tk)(z0) = a \u0338= 0. \u0110\u00f3 l\u00e0 m\u00e2u thu\u1eabn. K\u00e9o theo f n(f n1)(t1) . . . (f nk )(tk) = a c\u00f3 \u00edt nh\u1ea5t hai kh\u00f4ng \u0111i\u1ec3m ph\u00e2n bi\u1ec7t. Tr\u01b0\u1eddng h\u1ee3p 2. f l\u00e0 m\u1ed9t h\u00e0m h\u1eefu t\u1ef7 v\u00e0 kh\u00f4ng ph\u1ea3i l\u00e0 \u0111a th\u1ee9c. Ta xem x\u00e9t hai tr\u01b0\u1eddng h\u1ee3p c\u00f3 th\u1ec3 x\u1ea3y ra Tr\u01b0\u1eddng h\u1ee3p 2.1. f c\u00f3 kh\u00f4ng \u0111i\u1ec3m. Khi \u0111\u00f3 f c\u00f3 th\u1ec3 bi\u1ec3u di\u1ec5n \u0111\u01b0\u1ee3c d\u01b0\u1edbi d\u1ea1ng s (z \u2212 \u03b1i)mi f = Ai=1 , (3.1) t (z \u2212 \u03b2l)dl l=1 trong \u0111\u00f3 A \u0338= 0, mi \u2a7e 1, i = 1, . . . , s v\u00e0 dl \u2a7e 1, l = 1, . . . , t. \u0110\u1eb7t M = m1 + \u00b7 \u00b7 \u00b7 + ms \u2a7e s, N = d1 + \u00b7 \u00b7 \u00b7 + dt \u2a7e t. Khi \u0111\u00f3, v\u1edbi m\u1ed7i j = 1, 2, . . . , k, ta c\u00f3 s (z \u2212 \u03b1i)njmi f nj = Anj i=1 , j = 1, . . . , k. (3.2) (3.3) t (z \u2212 \u03b2l)njdl l=1 \u0110i\u1ec1u n\u00e0y k\u00e9o theo s (z \u2212 \u03b1i)njmi\u2212tj (f nj )(tj) = Anj i=1 gj (z ), t (z \u2212 \u03b2l)njdl+tj l=1 trong \u0111\u00f3 gj l\u00e0 m\u1ed9t \u0111a th\u1ee9c v\u1edbi deg gj(z) \u2a7d tj(s + t \u2212 1), j = 1, . . . , k.","68 K\u1ebft h\u1ee3p (3.1), (3.2) v\u00e0 (3.3), ta c\u00f3 kk s (n+ nj)mi\u2212 tj (z \u2212 \u03b1i) j=1 f n(f n1)(t1) . . . (f nk )(tk) = i=1 j=1 (3.4) P (z) g(z) = , kk Q(z) t (n+ nj)dl+ tj (z \u2212 \u03b2l) j=1 j=1 l=1 trong \u0111\u00f3 k k v\u1edbi Gi\u1ea3 s\u1eed r\u1eb1ng n+ nj gv(z) g(z) = A j=1 v=1 k deg g(z) \u2a7d (s + t \u2212 1) tj. j=1 f n(f n1)(t1) . . . (f nk )(tk) = a c\u00f3 m\u1ed9t kh\u00f4ng \u0111i\u1ec3m duy nh\u1ea5t z0. Khi \u0111\u00f3 z0 =\u0338 \u03b1i v\u1edbi m\u1ecdi i = 1, . . . , s. Th\u1ef1c v\u1eady, n\u1ebfu z0 = \u03b1i v\u1edbi i \u2208 {1, . . . , s}, khi \u0111\u00f3 0 = f n(z0)(f n1)(t1)(z0) . . . (f nk)(tk)(z0) = a \u0338= 0. \u0110\u00e2y l\u00e0 \u0111i\u1ec1u m\u00e2u thu\u1eabn. T\u1eeb (3.4) ta c\u00f3 f n(f n1)(t1) . . . (f nk)(tk) = a + B(z \u2212 z0)l , (3.5) k k t (n+ nj)dl+ tj (z \u2212 \u03b2l) j=1 j=1 l=1 trong \u0111\u00f3 B l\u00e0 m\u1ed9t h\u1eb1ng s\u1ed1 kh\u00e1c 0. \u0110i\u1ec1u \u0111\u00f3 k\u00e9o theo (f n(f n1)(t1) . . . (f nk )(tk))\u2032 = (z \u2212 z0)l\u22121G1(z) , (3.6) kk t (n+ nj)dl+ tj+1 (z \u2212 \u03b2l) j=1 j=1 l=1 trong \u0111\u00f3 kk G1(z) = B(l \u2212 (n + nj)N \u2212 t tj)zt + b1zt\u22121 + \u00b7 \u00b7 \u00b7 + bt. j=1 j=1","69 C\u0169ng t\u1eeb (3.4) ta c\u00f3 kk s (n+ nj)mi+ tj\u22121 (z \u2212 \u03b1i) j=1 j=1 G2(z) (f n(f n1)(t1) . . . (f nk )(tk))\u2032 = i=1 kk . (3.7) t (n+ nj)dl+ tj+1 (z \u2212 \u03b21) j=1 j=1 l=1 D\u1ec5 ki\u1ec3m tra \u0111\u01b0\u1ee3c k s + t \u2212 1 \u2a7d deg G2(z) \u2a7d ( tj + 1)(s + t \u2212 1). j=1 kk a) N\u1ebfu l =\u0338 (n + nj)N + t tj, khi \u0111\u00f3 deg P (z) \u2a7e deg Q(z). T\u1eeb (3.4), j=1 j=1 ta c\u00f3 sk k tk k (n + nj)mi \u2212 tj + deg g \u2a7e (n + nj)dl + tj . i=1 j=1 j=1 l=1 j=1 j=1 Ta ch\u00fa \u00fd r\u1eb1ng k deg g(z) \u2a7d ( tj)(s + t \u2212 1). j=1 \u0110i\u1ec1u n\u00e0y k\u00e9o theo k tj M \u2a7e N + j=1 , k n + nj j=1 do \u0111\u00f3 M > N. V\u00ec z0 =\u0338 \u03b1i v\u1edbi m\u1ecdi i = 1, . . . , s n\u00ean ta c\u00f3 sk k (n + nj)mi \u2212 tj \u2212 1 \u2a7d deg G1 = t. i=1 j=1 j=1 K\u00e9o theo k kk (3.8) (n + nj)M \u2a7d (1 + tj)s + t < ( tj + 2)M. j=1 j=1 j=1 kk Ta ch\u00fa \u00fd r\u1eb1ng n + nj \u2a7e tj + 2, nh\u01b0 v\u1eady (3.8) cho ta m\u00e2u thu\u1eabn. j=1 j=1","70 kk b) N\u1ebfu l = (n + nj)N + ( tj)t. j=1 j=1 N\u1ebfu M > N , l\u1eadp lu\u1eadn gi\u1ed1ng nh\u01b0 Tr\u01b0\u1eddng h\u1ee3p 1 ta c\u0169ng c\u00f3 m\u00e2u thu\u1eabn. N\u1ebfu M \u2a7d N . V\u00ec k l \u2212 1 \u2a7d deg G2 \u2a7d ( tj + 1)(s + t \u2212 1), j=1 n\u00ean kk k (n + tj)N = l \u2212 ( tj)t \u2a7d deg G2 + 1 \u2212 ( tj)t j=1 j=1 j=1 kk < (1 + tj)s + t \u2a7d ( tj + 2)N. (3.9) j=1 j=1 kk T\u1eeb \u0111i\u1ec1u ki\u1ec7n n + nj \u2a7e tj + 2 v\u00e0 (3.9), ta c\u00f3 m\u00e2u thu\u1eabn. j=1 j=1 Tr\u01b0\u1eddng h\u1ee3p 2.2. f kh\u00f4ng c\u00f3 kh\u00f4ng \u0111i\u1ec3m. Khi \u0111\u00f3 f \u0111\u01b0\u1ee3c bi\u1ec3u di\u1ec5n d\u01b0\u1edbi d\u1ea1ng A , dl \u2a7e 1, l = 1, . . . , t. (3.10) f= t (z \u2212 \u03b2l)dl l=1 Nh\u01b0 th\u1ebf, v\u1edbi m\u1ed7i j = 1, 2, . . . , k ta c\u00f3 (f nj )(tj) = Anj gj (z ), (3.11) t (z \u2212 \u03b2l)njdl+tj l=1 trong \u0111\u00f3 gj l\u00e0 m\u1ed9t \u0111a th\u1ee9c v\u1edbi deg gj(z) \u2a7d tj(t \u2212 1). \u0110i\u1ec1u n\u00e0y k\u00e9o theo f n(f n1)(t1) . . . (f nk )(tk) = g(z) g(z) =, (3.12) kk Q(z) t (n+ nj)dl+ tj (z \u2212 \u03b2l) j=1 j=1 l=1 k n+ nj k k trong \u0111\u00f3 g(z) = A j=1 gv(z) v\u1edbi deg g(z) \u2a7d ( tj)(t \u2212 1). Ta th\u1ea5y v=1 j=1 r\u1eb1ng f n(f n1)(t1) . . . (f nk)(tk) \u2212 a = g(z) \u2212 aQ(z) (3.13) . Q(z)","71 Do N = d1 + \u00b7 \u00b7 \u00b7 + dt \u2a7e t ta c\u00f3 kk deg Q \u2a7e (n + nj + tj)t > deg g, j=1 j=1 \u0111i\u1ec1u n\u00e0y k\u00e9o theo g(z) \u2212 aQ(z) = 0 l\u00e0 m\u1ed9t \u0111a th\u1ee9c b\u1eadc l\u1edbn h\u01a1n 1 n\u00ean c\u00f3 c\u00f3 \u00edt nh\u1ea5t m\u1ed9t kh\u00f4ng \u0111i\u1ec3m v\u00e0 kh\u00f4ng tr\u00f9ng v\u1edbi kh\u00f4ng \u0111i\u1ec3m c\u1ee7a Q(z). K\u00e9o theo f n(f n1)(t1) . . . (f nk)(tk) = a c\u00f3 \u00edt nh\u1ea5t m\u1ed9t nghi\u1ec7m. Ta gi\u1ea3 s\u1eed r\u1eb1ng f n(f n1)(t1) . . . (f nk )(tk) = a c\u00f3 m\u1ed9t kh\u00f4ng \u0111i\u1ec3m duy nh\u1ea5t z0, khi \u0111\u00f3 f n(f n1)(t1) . . . (f nk)(tk) = a + B(z \u2212 z0)l , (3.14) kk t (n+ nj)dl+ tj (z \u2212 \u03b2l) j=1 j=1 l=1 trong \u0111\u00f3 B l\u00e0 m\u1ed9t h\u1eb1ng s\u1ed1 kh\u00e1c kh\u00f4ng. \u0110i\u1ec1u n\u00e0y k\u00e9o theo (f n(f n1)(t1) . . . (f nk )(tk))\u2032 = (z \u2212 z0)l\u22121G1(z) , (3.15) kk t (n+ nj)dl+ tj+1 (z \u2212 \u03b2l) j=1 j=1 l=1 trong \u0111\u00f3 kk G1(z) = B(l \u2212 (n + nj)N \u2212 ( tj)t)zt + b1zt\u22121 + \u00b7 \u00b7 \u00b7 + bt. j=1 j=1 T\u1eeb (3.12), ta c\u00f3 (f n(f n1)(t1) . . . (f nk )(tk))\u2032 = G2(z) . (3.16) k k t (n+ nj)dl+ tj+1 (z \u2212 \u03b21) j=1 j=1 l=1 D\u1ec5 d\u00e0ng ki\u1ec3m tra \u0111\u01b0\u1ee3c k t \u2212 1 \u2a7d deg G2(z) \u2a7d ( tj + 1)(t \u2212 1). j=1 Ta xem x\u00e9t hai tr\u01b0\u1eddng h\u1ee3p c\u00f3 th\u1ec3 x\u1ea3y ra","72 kk a) N\u1ebfu l \u0338= (n + nj)N + ( tj)t, khi \u0111\u00f3 deg g(z) \u2a7e deg Q(z). T\u1eeb j=1 j=1 (3.12), ta c\u00f3 tk k kk deg g \u2a7e ((n + nj)dj + tj) = (n + nj)N + ( tj)t. j=1 j=1 j=1 j=1 j=1 k Ch\u00fa \u00fd r\u1eb1ng deg g(z) \u2a7d ( tj)(t \u2212 1). \u0110\u00f3 l\u00e0 \u0111i\u1ec1u m\u00e2u thu\u1eabn. j=1 kk b) N\u1ebfu l = (n + nj)N + ( tj)t. T\u1eeb j=1 j=1 k l \u2212 1 \u2a7d deg G2 \u2a7d ( tj + 1)(t \u2212 1), j=1 ta c\u00f3 kk k (n + nj)N = l \u2212 ( tj)t \u2a7d deg G2 + 1 \u2212 ( tj)t j=1 j=1 j=1 k = t \u2212 tj. (3.17) j=1 kk T\u1eeb \u0111i\u1ec1u ki\u1ec7n n + nj \u2a7e tj + 2 v\u00e0 t \u2a7d N, ta c\u00f3 j=1 j=1 kk ( tj + 2)N + tj \u2a7d N. j=1 j=1 \u0110\u00e2y l\u00e0 \u0111i\u1ec1u m\u00e2u thu\u1eabn. Nh\u01b0 v\u1eady ta thu \u0111\u01b0\u1ee3c f n(f n1)(t1) . . . (f nk )(tk) = a c\u00f3 \u00edt nh\u1ea5t hai kh\u00f4ng \u0111i\u1ec3m ph\u00e2n bi\u1ec7t. M\u1ec7nh \u0111\u1ec1 \u0111\u01b0\u1ee3c ch\u1ee9ng minh trong c\u00e1c tr\u01b0\u1eddng h\u1ee3p. Cho f v\u00e0 g l\u00e0 hai h\u00e0m ph\u00e2n h\u00ecnh v\u00e0 a v\u00e0 b l\u00e0 hai s\u1ed1 ph\u1ee9c ph\u00e2n bi\u1ec7t. Ta nh\u1eafc l\u1ea1i, n\u1ebfu g \u2212 b = 0 m\u1ed7i khi f \u2212 a = 0 th\u00ec ta vi\u1ebft f = a \u21d2 g = b. N\u1ebfu f = a \u21d2 g = b v\u00e0 g = b \u21d2 f = a th\u00ec ta vi\u1ebft f = a \u21d4 g = b. N\u1ebfu","73 f \u2212 a v\u00e0 g \u2212 b c\u00f3 chung kh\u00f4ng \u0111i\u1ec3m v\u00e0 c\u1ef1c \u0111i\u1ec3m k\u1ec3 c\u1ea3 b\u1ed9i th\u00ec ta k\u00ed hi\u1ec7u f \u2212 a \u21cc g \u2212 b. S\u1eed d\u1ee5ng kh\u00e1i ni\u1ec7m n\u00e0y ch\u00fang t\u00f4i \u0111\u00e3 ch\u1ee9ng minh k\u1ebft qu\u1ea3 sau v\u1ec1 ti\u00eau chu\u1ea9n chu\u1ea9n t\u1eafc c\u1ee7a m\u1ed9t h\u1ecd c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh. \u0110\u1ecbnh l\u00fd 3.2.3 ([47]). Cho F l\u00e0 m\u1ed9t h\u1ecd c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh tr\u00ean mi\u1ec1n D \u2282 C. Cho a v\u00e0 b l\u00e0 hai s\u1ed1 ph\u1ee9c th\u1ecfa m\u00e3n b \u0338= 0, g\u1ecdi n \u2208 N, nj, tj, k \u2208 N\u2217, (j = 1, 2, . . . , k) th\u1ecfa m\u00e3n kk (3.18) nj \u2a7e tj, n + nj \u2a7e tj + 3 j=1 j=1 v\u00e0 f n+n1+\u00b7\u00b7\u00b7+nk = a \u21d4 f n(f n1)(t1) . . . (f nk )(tk) = b (3.19) \u0111\u1ed1i v\u1edbi f \u2208 F . Khi \u0111\u00f3 F l\u00e0 m\u1ed9t h\u1ecd chu\u1ea9n t\u1eafc. Ngo\u00e0i ra, n\u1ebfu F l\u00e0 m\u1ed9t h\u1ecd c\u00e1c h\u00e0m ch\u1ec9nh h\u00ecnh th\u00ec kh\u1eb3ng \u0111\u1ecbnh \u0111\u00fang khi (3.18) \u0111\u01b0\u1ee3c thay th\u1ebf b\u1edfi m\u1ed9t trong c\u00e1c \u0111i\u1ec1u ki\u1ec7n sau: k = 1, n = 0, n1 \u2a7e t1 + 1; (3.20) (3.21) kk n \u2a7e 1 ho\u1eb7c k \u2a7e 2, nj \u2a7e tj, n + nj \u2a7e tj + 2. j=1 j=1 Ch\u1ee9ng minh. Kh\u00f4ng m\u1ea5t t\u00ednh t\u1ed5ng qu\u00e1t, ta c\u00f3 th\u1ec3 gi\u1ea3 thi\u1ebft r\u1eb1ng D l\u00e0 \u0111\u0129a \u0111\u01a1n v\u1ecb. Gi\u1ea3 s\u1eed r\u1eb1ng F kh\u00f4ng chu\u1ea9n t\u1eafc t\u1ea1i z0 \u2208 D. S\u1eed d\u1ee5ng M\u1ec7nh \u0111\u1ec1 3.1.18 kk v\u1edbi \u03b1 = tj\/ n + nj), ta suy ra t\u1ed3n t\u1ea1i m\u1ed9t d\u00e3y \u0111i\u1ec3m j=1 j=1 zv : |zv| < r, v = 1, 2, . . . , \u221e, zv \u2192 z0, t\u1ed3n t\u1ea1i d\u00e3y s\u1ed1 d\u01b0\u01a1ng \u03c1v, v = 1, 2, . . . , \u221e, \u03c1v \u2192 0+ v\u00e0 d\u00e3y c\u00e1c h\u00e0m fv, fv \u2208 F , v = 1, 2, . . . , \u221e sao cho h\u00e0m gv(\u03be) = fv(zv + \u03c1v\u03be) \u03c1v\u03b1","74 h\u1ed9i t\u1ee5 c\u1ea7u \u0111\u1ec1u \u0111\u1ebfn h\u00e0m g(\u03be) tr\u00ean c\u00e1c t\u1eadp con compact c\u1ee7a C, trong \u0111\u00f3 g(\u03be) l\u00e0 m\u1ed9t h\u00e0m ph\u00e2n h\u00ecnh kh\u00e1c h\u1eb1ng. T\u1eeb c\u00e1ch x\u00e1c \u0111\u1ecbnh h\u00e0m gv ta c\u00f3 fvn(zv + \u03c1v\u03be)(fvn1)(t1)(zv + \u03c1v\u03be) . . . (fvnk)(tk)(zv + \u03c1v\u03be) \u2212 b = gvn(\u03be)(gvn1(\u03be))(t1) . . . (gvnk(\u03be))(tk) \u2212 b. \u0110i\u1ec1u n\u00e0y k\u00e9o theo fvn(zv + \u03c1v\u03be)(fvn1)(t1)(zv + \u03c1v\u03be) . . . (fvnk)(tk)(zv + \u03c1v\u03be) \u2212 b (3.22) \u2192 gn(\u03be)(gn1(\u03be))(t1) . . . (gnk(\u03be))(tk) \u2212 b \u0111\u1ec1u (v\u1edbi kho\u1ea3ng c\u00e1ch c\u1ea7u) tr\u00ean m\u1ed7i t\u1eadp con compact c\u1ee7a C \\\\{c\u1ef1c \u0111i\u1ec3m c\u1ee7a g}. Ta x\u00e9t hai tr\u01b0\u1eddng h\u1ee3p: Tr\u01b0\u1eddng h\u1ee3p 1. a \u0338= 0. L\u1ea5y M l\u00e0 m\u1ed9t h\u1eb1ng s\u1ed1 d\u01b0\u01a1ng sao cho 1 M |a| .\u2a7d n+n1+\u00b7\u00b7\u00b7+nk V\u1edbi m\u1ed7i f \u2208 F , ta k\u00ed hi\u1ec7u Ef b\u1edfi (3.23) Ef = z \u2208 D : f n(f n1)(t1) . . . (f nk)(tk) = b . Khi \u0111\u00f3 |f (z)| \u2a7e M v\u1edbi m\u1ed7i f \u2208 F v\u00e0 z \u2208 Ef . B\u00e2y gi\u1edd ta ch\u1ee9ng minh ph\u01b0\u01a1ng tr\u00ecnh gn(\u03be)(gn1(\u03be))(t1) . . . (gnk(\u03be))(tk) = b c\u00f3 \u00edt nh\u1ea5t m\u1ed9t kh\u00f4ng \u0111i\u1ec3m l\u00e0 \u03be0. Th\u1ef1c v\u1eady, ta x\u00e9t hai tr\u01b0\u1eddng h\u1ee3p con: Tr\u01b0\u1eddng h\u1ee3p con 1.1. g l\u00e0 m\u1ed9t h\u00e0m ph\u00e2n h\u00ecnh. N\u1ebfu g l\u00e0 h\u00e0m ph\u00e2n h\u00ecnh si\u00eau vi\u1ec7t, ta th\u1ea5y r\u1eb1ng ph\u01b0\u01a1ng tr\u00ecnh (3.23) c\u00f3 v\u00f4 s\u1ed1 nghi\u1ec7m theo M\u1ec7nh \u0111\u1ec1 3.2.1. N\u1ebfu g l\u00e0 h\u00e0m h\u1eefu t\u1ef7, ph\u01b0\u01a1ng tr\u00ecnh (3.23) c\u00f3 \u00edt nh\u1ea5t m\u1ed9t kh\u00f4ng \u0111i\u1ec3m theo M\u1ec7nh \u0111\u1ec1 3.2.2. Tr\u01b0\u1eddng h\u1ee3p con 1.2. g l\u00e0 m\u1ed9t h\u00e0m nguy\u00ean. Ta x\u00e9t hai kh\u1ea3 n\u0103ng a) g l\u00e0 h\u00e0m nguy\u00ean si\u00eau vi\u1ec7t: N\u1ebfu n = 0 v\u00e0 k = 1 th\u00ec v\u1edbi n1 = t1 + 1 (xem [20]) ho\u1eb7c n1 \u2a7e t1 + 2 (theo M\u1ec7nh \u0111\u1ec1 3.2.1), h\u00e0m (gn1)t1 \u2212 b c\u00f3 v\u00f4 s\u1ed1 kh\u00f4ng \u0111i\u1ec3m.","75 kk N\u1ebfu n \u2a7e 1 ho\u1eb7c k \u2a7e 2, nj \u2a7e tj, n+ nj \u2a7e tj +2, theo M\u1ec7nh \u0111\u1ec1 3.2.1, j=1 j=1 ph\u01b0\u01a1ng tr\u00ecnh (3.23) c\u00f3 v\u00f4 s\u1ed1 kh\u00f4ng \u0111i\u1ec3m. b) g l\u00e0 \u0111a th\u1ee9c. Do k, n, nj, tj th\u1ecfa m\u00e3n gi\u1ea3 thi\u1ebft c\u1ee7a \u0110\u1ecbnh l\u00fd 3.2.3 k\u00e9o theo gn(\u03be)(gn1(\u03be))(t1) . . . (gnk(\u03be))(tk) \u2212 b l\u00e0 \u0111a th\u1ee9c b\u1eadc \u00edt nh\u1ea5t 1, suy ra ph\u01b0\u01a1ng tr\u00ecnh (3.23) c\u00f3 \u00edt nh\u1ea5t m\u1ed9t nghi\u1ec7m. Nh\u01b0 v\u1eady, trong m\u1ecdi tr\u01b0\u1eddng h\u1ee3p Ph\u01b0\u01a1ng tr\u00ecnh (3.23) lu\u00f4n c\u00f3 nghi\u1ec7m, t\u1ee9c l\u00e0 lu\u00f4n t\u1ed3n t\u1ea1i \u03be0 \u2208 C th\u1ecfa m\u00e3n gn(\u03be0)(gn1)(t1)(\u03be0) . . . (gnk)(tk)(\u03be0) = b. (3.24) Ta th\u1ea5y r\u1eb1ng g(\u03be0) \u0338= 0, \u221e, n\u00ean gv(\u03be) h\u1ed9i t\u1ee5 \u0111\u1ec1u \u0111\u1ebfn g(\u03be) trong m\u1ed9t l\u00e2n c\u1eadn c\u1ee7a \u03be0. T\u1eeb (3.22) v\u00e0 \u0111\u1ecbnh l\u00fd Hurwitz, t\u1ed3n t\u1ea1i m\u1ed9t d\u00e3y \u03bev \u2192 \u03be0 th\u1ecfa m\u00e3n fvn(zv + \u03c1v\u03bev)(fvn1)(t1)(zv + \u03c1v\u03bev) . . . (fvnk)(tk)(zv + \u03c1v\u03bev) = b v\u1edbi m\u1ed7i s\u1ed1 v \u0111\u1ee7 l\u1edbn. Hi\u1ec3n nhi\u00ean \u03b6v = zv + \u03c1v\u03bev \u2208 Efv. \u0110i\u1ec1u \u0111\u00f3 k\u00e9o theo |gv(\u03bev)| = |fv(\u03b6v)| \u2a7e M (3.25) \u03c1\u03b1v \u03c1\u03b1v . Do \u03be0 kh\u00f4ng ph\u1ea3i l\u00e0 c\u1ef1c \u0111i\u1ec3m c\u1ee7a g, n\u00ean g(\u03be) b\u1ecb ch\u1eb7n trong m\u1ed9t l\u00e2n c\u1eadn \u03be0. Tuy nhi\u00ean, cho v \u2192 \u221e trong (3.25), ta c\u00f3 m\u00e2u thu\u1eabn v\u1edbi vi\u1ec7c h\u00e0m g(\u03be) b\u1ecb ch\u1eb7n trong m\u1ed9t l\u00e2n c\u1eadn \u03be0. Tr\u01b0\u1eddng h\u1ee3p 2. a = 0. V\u1edbi m\u1ed7i f \u2208 F, n\u1ebfu t\u1ed3n t\u1ea1i z0 \u2208 D sao cho f (z0) = 0. Gi\u1ea3 s\u1eed z0 l\u00e0 kh\u00f4ng \u0111i\u1ec3m b\u1ed9i m \u2a7e 1 c\u1ee7a f , khi \u0111\u00f3 z0 l\u00e0 kh\u00f4ng \u0111i\u1ec3m b\u1ed9i kk m n + nj \u2212 tj \u2a7e 2 j=1 j=1 c\u1ee7a f n(f n1)(t1) . . . (f nk)(tk), do \u0111\u00f3 f n(z0)(f n1)(t1)(z0) . . . (f nk)(tk)(z0) = 0, m\u1eabu thu\u1eabn v\u1edbi gi\u1ea3 thi\u1ebft v\u00ec b \u0338= 0. Nh\u01b0 v\u1eady v\u1edbi m\u1ecdi f \u2208 F ta lu\u00f4n c\u00f3 f (z) =\u0338 0 v\u1edbi m\u1ecdi z \u2208 D. H\u01a1n n\u1eefa, n\u1ebfu f n(z0)(f n1)(t1)(z0) . . . (f nk)(tk)(z0) = b,","76 v\u1edbi z0 \u2208 D th\u00ec t\u1eeb gi\u1ea3 thi\u1ebft ta suy ra f (z0)n+n1+\u00b7\u00b7\u00b7+nk = 0, do \u0111\u00f3 f (z0) = 0, k\u00e9o theo b = 0. \u0110\u00f3 ch\u00ednh l\u00e0 m\u00e2u thu\u1eabn. Nh\u01b0 th\u1ebf f =\u0338 0 v\u00e0 f n(f n1)(t1) . . . (f nk)(tk) =\u0338 b v\u1edbi m\u1ecdi f \u2208 F . Theo \u0111\u1ecbnh l\u00fd Hurwitz, ta c\u00f3 g =\u0338 0, gn(gn1)(t1) . . . (gnk)(tk) \u0338= b ho\u1eb7c gn(gn1)(t1) . . . (gnk)(tk) \u2261 b. N\u1ebfu gn(gn1)(t1) . . . (gnk)(tk) \u2261 b. Theo M\u1ec7nh \u0111\u1ec1 3.1.19, b\u1eadc c\u1ee7a g cao nh\u1ea5t l\u00e0 1. Do \u0111\u00f3 ta c\u00f3 g(z) = eP (z) theo M\u1ec7nh \u0111\u1ec1 3.1.4, trong \u0111\u00f3 P l\u00e0 m\u1ed9t \u0111a th\u1ee9c v\u1edbi b\u1eadc cao nh\u1ea5t l\u00e0 1. Nh\u01b0 v\u1eady g(\u03be) = ec\u03be+d, trong \u0111\u00f3 c l\u00e0 m\u1ed9t h\u1eb1ng s\u1ed1 kh\u00e1c kh\u00f4ng. \u0110i\u1ec1u n\u00e0y k\u00e9o theo kk g n (\u03be )(g n1 (\u03be ))(t1) (gnk (\u03be ))(tk ) (n1c)t1 c)tk (n+ nj )c\u03be +(n+ nj )d \u2261 . . . = . . . (nk e j=1 j=1 b. \u0110\u00e2y ch\u00ednh l\u00e0 \u0111i\u1ec1u m\u00e2u thu\u1eabn. Nh\u01b0 v\u1eady (3.26) gn(gn1)(t1) . . . (gnk)(tk) \u0338= b. Ta x\u00e9t hai tr\u01b0\u1eddng h\u1ee3p con nh\u01b0 sau: Tr\u01b0\u1eddng h\u1ee3p con 2.1. g l\u00e0 h\u00e0m ph\u00e2n h\u00ecnh. T\u1eeb \u0111i\u1ec1u ki\u1ec7n kk nj \u2a7e tj, n + nj \u2a7e tj + 3, j=1 j=1 ta th\u1ea5y gn(gn1)(t1) . . . (gnk)(tk) \u2212 b c\u00f3 m\u1ed9t kh\u00f4ng \u0111i\u1ec3m theo M\u1ec7nh \u0111\u1ec1 3.2.1 v\u00e0 M\u1ec7nh \u0111\u1ec1 3.2.2. \u0110i\u1ec1u n\u00e0y m\u00e2u thu\u1eabn v\u1edbi (3.26). Tr\u01b0\u1eddng h\u1ee3p con 2.2. N\u1ebfu g l\u00e0 m\u1ed9t h\u00e0m nguy\u00ean si\u00eau vi\u1ec7t (ch\u00fa \u00fd r\u1eb1ng g =\u0338 0). Th\u1ee9 nh\u1ea5t, n = 0, k = 1, n1 = t1 + 1 (xem [20]) v\u00e0 n1 \u2a7e t1 + 2 (theo M\u1ec7nh \u0111\u1ec1 3.2.1 v\u00e0 M\u1ec7nh \u0111\u1ec1 3.2.2), th\u00ec (gn1)t1 \u2212 b c\u00f3 m\u1ed9t kh\u00f4ng \u0111i\u1ec3m. Th\u1ee9 hai, kk n \u2a7e 1 or k \u2a7e 2, nj \u2a7e tj, n + nj \u2a7e tj + 2, theo M\u1ec7nh \u0111\u1ec1 3.2.1, ta th\u1ea5y j=1 j=1 r\u1eb1ng gn(gn1)(t1) . . . (gnk)(tk) \u2212 b c\u00f3 m\u1ed9t kh\u00f4ng \u0111i\u1ec3m. \u0110i\u1ec1u nay m\u00e2u thu\u1eabn v\u1edbi (3.26). N\u1ebfu g l\u00e0 m\u1ed9t \u0111a th\u1ee9c, th\u00ec t\u1eeb k, n, nj, tj th\u1ecfa m\u00e3n gi\u1ea3 thi\u1ebft c\u1ee7a \u0110\u1ecbnh l\u00fd 3.2.3, ta c\u00f3 gn(gn1)(t1) . . . (gnk)(tk) \u2212 b c\u00f3 kh\u00f4ng \u0111i\u1ec3m. \u0110i\u1ec1u n\u00e0y m\u00e2u thu\u1eabn v\u1edbi (3.26). Nh\u01b0 v\u1eady \u0110\u1ecbnh l\u00fd 3.2.3 \u0111\u01b0\u1ee3c ch\u1ee9ng minh.","77 3.2.2. \u0110\u1ecbnh l\u00fd duy nh\u1ea5t Nh\u01b0 \u0111\u00e3 n\u00f3i trong ph\u1ea7n m\u1edf \u0111\u1ea7u, n\u0103m 1996 Bru\u00a8ck ([2]) \u0111\u00e3 \u0111\u1eb7t ra gi\u1ea3 thuy\u1ebft: cho f l\u00e0 m\u1ed9t h\u00e0m nguy\u00ean th\u1ecfa m\u00e3n \u03c32(f ) kh\u00f4ng l\u00e0 m\u1ed9t s\u1ed1 nguy\u00ean hay \u221e. N\u1ebfu f v\u00e0 f \u2032 chung nhau m\u1ed9t gi\u00e1 tr\u1ecb h\u1eefu h\u1ea1n a \u2208 C k\u1ec3 c\u1ea3 b\u1ed9i th\u00ec f\u2032 \u2212 a (3.27) = c, f \u2212a trong \u0111\u00f3 c l\u00e0 m\u1ed9t h\u1eb1ng s\u1ed1 n\u00e0o \u0111\u00f3. Gi\u1ea3 thuy\u1ebft n\u00e0y \u0111\u00e3 \u0111\u01b0\u1ee3c Bru\u00a8ck ch\u1ee9ng minh n\u0103m 1996 cho tr\u01b0\u1eddng h\u1ee3p a = 0 (xem [2]), v\u1ec1 sau \u0111\u00e3 h\u00fat \u0111\u01b0\u1ee3c s\u1ef1 quan t\u00e2m c\u1ee7a nhi\u1ec1u t\u00e1c gi\u1ea3 v\u00e0 c\u00f3 nhi\u1ec1u c\u00f4ng tr\u00ecnh \u0111\u01b0\u1ee3c c\u00f4ng b\u1ed1. V\u1edbi m\u1ed9t h\u00e0m ph\u00e2n h\u00ecnh f , k\u00ed hi\u1ec7u M [f ] := f n(f n1)(t1) . . . (f nk)(tk) v\u00e0 F = f n+n1+\u00b7\u00b7\u00b7+nk, trong \u0111\u00f3 n, n1, ..., nk, t1, ..., tk l\u00e0 c\u00e1c s\u1ed1 nguy\u00ean d\u01b0\u01a1ng. \u0110\u1ecbnh l\u00fd sau \u0111\u00e2y c\u1ee7a ch\u00fang t\u00f4i l\u00e0 m\u1ed9t k\u1ebft qu\u1ea3 v\u1ec1 v\u1ea5n \u0111\u1ec1 duy nh\u1ea5t cho c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh li\u00ean quan \u0111\u1ebfn gi\u1ea3 thuy\u1ebft Bru\u00a8ck khi thay th\u1ebf f b\u1edfi F v\u00e0 f \u2032 b\u1edfi M [f ]. K\u1ef9 thu\u1eadt ch\u1ee9ng minh c\u1ee7a \u0111\u1ecbnh l\u00fd d\u1ef1a v\u00e0o ti\u00eau chu\u1ea9n chu\u1ea9n t\u1eafc c\u1ee7a h\u1ecd c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh \u0111\u00e3 \u0111\u01b0\u1ee3c ch\u00fang t\u00f4i ch\u1ee9ng minh trong M\u1ee5c 3.2.1. \u0110\u1ecbnh l\u00fd 3.2.4 ([47]). Cho n \u2208 N v\u00e0 k, ni, ti \u2208 N\u2217, i = 1, . . . , k th\u1ecfa m\u00e3n m\u1ed9t trong c\u00e1c \u0111i\u1ec1u ki\u1ec7n sau: 1) k = 1, n = 0, n1 \u2a7e t1 + 1; kk 2) n \u2a7e 1 ho\u1eb7c k \u2a7e 2, nj \u2a7e tj, n + nj \u2a7e tj + 2. j=1 j=1 Cho a v\u00e0 b l\u00e0 hai gi\u00e1 tr\u1ecb h\u1eefu h\u1ea1n kh\u00e1c 0 v\u00e0 f l\u00e0 m\u1ed9t h\u00e0m nguy\u00ean kh\u00e1c h\u1eb1ng. N\u1ebfu F = a \u21cc M [f ] = b th\u00ec M [f ] \u2212 b F \u2212 a = c, trong \u0111\u00f3 c l\u00e0 m\u1ed9t h\u1eb1ng s\u1ed1. \u0110\u1eb7c bi\u1ec7t, n\u1ebfu a = b th\u00ec f = c1etz, trong \u0111\u00f3 c1 v\u00e0 t l\u00e0 c\u00e1c h\u1eb1ng s\u1ed1 kh\u00e1c 0 v\u00e0 t th\u1ecfa m\u00e3n \u0111i\u1ec1u ki\u1ec7n (tn1)t1 . . . (tnk)tk = 1.","78 Ch\u1ee9ng minh. \u0110\u1eb7t F = {g\u03c9(z) = f (z + \u03c9), \u03c9 \u2208 C}, z \u2208 D = \u2206, trong \u0111\u00f3 \u2206 l\u00e0 m\u1ed9t \u0111\u0129a \u0111\u01a1n v\u1ecb. S\u1eed d\u1ee5ng \u0110\u1ecbnh l\u00fd 3.2.3, ta c\u00f3 h\u1ecd h\u00e0m F l\u00e0 chu\u1ea9n t\u1eafc tr\u00ean D. Theo \u0111\u1ecbnh l\u00fd Marty, t\u1ed3n t\u1ea1i m\u1ed9t h\u1eb1ng s\u1ed1 M > 0 th\u1ecfa m\u00e3n |f \u2032(\u03c9)| |g\u03c9\u2032 (0)| 1 + |f (\u03c9)|2 1 + |g\u03c9(0)|2 f #(\u03c9) = = \u2a7d M, v\u1edbi m\u1ecdi \u03c9 \u2208 C. Theo M\u1ec7nh \u0111\u1ec1 3.1.19, b\u1eadc c\u1ee7a f cao nh\u1ea5t l\u00e0 1. T\u1eeb \u0111i\u1ec1u ki\u1ec7n f n+n1+\u00b7\u00b7\u00b7+nk = a \u21cc f n(f n1)(t1) . . . (f nk )(tk) = b, ta suy ra f l\u00e0 h\u00e0m nguy\u00ean si\u00eau vi\u1ec7t v\u00e0 f n(f n1)(t1) . . . (f nk )(tk) \u2212 b = e\u03b1(z). (3.28) f n+n1+\u00b7\u00b7\u00b7+nk \u2212 a T\u1eeb (3.28), ta c\u00f3 T (r, e\u03b1(z)) = O(T (r, f )). Do \u0111\u00f3 \u03c3(e\u03b1) \u2a7d \u03c3(f ) \u2a7d 1. \u0110i\u1ec1u \u0111\u00f3 k\u00e9o theo \u03b1(z) l\u00e0 \u0111a th\u1ee9c v\u00e0 deg(\u03b1) \u2a7d 1. Do f l\u00e0 m\u1ed9t h\u00e0m nguy\u00ean si\u00eau vi\u1ec7t n\u00ean M (r, f ) \u2192 \u221e khi r \u2192 \u221e. \u0110\u1eb7t M (rn, f ) = |f (zn)|, trong \u0111\u00f3 zn = rnei\u03b8n, \u03b8n \u2208 [0, 2\u03c0), |zn| = rn. Ta th\u1ea5y r\u1eb1ng 11 lim = lim = 0. (3.29) rn\u2192\u221e |f (zn)| rn\u2192\u221e M (rn, f ) Theo M\u1ec7nh \u0111\u1ec1 3.1.3, t\u1ed3n t\u1ea1i m\u1ed9t t\u1eadp h\u1ee3p F \u2282 R+ c\u00f3 \u0111\u1ed9 \u0111o logarit h\u1eefu h\u1ea1n th\u1ecfa m\u00e3n f (m)(z) \u03bd(r, f ) m = (1 + o(1)) (3.30) f (z) z \u0111\u00fang v\u1edbi m\u1ecdi m \u2a7e 1 v\u00e0 r \u0338\u2208 F. T\u00ednh to\u00e1n \u0111\u01a1n gi\u1ea3n ta c\u00f3 (f n)(k) = cm0,m1,...,mk f m0(f \u2032)m1 . . . (f (k))mk , (3.31)","79 trong \u0111\u00f3 cm0,m1,...,mk l\u00e0 c\u00e1c h\u1eb1ng s\u1ed1 v\u00e0 m0, m1, . . . , mk l\u00e0 c\u00e1c s\u1ed1 nguy\u00ean kh\u00f4ng k \u00e2m th\u1ecfa m\u00e3n m0 + m1 + \u00b7 \u00b7 \u00b7 + mk = n, jmj = k. j=1 T\u1eeb (3.31), ta c\u00f3 (f n)(k) f m0 (f \u2032)m1 (f (k))mk fn = cm0,m1,...,mk f m0 f m1 . . . f mk . \u0110i\u1ec1u n\u00e0y k\u00e9o theo (f n)(k)(zj) = cm0,m1,...,mk (f \u2032)m1(zj) . . . (f (k))mk(zj) (3.32) f n(zj) f m1(zj) f mk(zj) = cm0,m1,...,mk \u03bd(rj, f ) m1+\u00b7\u00b7\u00b7+mk zj (1 + o(1)). T\u1eeb (3.28), ta c\u00f3 \u2212(f n1 )(t1)...(f nk )(tk) b = e\u03b1(z). (3.33) f n+n1+\u00b7\u00b7\u00b7+nk f n1 ...f nk 1\u2212 a f n+n1+\u00b7\u00b7\u00b7+nk \u00c1p d\u1ee5ng (3.32) v\u00e0o (3.33), s\u1eed d\u1ee5ng (3.30) v\u00e0 M\u1ec7nh \u0111\u1ec1 3.1.2, ta c\u00f3 |\u03b1(zn)| = | log e\u03b1(zn)| (f n1 )(t1)(zn)...(f nk )(tk)(zn) b f n+n1+\u00b7\u00b7\u00b7+nk (zn) \u2212f n1 (zn)...f nk (zn) = log a 1 \u2212 f n+n1+\u00b7\u00b7\u00b7+nk (zn) \u2a7d O(log \u03bd(rn, f )) + O(log rn) + O(1) = O(log rn), (3.34) khi rn \u2192 \u221e. T\u1eeb (3.34), ta thu \u0111\u01b0\u1ee3c \u03b1(z) l\u00e0 m\u1ed9t h\u1eb1ng s\u1ed1 v\u00ec \u03b1(z) l\u00e0 m\u1ed9t \u0111a th\u1ee9c. Theo \u0111\u1eb3ng th\u1ee9c (3.28), ta c\u00f3 f n(f n1)(t1) . . . (f nk )(tk) \u2212 b = c. f n+n1+\u00b7\u00b7\u00b7+nk \u2212 a N\u1ebfu a = b, ta s\u1ebd ch\u1ec9 ra s\u1ef1 t\u1ed3n t\u1ea1i c\u1ee7a \u03be0 th\u1ecfa m\u00e3n f n(\u03be0)(f n1)(t1)(\u03be0) . . . (f nk)(tk)(\u03be0) = b. V\u00ec f l\u00e0 m\u1ed9t h\u00e0m nguy\u00ean si\u00eau vi\u1ec7t, do \u0111\u00f3 theo [20] n\u1ebfu n = 0, k = 1, n1 = t1 + 1 v\u00e0 theo M\u1ec7nh \u0111\u1ec1 3.2.1 v\u00e0 M\u1ec7nh \u0111\u1ec1 3.2.2 n\u1ebfu n1 \u2a7e t1 + 2, ta suy","80 ra (f n1)t1 \u2212 b c\u00f3 v\u00f4 s\u1ed1 kh\u00f4ng \u0111i\u1ec3m. N\u1ebfu n \u2a7e 1 ho\u1eb7c k \u2a7e 2, t\u1eeb \u0111i\u1ec1u ki\u1ec7n kk n + nj \u2a7e tj + 2 v\u00e0 M\u1ec7nh \u0111\u1ec1 3.2.1, ta c\u00f3 f n(f n1)(t1) . . . (f nk)(tk) = b c\u00f3 j=1 j=1 v\u00f4 s\u1ed1 kh\u00f4ng \u0111i\u1ec3m. Nh\u01b0 v\u1eady, trong m\u1ecdi tr\u01b0\u1eddng h\u1ee3p f n(f n1)(t1) . . . (f nk )(tk) = b \u0111\u1ec1u c\u00f3 nghi\u1ec7m. Ta g\u1ecdi \u03be0 l\u00e0 m\u1ed9t kh\u00f4ng \u0111i\u1ec3m b\u1ed9i m \u2a7e 1 c\u1ee7a f n(f n1)(t1) . . . (f nk)(tk) \u2212 b, khi \u0111\u00f3 theo gi\u1ea3 thi\u1ebft, ta suy ra \u03be0 l\u00e0 m\u1ed9t kh\u00f4ng \u0111i\u1ec3m c\u1ee7a f n+n1+\u00b7\u00b7\u00b7+nk \u2212 b v\u1edbi b\u1ed9i m. \u0110i\u1ec1u n\u00e0y k\u00e9o theo 1 = f n(\u03be0)(f n1)(t1)(\u03be0) . . . (f nk)(tk)(\u03be0) \u2212 b = c. f n+n1+\u00b7\u00b7\u00b7+nk (\u03be0) \u2212 b Do \u0111\u00f3 f n(f n1)(t1) . . . (f nk)(tk) = f n+n1+\u00b7\u00b7\u00b7+nk, k\u00e9o theo f kh\u00f4ng c\u00f3 kh\u00f4ng \u0111i\u1ec3m v\u00e0 b\u1eadc c\u1ee7a f nhi\u1ec1u nh\u1ea5t l\u00e0 1. \u0110i\u1ec1u n\u00e0y k\u00e9o theo f = c1etz, trong \u0111\u00f3 c1 v\u00e0 t l\u00e0 c\u00e1c h\u1eb1ng s\u1ed1 v\u00e0 t th\u1ecfa m\u00e3n (tn1)t1 . . . (tnk)tk = 1. \u0110\u1ecbnh l\u00fd \u0111\u01b0\u1ee3c ch\u1ee9ng minh. Tr\u01b0\u1eddng h\u1ee3p \u0111\u1eb7c bi\u1ec7t c\u1ee7a \u0110\u1ecbnh l\u00fd 3.2.4, n\u1ebfu ta ch\u1ecdn n = 0, k = 1, t1 = 1 trong \u0110\u1ecbnh l\u00fd 3.2.4, th\u00ec ta c\u00f3: H\u1ec7 qu\u1ea3 3.2.5. Cho f l\u00e0 m\u1ed9t h\u00e0m nguy\u00ean kh\u00e1c h\u1eb1ng, n \u2a7e 2 l\u00e0 m\u1ed9t s\u1ed1 nguy\u00ean v\u00e0 F = f n. N\u1ebfu F v\u00e0 F \u2032 chung nhau gi\u00e1 tr\u1ecb 1 CM th\u00ec F \u2261 F \u2032 v\u00e0 f c\u00f3 d\u1ea1ng f = cez\/n, trong \u0111\u00f3 c l\u00e0 m\u1ed9t h\u1eb1ng s\u1ed1 kh\u00e1c 0. Ch\u00fa \u00fd. Nh\u01b0 \u0111\u00e3 n\u00f3i trong ph\u1ea7n m\u1edf \u0111\u1ea7u, n\u0103m 2008, L. Z. Yang v\u00e0 J. L. Zhang ([52]) \u0111\u00e3 ch\u1ee9ng minh m\u1ed9t k\u1ebft qu\u1ea3 t\u01b0\u01a1ng t\u1ef1 H\u1ec7 qu\u1ea3 3.2.5 v\u1edbi \u0111i\u1ec1u ki\u1ec7n n \u2a7e 7. Nh\u01b0 v\u1eady \u0110\u1ecbnh l\u00fd 3.2.4 l\u00e0 m\u1ed9t c\u1ea3i ti\u1ebfn th\u1ef1c s\u1ef1 k\u1ebft qu\u1ea3 c\u1ee7a Yang v\u00e0 Zhang.","81 K\u1ebft lu\u1eadn Ch\u01b0\u01a1ng 3 Trong Ch\u01b0\u01a1ng 3, ngo\u00e0i vi\u1ec7c gi\u1edbi thi\u1ec7u m\u1ed9t s\u1ed1 ki\u1ebfn th\u1ee9c c\u01a1 b\u1ea3n trong l\u00fd thuy\u1ebft ph\u00e2n b\u1ed1 gi\u00e1 tr\u1ecb Nevanlinna v\u00e0 t\u00ednh chu\u1ea9n t\u1eafc c\u1ee7a h\u1ecd c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh, lu\u1eadn \u00e1n \u0111\u00e3 thu \u0111\u01b0\u1ee3c c\u00e1c k\u1ebft qu\u1ea3 ch\u00ednh sau : - Ph\u00e1t bi\u1ec3u v\u00e0 ch\u1ee9ng minh m\u1ed9t s\u1ed1 k\u1ebft qu\u1ea3 b\u1ed5 tr\u1ee3 v\u1ec1 nghi\u1ec7m c\u1ee7a f n(f n1)(t1) . . . (f nk)(tk) = a trong c\u00e1c tr\u01b0\u1eddng h\u1ee3p f l\u00e0 h\u00e0m ph\u00e2n h\u00ecnh si\u00eau vi\u1ec7t hay h\u1eefu t\u1ef7. - Ph\u00e1t bi\u1ec3u v\u00e0 ch\u1ee9ng minh m\u1ed9t ti\u00eau chu\u1ea9n chu\u1ea9n t\u1eafc cho m\u1ed9t h\u1ecd c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh (\u0110\u1ecbnh l\u00fd 3.2.3). \u0110i\u1ec1u ki\u1ec7n \u0111\u1ea1i s\u1ed1 trong \u0111\u1ecbnh l\u00fd n\u00e0y li\u00ean quan \u0111\u1ebfn l\u0169y th\u1eeba c\u1ee7a h\u00e0m ph\u00e2n h\u00ecnh c\u00f3 c\u00f9ng s\u1ed1 kh\u00f4ng \u0111i\u1ec3m v\u1edbi m\u1ed9t \u0111\u01a1n th\u1ee9c vi ph\u00e2n c\u1ee7a h\u00e0m ph\u00e2n h\u00ecnh \u0111\u00f3. - Ph\u00e1t bi\u1ec3u v\u00e0 ch\u1ee9ng minh \u0110\u1ecbnh l\u00fd 3.2.4 v\u1ec1 v\u1ea5n \u0111\u1ec1 duy nh\u1ea5t cho c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh li\u00ean quan \u0111\u1ebfn gi\u1ea3 thuy\u1ebft Bru\u00a8ck, trong \u0111\u00f3 ch\u00fang t\u00f4i thay th\u1ebf f b\u1edfi F = f n+n1+\u00b7\u00b7\u00b7+nk v\u00e0 f \u2032 b\u1edfi m\u1ed9t \u0111a th\u1ee9c vi ph\u00e2n c\u1ee7a f d\u1ea1ng M [f ] := f n(f n1)(t1) . . . (f nk)(tk).","82 K\u1ebft lu\u1eadn chung v\u00e0 \u0111\u1ec1 ngh\u1ecb Lu\u1eadn \u00e1n \u0111\u00e3 nghi\u00ean c\u1ee9u v\u1ec1 m\u1ed9t s\u1ed1 d\u1ea1ng \u0111\u1ecbnh l\u00fd c\u01a1 b\u1ea3n trong l\u00fd thuy\u1ebft Nevanlinna - Cartan cho \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u1ec9nh tr\u00ean h\u00ecnh v\u00e0nh khuy\u00ean trong tr\u01b0\u1eddng h\u1ee3p c\u00e1c si\u00eau m\u1eb7t v\u00e0 v\u1ea5n \u0111\u1ec1 duy nh\u1ea5t cho \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh tr\u00ean h\u00ecnh v\u00e0nh khuy\u00ean v\u00e0 h\u00e0m nguy\u00ean li\u00ean quan \u0111\u1ebfn gi\u1ea3 thuy\u1ebft Bru\u00a8ck. C\u00e1c k\u1ebft qu\u1ea3 ch\u00ednh c\u1ee7a lu\u1eadn \u00e1n bao g\u1ed3m: 1. Ph\u00e1t bi\u1ec3u v\u00e0 ch\u1ee9ng minh hai d\u1ea1ng \u0111\u1ecbnh l\u00fd c\u01a1 b\u1ea3n: \u0110\u1ecbnh l\u00fd c\u01a1 b\u1ea3n th\u1ee9 nh\u1ea5t v\u00e0 \u0110\u1ecbnh l\u00fd c\u01a1 b\u1ea3n th\u1ee9 hai cho \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh tr\u00ean h\u00ecnh v\u00e0nh khuy\u00ean trong c\u00e1c tr\u01b0\u1eddng h\u1ee3p m\u1ee5c ti\u00eau l\u00e0 c\u00e1c si\u00eau m\u1eb7t. 2. \u0110\u01b0a ra hai \u0111\u1ecbnh l\u00fd duy nh\u1ea5t cho \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh tr\u00ean h\u00ecnh v\u00e0nh khuy\u00ean t\u1eeb \u2206 v\u00e0o Pn(C) trong tr\u01b0\u1eddng h\u1ee3p m\u1ee5c ti\u00eau l\u00e0 si\u00eau m\u1eb7t \u1edf v\u1ecb tr\u00ed t\u1ed5ng qu\u00e1t \u0111\u1ed1i v\u1edbi ph\u00e9p nh\u00fang Veronese. 3. \u0110\u01b0a ra m\u1ed9t ti\u00eau chu\u1ea9n chu\u1ea9n t\u1eafc m\u1edbi cho h\u1ecd c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh tr\u00ean m\u1eb7t ph\u1eb3ng ph\u1ee9c C v\u00e0 ch\u1ee9ng minh m\u1ed9t k\u1ebft qu\u1ea3 v\u1ec1 v\u1ea5n \u0111\u1ec1 duy nh\u1ea5t cho c\u00e1c h\u00e0m ph\u00e2n h\u00ecnh li\u00ean quan \u0111\u1ebfn gi\u1ea3 thuy\u1ebft Bru\u00a8ck. Ch\u00fang t\u00f4i \u0111\u1ec1 xu\u1ea5t m\u1ed9t s\u1ed1 h\u01b0\u1edbng nghi\u00ean c\u1ee9u ti\u1ebfp theo cho k\u1ebft qu\u1ea3 c\u1ee7a lu\u1eadn \u00e1n nh\u01b0 sau: 1. Nghi\u00ean c\u1ee9u m\u1ed9t s\u1ed1 \u0110\u1ecbnh l\u00fd c\u01a1 b\u1ea3n th\u1ee9 hai cho \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh tr\u00ean h\u00ecnh v\u00e0nh khuy\u00ean v\u00e0o m\u1ed9t \u0111a t\u1ea1p \u0111\u1ea1i s\u1ed1 trong Pn(C) trong c\u00e1c tr\u01b0\u1eddng h\u1ee3p m\u1ee5c ti\u00eau l\u00e0 si\u00eau ph\u1eb3ng hay si\u00eau m\u1eb7t. 2. Nghi\u00ean c\u1ee9u v\u1ea5n \u0111\u1ec1 duy nh\u1ea5t cho \u0111\u01b0\u1eddng cong ch\u1ec9nh h\u00ecnh tr\u00ean h\u00ecnh v\u00e0nh khuy\u00ean trong tr\u01b0\u1eddng h\u1ee3p si\u00eau m\u1eb7t \u1edf v\u1ecb tr\u00ed t\u1ed5ng qu\u00e1t. 3. 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