On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity

We prove some uniqueness theorems for algebraically nondegenerate holomorphic curves sharing hyper-surfaces ignoring multiplicity.

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Date:2011
Main Author: Phuong Ha Tran
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Cite this:On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity / Phuong Ha Tran // Український математичний журнал. — 2011. — Т. 63, № 4. — С. 556–565. — Бібліогр.: 8 назв. — англ.

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spelling irk-123456789-1660312020-02-19T01:25:21Z On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity Phuong Ha Tran Статті We prove some uniqueness theorems for algebraically nondegenerate holomorphic curves sharing hyper-surfaces ignoring multiplicity. Доведено деякі теореми єдності для алгебраічих голоморфних кривих, що розділяють гіперплощини без врахування кратності. 2011 Article On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity / Phuong Ha Tran // Український математичний журнал. — 2011. — Т. 63, № 4. — С. 556–565. — Бібліогр.: 8 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/166031 517.5 en Український математичний журнал Інститут математики НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
topic Статті
Статті
spellingShingle Статті
Статті
Phuong Ha Tran
On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity
Український математичний журнал
description We prove some uniqueness theorems for algebraically nondegenerate holomorphic curves sharing hyper-surfaces ignoring multiplicity.
format Article
author Phuong Ha Tran
author_facet Phuong Ha Tran
author_sort Phuong Ha Tran
title On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity
title_short On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity
title_full On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity
title_fullStr On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity
title_full_unstemmed On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity
title_sort on uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity
publisher Інститут математики НАН України
publishDate 2011
topic_facet Статті
url http://dspace.nbuv.gov.ua/handle/123456789/166031
citation_txt On uniqueness theorems for holomorphic curves sharing hypersurfaces without counting multiplicity / Phuong Ha Tran // Український математичний журнал. — 2011. — Т. 63, № 4. — С. 556–565. — Бібліогр.: 8 назв. — англ.
series Український математичний журнал
work_keys_str_mv AT phuonghatran onuniquenesstheoremsforholomorphiccurvessharinghypersurfaceswithoutcountingmultiplicity
first_indexed 2025-07-14T20:31:03Z
last_indexed 2025-07-14T20:31:03Z
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fulltext UDC 517.5 Ha Tran Phuong (Thai Nguyen Univ. Education, Vietnam) ON UNIQUENESS THEOREMS FOR HOLOMORPHIC CURVES SHARING HYPERSURFACES WITHOUT COUNTING MULTIPLICITY∗∗∗∗ PRO TEOREMY {DYNOSTI DLQ HOLOMORFNYX KRYVYX, WO ROZDILQGT| HIPERPLOWYNY BEZ VRAXUVANNQ KRATNOSTI We prove some uniqueness theorems for algebraically nondegenerate holomorphic curves sharing hyper- surfaces ignoring multiplicity. Dovedeno deqki teoremy [dynosti dlq alhebra]çno nevyrodΩenyx holomorfnyx kryvyx, wo roz- dilqgt\ hiperplowyny bez vraxuvannq kratnosti. 1. Introduction. In 1926, R. Nevanlinna proved that two non-constant meromorphic maps f, g : C → P C1( ) satisfying f a j −1( ) = g a j −1( ) , for a1 , … , a5 ∈ P C1( ) dis- tinct, we must have f ≡ g. In 1975, H. Fujimoto [5] generalized Nevanlinna’s result to the case of meromorphic mappings of Cm into P Cn ( ) . Since that time, this problem has been studied intensively. In this paper we give some uniqueness results for algeb- raically nondegenerate holomorphic curves sharing sufficiently many nonlinear hyper- surfaces in a projective space. To state our results, we first introduce some notations. Let f : C → P Cn ( ) be a holomorphic map, let D be a hypersurface in P Cn ( ) of degree d and Q be the homogeneous polynomial of degree d in n + 1 variables with coefficients in C defining D, we define E D z Q f zf ( ) : ( )= ∈ ={ }C � 0 ignoring multiplicity , E D z m Q f z z mf Q f( ) : ( , ) ( ) ( )= ∈ × = ={ }C N � �0 and ord , where ordγ ( )z is the order of holomorphic functions γ at z. Let D = D Dq1, ,…{ } be a collection of hypersurfaces, we define E E Df f D ( ) : ( )D D = ∈ ∪ and E E Df f D ( ) : ( )D D = ∈ ∪ . In 1975, H. Fujimoto [5] proved the following theorem. Theorem A. Let H = H H n1 3 2, ,…{ }+ be a collection of 3n + 2 hyperplanes in general position in P Cn ( ) , and f, g : Cm → P Cn ( ) be meromorphic maps such that f Hm( )C ⊄ and g Hm( )C ⊄ for any H ∈H . If E H E Hf j g j( ) ( )= for any H j ∈H then f ≡ g. By Theorem A, the linearly nondegenerate meromorphic maps are uniquely deter- mined by 3n + 2 hyperplanes in general position. In the last years, many uniqueness theorems for holomorphic maps with hyperplanes have been established. For the case hypersurfaces, in 2008, by using the second main theorem with ramification of An- Phuong [1], H. T. Phuong [7], M. Dulock and M. Ru [4] proved some uniqueness ∗ Financial support provided to the author by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED). © HA TRAN PHUONG, 2011 556 ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4 ON UNIQUENESS THEOREMS FOR HOLOMORPHIC … 557 theorems for algebraically nondegenerate holomorphic curves. Recently, G. Dethloff and T. V. Tan [3] proved one uniqueness theorem for meromorphic maps in the case moving hypersurfaces. Our contribution is to give some unicity results for algedraical- ly nondegenerate holomorphic curves sharing sufficiently many hypersurfaces in gene- ral position for Veronese embedding. Now let Dj be a hypersurface of degree d in P Cn ( ) , which is defined by a ho- mogeneous polynomial Qj of degree d. Then Q z zj n( , , )0 … = a z zk i n i k n k kn d 0 0 0 … = ∑ , where ik0 + … + ikn = d for k = 0, 1, … , nd and nd = n d n +    – 1. We denote by a = ( , , )a and0 … the vector associated with Dj (or with Qj ). Let D = D Dq1, ,…{ } be a collection of arbitrary hypersurfaces and Qj be the homogeneous polynomial in C z zn0, ,… of degree d j defining Dj for j = = 1, … , q. We define the minimal index of degrees of D by δD : min , ,= …{ }d dq1 . Let mD be the least common multiple of the d j for j = 1, … , q and denote nD = n m n +    D – 1. For j = 1, … , q, we set Qj ∗ = Qj m d jD / and let a j ∗ be the vector associated with Qj ∗ . The collection D is said to be in general position for Veronese embedding if q > nD and for any distinct i1 , … , inD +1 ∈ 1, ,…{ }q , the vectors ai1 ∗ ,…, a inD + ∗ 1 are linearly independent. Recall that the collection D = D Dq1, ,…{ } is said to be in N-subgeneral position if q > N and for any distinct i1 , … , iN+1 ∈ 1, ,…{ }q Di k N k = + = ∅ 1 1 ∩ , where N be a positive integer such that N ≥ n. It is seen that, for hyperplanes, general position for Veronese embedding is equivalent to the usual concept of hyperplanes in general position (namely n-subgeneral position). For hypersurfaces, general position for Veronese embedding implies nD -subgeneral position. The following results we obtained in this paper. Theorem 1.1. Let f and g be algebraically nondegenerate holomorphic curves from C into P Cn ( ) . Let D = D Dq1, ,…{ } be a collection of q ≥ nD + 2 + + 2 2nD D/δ hypersurfaces in general position for Veronese embedding in P Cn ( ) such that f z( ) = g z( ) for all z ∈ E f ( )D ∪ Eg ( )D . Then f ≡ g. Theorem 1.2. Let f and g be algebraically nondegenerate holomorphic curves from C into P Cn ( ) . Let D = D Dq1, ,…{ } be a collection of q ≥ nD + 2 + ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4 558 HA TRAN PHUONG + 2nD D/δ hypersurfaces in general position for Veronese embedding in P Cn ( ) such that (a) f z( ) = g z( ) for all z ∈ E f ( )D ∪ Eg ( )D , (b) E Df i( ) ∩ E Df j( ) = ∅ and E Dg i( ) ∩ E Dg j( ) = ∅ for all i ≠ j ∈ ∈ 1, ,…{ }q . Then f ≡ g. Theorems 1.1 and 1.2 are uniqueness theorems for algebraically nondegenerate me- romorphic curves in the case hypersurface, they have shown the sufficient conditions for two algebraically nondegenerate meromorphic curves being equivalent. Note that, if n = 1, hypersurfaces are distinct points, Theorem 1.2 becomes Nevanlinna’s five theorem. And in Theorem 1.2 too, if mD = 1, δD = 1, nD = n and the number of hyperplanes is 3n + 2 as in Fujimoto’s result. Furthermore, the number of hypersurfa- ces in our results is smaller that in Dulock – Ru’s result. 2. Preliminaries from Nevanlinna – Cartan theory. In this section, we introduce some notations in Nevanlinna – Cartan theory and recall some results, which are necessary for proofs of our main results. Let f : C → P Cn ( ) be a holomorphic map and f = ( , , )f fn0 … be a reduced rep- resentative of f. The Nevanlinna – Cartan characteristic function T rf ( ) is defined by T rf ( ) = 1 2 0 2 π θθ π log f re di( )∫ , where f z( ) = max ( )f z0{ , … , f zn ( ) } . The above definition is independent, up to an additive constant, of the choice of the reduced representation of f. Let D be a hypersurface in P Cn ( ) of degree d. Let Q be the homogeneous po- lynomial of degree d defining D. The proximity function of f is defined by m r Df ( , ) = 1 2 0 2 π θ π θ θ log∫ ( ) ( ) f re Q f re d i d i� . Let n r Df ( , ) be the number of zeros of Q f� in the disk z ≤ r, counting multi- plicity. For any positive integer k, let n r D kf ( , , )≤ be the number of zeros having multiplicity ≤ k of Q f� in the disk z ≤ r, counting multiplicity and let n r D kf ( , , )> be the number of zeros having multiplicity > k of Q f� in the disk z ≤ r, counting multiplicity. The integrated counting functions are defined by N r Df ( , ) = n t D n D t dt f f r ( , ) ( , )− ∫ 0 0 + n D rf ( , ) log0 , N r Df k, ( , )≤ = n t D k n D k t dt f f r ( , , ) ( , , )≤ − ≤ ∫ 0 0 + n D k rf ( , , ) log0 ≤ , N r Df k, ( , )> = n t D k n D k t dt f f r ( , , ) ( , , )> − > ∫ 0 0 + n D k rf ( , , ) log0 > . For any positive integers ∆, k, let n r Df ∆ ( , ) be the number of zeros of Q f� in the ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4 ON UNIQUENESS THEOREMS FOR HOLOMORPHIC … 559 disk z ≤ r, where any zero is counted with multiplicity if its multiplicity is less than or equal to ∆, and ∆ times otherwise. Let n r D kf ∆ ( , , )≤ (resp. n r D kf ∆ ( , , )> ) be the number of zeros having multiplicity ≤ k (resp. > k) of Q f� in the disk z ≤ r, where any zero is counted if its multiplicity is less than or equal to ∆, and ∆ times otherwise, too. The integrates truncated counting functions are defined by N r Df ∆ ( , ) = n t D n D t dt f f r ∆ ∆( , ) ( , )− ∫ 0 0 + n D rf ∆ ( , ) log0 , N r Df k ∆ , ( , )≤ = n t D k n D k t dt f f r ∆ ∆( , , ) ( , , )≤ − ≤ ∫ 0 0 + n D k rf ∆ ( , , ) log0 ≤ , N r Df k ∆ , ( , )> = n t D k n D k t dt f f r ∆ ∆( , , ) ( , , )> − > ∫ 0 0 + n D k rf ∆ ( , , ) log0 > . Next, we will recall the following lemma which show properties of integrated coun- ting functions. The proof can be found in [7]. Lemma 2.1. With the above notations we have 1) N r Df ( , ) = N r Df k, ( , )≤ + N r Df k, ( , )> , 2) N r Df ∆ ( , ) = N r Df k, ( , )≤ ∆ + N r Df k, ( , )> ∆ , 3) N r Df ∆ ( , ) ≤ N r Df ( , ) , 4) N r Df 1 ( , ) ≤ N r Df ∆ ( , ) ≤ ∆N r Df 1 ( , ) , 5) N r Df k, ( , )≤ 1 ≤ N r Df k, ( , )≤ ∆ ≤ ∆ N r Df k, ( , )≤ 1 , 6) N r Df k, ( , )> 1 ≤ N r Df k, ( , )> ∆ ≤ ∆ N r Df k, ( , )> 1 , 7) 1 1k N r Df k+ ≤, ( , )∆ + N r Df k, ( , )> ∆ ≤ ∆ k N r Df+ 1 ( , ) . First main theorem. Let f : C → P Cn ( ) be a holomorphic map, and D be a hy- persurface in P Cn ( ) of degree d. If f D( )C ⊄ , then for every real number r with 0 < r < ∞, m r Df ( , ) + N r Df ( , ) = dT rf ( ) + O( )1 , where O( )1 is a constant independent of r. In 1933, H. Cartan [2] proved the following theorem. Theorem 2.1. Let f : C → P Cn ( ) be a linearly nondegenerate holomorphic map, and let H j , 1 ≤ j ≤ q, be hyperplanes in P Cn ( ) in general position. Then q n T rf− + −( )( ) ( )1 ε ≤ N r Hf n j j q ( , ) = ∑ 1 + S rf ( ) , where S rf ( ) = o T rf ( )( ) and inequality holds for all large r outside a set of finite Lebesgue measure. 3. Proofs of Theorems 1.1 and 1.2. To prove our theorems we need the following lemma. Lemma 3.1. Let f : C → P Cn ( ) be an algebraically nondegenerate holomorphic ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4 560 HA TRAN PHUONG map and D1 , … , Dq be hypersurfaces in P Cn ( ) of degree d j , j = 1, … , q, such that the collection D = D Dq1, ,…{ } is in general position for Veronese embedding in P Cn ( ) . Then for every ε > 0, q n T rf− − −( )D 1 ε ( ) ≤ 1 1 d N r D j f n j j q D ( , ) = ∑ + S rf ( ) , where inequality holds for all large r outside a set of finite Lebesgue measure. Proof. Let f = ( , , )f fn0 … be a reduced representative of f, where f0 , … , fn are entire functions on C without common zeros, and Qj , j ≤ 1 ≤ q , be the homogeneous polynomials in C[ , , ]z zn0 … of degree d j defining Dj . Of course we may assume that q ≥ nD + 1. We first claim that it suffices to prove the lemma in the case that all of the d j are equal to mD . Indeed, if we have the lemma in that case, then we know that for ε > 0 as in statement of the lemma that q n T rf− − −( )D 1 ε ( ) ≤ 1 1 m N r Qf n j m d j q j D D D( , ) / = ∑ + S rf ( ) . Note that if z ∈C is the zero of Q fj � with multiplicity a then z is the zero of Q fj m d jD / � with multiplicity a m d j D . This implies that N r Qf n j m d jD D, /( ) ≤ m d N r Q j f n j D D ( , ) . Hence q n T rf− − −( )D 1 ε ( ) ≤ 1 1 m N r Qf n j m d j q j D D D( , ) / = ∑ + S rf ( ) ≤ ≤ 1 1 d N r Q j f n j j q D ( , ) = ∑ + S rf ( ) . Therefore, without loss of generality, we can assume that D1 , … , Dq have a same degree of mD . We recall the lexicographic ordering on the n-tuples of natural numbers: let J = = j1{ , … , jn} , I = i1{ , … , in} ∈ Nn , J < I if and only if for some b ∈ 1{ , … , n} we have j il l= for l < b and j ib b< . With the n-tuples I = i1{ , … , in} of non- negative integers, we denote σ( ) :I i jj = ∑ . Let ( : : )z zn0 … be a homogeneous coordinates in P Cn ( ) and let I In0, ,…{ }D be a set of (n + 1)-tuples such that σ( )I j = mD , j = 0, … , nD , and I Ii j< for i < j ∈ ∈ 0, ,…{ }nD . For z = ( : : )z zn0 … ∈ P Cn ( ) , we write z I as z zi n in 0 0 … where I = i in0, ,…{ } ∈ I In0, ,…{ }D . Then we may write the set of mo- nomials of degree mD as z zI In0 , ,…{ }D . Denote ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4 ON UNIQUENESS THEOREMS FOR HOLOMORPHIC … 561 � Dm : P Cn ( ) → P CnD ( ) be the Veronese embedding of degree mD . Let ( : : )w wn0 … D be a homogeneous coordinate in P CnD ( ) . Then �d is given by � Dm ( )z = w wm0( ) : : ( )z z… D( ) , where w j I j( )z z= , j = 0, … , nD . Now we set F = ( : : )F Fn0 … D = � Dm f� , then Fj I j= f , j = 0, … , nD . Then F is a holomorphic map from C to P CnD ( ) and F = ( , , )F Fn0 … D is a reduced representative of F. We know from the assump- tion that f is algebraically nondegenerate hence F is linearly nondegenerate. For any hypersurface Dj ∈ ( , , )D Dq1 … , let a j = ( , , )a aj jn0 … D be the vec- tor associated with Dj , we set L j = a wj0 0 + … + a win nD D . Then L j is a linear form in P CnD ( ) . Let H j be a hyperplane in P CnD ( ) , which is made by L j , we say that the hyperplane H j associated with Dj . Hence for the collection of hypersurfaces ( , , )D Dq1 … in P Cn ( ) , we have the collection of hy- perplanes ( , , )H Hq1 … of associate hyperplanes in P CnD ( ) . By the assumption that (D1 , … , Dq ) is in general position for Veronese embedding in P Cn ( ) , we have that (H1 , … , Hq ) is in general position in P CnD ( ) . By the definition of F we have for any j = 1, … , q, a Fj j k k n a F. : .= = ∑ 0 D = Dj ( )f . So N r Df j( , ) = N r HF j( , ) and N r Df n j D ( , ) = N r HF n j D ( , ) . (3.1) Furthermore by the first main theorem, T rF ( ) = m T rfD ( ) + O( )1 , S rF ( ) = S rf ( ) . (3.2) With the assumption of Lemma 3.1, applying Theorem 2.1 to holomorphic map F : C → P CnD ( ) and the hyperplanes H j , j = 1, … , q, we have ( ) ( )q n T rF− − −D 1 ε ≤ N r HF n j q j D = ∑ 1 ( , ) + S rF ( ) (3.3) holds for all large r outside a set of finite Lebesgue measure. Combining with (3.1), (3.2) and (3.3) together, we have ( ) ( )q n m T rf− − −D D1 ε ≤ N r Df n j q j D = ∑ 1 ( , ) + S rf ( ) . This concludes the proof of the lemma. ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4 562 HA TRAN PHUONG Proof of Theorem 1.1. Assume for the sake contradiction that f � g. Then there are two numbers, α, β ∈ 0, ,…{ }n , α ≠ β such that f gα β � f gβ α . Let k be a sufficiently large positive integer, which will be chosen later, and ε be a real number such that 0 < ε < 1. For any Dj ∈D , for all large r outside a set of finite Lebesgue measure, by the first main theorem we have N r Df n j D ( , ) = N r Df k n j, ( , )≤ D + N r Df k n j, ( , )> D = = k k N r Df k n j+ ≤1 , ( , )D + 1 1k N r Df k n j+ ≤, ( , )D + N r Df k n j, ( , )> D ≤ ≤ k k N r Df k n j+ ≤1 , ( , )D + n k N r Df k j D + ≤ 1 1 , ( , ) + n N r Df k jD , ( , )> 1 ≤ ≤ k k N r Df k n j+ ≤1 , ( , )D + n k N r Df k j D + ≤ 1 , ( , ) + n k N r Df k j D + > 1 , ( , ) ≤ ≤ k k N r Df k n j+ ≤1 , ( , )D + n k N r Df j D + 1 ( , ) ≤ ≤ k k N r Df k n j+ ≤1 , ( , )D + d n k T r j f D + 1 ( ) + O( )1 , where d j is degree of Dj . So 1 d N r D j f n j D ( , ) ≤ k d k N r D j f k n j ( ) ( , ),+ ≤1 D + n k T rf D + 1 ( ) + O( )1 . It implies that 1 1 d N r D j f n j q j D = ∑ ( , ) ≤ k k d N r D jj q f k n j+ = ≤∑ 1 1 1 , ( , )D + qn k T rf D + 1 ( ) + O( )1 . (3.4) From Lemma 3.1, the inequality (3.4) becomes ( – ) ( )q n T rf− −D 1 ε ≤ k k N r Df k n j q j+ ≤ = ∑ 1 1 , ( , )D + qn k T rf D + 1 ( ) + S rf ( ) ≤ ≤ k k N r Df k n j q jδD D ( ) ( , ),+ ≤ = ∑ 1 1 + qn k T rf D + 1 ( ) + S rf ( ) . This is equivalent to q qn k n T rf− + − −     D D 1 1 – ( )ε ≤ k k N r Df k n j q jδD D ( ) ( , ),+ ≤ = ∑ 1 1 + S rf ( ) . So q k n n k T rf( ) ( ) ( ) ( )+ − − + + +( )1 1 1D D ε ≤ ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4 ON UNIQUENESS THEOREMS FOR HOLOMORPHIC … 563 ≤ k N r Df k n j q jδD D , ( , )≤ = ∑ 1 + S rf ( ) ≤ ≤ n k N r Df k j q j D Dδ , ( , )≤ = ∑ 1 1 + S rf ( ) . (3.5) Assume that z0 ∈C is a zero of D fj � with multiplicity less than or equal to k, then z E f0 ∈ ( )D ∪ Eg ( )D . This implies that g z( )0 = f z( )0 , so f z f z α β ( ) ( ) 0 0 = = g z g z α β ( ) ( ) 0 0 , namely z0 is a zero of the function f f α β – g g α β . Note that by the hypo- thesis D is in general position for Veronese embedding then there exist at most nD hypersurfaces Dj in D such that D f zj � ( )0 = 0. This implies that N r Df k j q j, ( , )≤ = ∑ 1 1 ≤ n N rf f g gD α β α β/ / ( )− . Therefore, by properties of counting function, (3.5) becomes q k n n k T rf( ) ( ) ( ) ( )+ − − + + +( )1 1 1D D ε ≤ ≤ n k N r S rf f g g f D D 2 δ α β α β/ / ( ) ( )− + ≤ n k T r T rf g D D 2 δ ( ) ( )+( ) + S rf ( ) . (3.6) Similarly for the holomorphic map g, we have q k n n k T rg( ) ( ) ( ) ( )+ − − + + +( )1 1 1D D ε ≤ ≤ n k T r T rf g D D 2 δ ( ) ( )+( ) + S rg ( ) . (3.7) Adding the inequalities (3.6) and (3.7), we have q k n n k T r T rf g( ) ( ) ( ) ( ) ( )+ − − + + +( ) +( )1 1 1D D ε ≤ ≤ 2 2n k T r T rf g D Dδ ( ) ( )+( ) + S rf ( ) + S rr ( ) . This concludes that q k n( )+ −1 D – ( ) ( )n kD + + +1 1ε – 2 2n kD Dδ ≤ S r S r T r T r f g f g ( ) ( ) ( ) ( ) + + holds for all large r. Let r → ∞, we have q k n( )+ −1 D – ( ) ( )n kD + + +1 1ε – 2 2n kD Dδ ≤ 0. This is equivalent to k q n nδ ε δD D D D− + + −( )( )1 2 2 + q qn n− − + +( )D D D( )1 ε δ ≤ 0. (3.8) If we take ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4 564 HA TRAN PHUONG k > ( ) ( ) qn q n n n D D D D D D D − + + + − + + − 1 1 2 2 ε δ δ ε δ , then since the hypothesis that q ≥ nD + 2 + 2 2nD Dδ , we have a contradiction. Hence f g f gi j j i≡ for any i ≠ j ∈ 0, ,…{ }n , namely f g≡ . This is the conclusion of the proof of Theorem 1.1. Proof of Theorem 1.2. Assume for the sake contradiction that f � g. Then there are two numbers α, β ∈ 0, ,…{ }n , α ≠ β such that f gα β � f gβ α . Let k be a suf- ficiently large positive integer, which will be chosen later, and ε be a real number such that 0 < ε < 1. With the hypothesis in Theorem 1.2 and the proof of Theorem 1.1, we have q k n n k T rf( ) ( ) ( ) ( )+ − − + + +( )1 1 1D D ε ≤ ≤ n k N r Df k j q j D Dδ , ( , )≤ = ∑ 1 1 + S rf ( ) . (3.9) We know that, if z0 ∈C is a zero of D fj � with multiplicity less than or equal to k, then z0 will be a zero of the function f f α β – g g α β . By the hypothesis we have E Df i( ) ∩ E Df j( ) = ∅ for any pair i ≠ j ∈ 1, ,…{ }q . So if z0 is a zero of D fj � then z0 will be not a zero of D fj � for all i ≠ j ∈ 1, ,…{ }q . Hence N r Df k j q j, ( , )≤ = ∑ 1 1 ≤ N rf f g g( / ) ( / )( )α β α β− ≤ T rf ( ) + T rg ( ) + O( )1 . Therefore, (3.9) becomes q k n n k T rf( ) ( ) ( ) ( )+ − − + + +( )1 1 1D D ε ≤ ≤ n k T r T rf g D Dδ ( ) ( )+( ) + S rf ( ) . (3.10) Similarly for the holomorphic map g, we have q k n n k T rg( ) ( ) ( ) ( )+ − − + + +( )1 1 1D D ε ≤ ≤ n k T r T rf g D Dδ ( ) ( )+( ) + S rg ( ) . (3.11) Since the inequalities (3.10) and (3.11), we have q k n n k T r T rf g( ) ( ) ( ) ( ) ( )+ − − + + +( ) +( )1 1 1D D ε ≤ ≤ 2 n k T r T rf g D Dδ ( ) ( )+( ) + S rf ( ) + S rg ( ) . Hence ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4 ON UNIQUENESS THEOREMS FOR HOLOMORPHIC … 565 q k n n kδ δ εD D D D( ) ( ) ( )+ − − + + +1 1 1 – 2 n kD ≤ S r S r T r T r f g f g ( ) ( ) ( ) ( ) + + holds for all large r. Let r → ∞, we have k q n nδ ε δD D D D− + + −( )( )1 2 + q n n− − + +( )D D D( )1 ε δ ≤ 0. If we take k > ( ) ( ) qn q n q n n D D D D D D D − + + + − + + − 1 1 2 ε δ δ ε δ , then since the hypothesis that q ≥ nD + 2 + 2 nD Dδ , we have a contradiction. Hence f g f gi j j i≡ for any i ≠ j ∈ 0, ,…{ }n , namely f g≡ . This is the conclusion of the proof of Theorem 1.2. 1. An T. T. H., Phuong H. T. An explicit estimate on multiplicity truncation in the second main theo- rem for holomorphic curves encountering hypersurfaces in general position in projective space // Houston J. 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Geometric conditions for unicity of holomorphic curves // Contemp. Math. – Providence, RI: Amer. Math. Soc., 1983. – 25. Received 28.10.10 ISSN 1027-3190. Ukr. mat. Ωurn., 2011, t. 63, # 4