Higher dimensional generalizations of some theorems on normality of meromorphic functions
aa r X i v : . [ m a t h . C V ] S e p Higher dimensional generalizations of sometheorems on normality of meromorphicfunctions
Tran Van Tan
Abstract
In [Israel J. Math, 2014], Grahl and Nevo obtained a significantimprovement for the well-known normality criterion of Montel. Theyproved that for a family of meromorphic functions F in a domain D ⊂ C , and for a positive constant ǫ , if for each f ∈ F there existmeromorphic functions a f , b f , c f such that f omits a f , b f , c f in D andmin { ρ ( a f ( z ) , b f ( z )) , ρ ( b f ( z ) , c f ( z )) , ρ ( c f ( z ) , a f ( z )) } ≥ ǫ, for all z ∈ D , then F is normal in D . Here, ρ is the spherical metricin b C . In this paper, we establish the high-dimensional versions forthe above result and for the following well-known result of Lappan:A meromorphic function f in the unit disc △ := { z ∈ C : | z | < } isnormal if there are five distinct values a , . . . , a such thatsup { (1 − | z | ) | f ′ ( z ) | | f ( z ) | : z ∈ f − { a , . . . , a }} < ∞ . Mathematics Subject Classification.
Key words.
Normal family, Nevanlinna theory.
Perhaps the most celebrated theorem in the theory of normal families is thefollowing criterion of Montel [11]. 1 heorem A.
Let F be a family of meromorphic functions in a domain D ⊂ C , and let a, b, c be three distinct points in b C . Assume that all functionsin F omit three points a, b, c in D. Then F is a normal family in D. In [2], Carath´eodory extended Theorem A to the case where the omittedpoints may depend on the function in the family and satisfy a condition onthe spherical distance. In 2014, Grahl and Nevo [5] generalized the result ofCarath´eodory to the case where all functions in the family omit three func-tions, and obtained the following theorem.
Theorem B.
Let F be a family of meromorphic functions in a domain D ⊂ C , and let ǫ be a positive constant. Denote by ρ the spherical metric in b C . Assume that for each f ∈ F there exist meromorphic functions a f , b f , c f in D such that f omits a f , b f , c f in D and min { ρ ( a f ( z ) , b f ( z )) , ρ ( b f ( z ) , c f ( z )) , ρ ( c f ( z ) , a f ( z )) } ≥ ǫ, for all z ∈ D , f ∈ F . Then F is a normal family in D . In section 3, we shall establish the higher dimensional version of Theorem B.A well-known result of Lehto and Virtanen [8] states that the meromor-phic function f in the unit disc △ := { z ∈ C : | z | < } is normal if and onlyif sup z ∈△ (1 − | z | ) f ( z ) < ∞ , where f := | f ′ | | f | is the spherical derivativeof f. In 1972, Pommerenke [14] gave an open question: if
M > E ⊂ b C such that if f is a meromorphic in △ thenthe condition that (1 − | z | ) f ( z ) M for each z ∈ f − ( E ) implies that f isa normal function? This question was answered by the following well-knownresult of Lappan [10]. Theorem C.
Let E ⊂ b C be any set consisting of five distinct points. If f isa meromorphic function in △ such thatsup { (1 − | z | ) f ( z ) : z ∈ f − ( E ) } < ∞ (1.1)then f is a normal function.In 1986, Hahn [7] generalized Theorem C to the case of high dimension,however, unfortunately his proof based on a false lemma (Lemma 2, [7]).2n section 4, we shall establish the higher dimensional version for theabove five-point theorem of Lappan.In the case of dimension one, if f is not normal, then by a result ofLohwater and Pommerenke [9], there exist sequences { z k } ⊂ △ , { r k } ⊂ R ,r k > , with lim k →∞ r k −| z k | = 0 such that g k ( ξ ) := f ( z k + r k ξ ) convergesuniformly on compact subsets of C to a non-constant meromorphic function g. Then condition (1.1) implies that all zero points of g − a ( a ∈ E ) havemultiplicity at least 2; this is impossible because that g is non-constant and E = 5 . In the high dimensional case ( n ≥ , from the view of Nevanlinna theory,the most difficulty comes from the fact that for any q ≥ n + 1, there arehyperplanes H , . . . , H q in general position in P n ( C ), and a non-constantentire curve g in P n ( C ) such that all zero points of H j ( g ) ( j = 1 , . . . , q )have multiplicity not less than 2 (even not less than n ). Indeed, let u bea non-constant holomorphic function nowhere vanishing on C . We consider g = ( (cid:0) n (cid:1) u n : (cid:0) n (cid:1) u n − : · · · : (cid:0) nn (cid:1) u ) and q ( q ≥ n + 1) hyperplanes H j : a j x + a j x + · + a nj x n = 0 , ( j = 1 , . . . , q ), where a , . . . , a q are q distinctcomplex numbers. Then they satisfy:i) g is linearly non-degenerate;ii) For any 1 j < j < · · · < j n q, the Vandermonde determinantdet( a sj i ) i,s n = Q t Acknowledgements: This research was supported by Vietnam NationalFoundation for Science and Technology Development (NAFOSTED) undergrant number 101.02-2016.17, and was done during a stay of the author atthe Vietnam Institute for Advanced Studies in Mathematics. He wishes toexpress his gratitude to this institute. The author also would like to thankthe referee for valuable comments and suggestions. Let ν be a nonnegative divisor on C . For each positive integer (or + ∞ ) p, we define the counting function of ν (where multiplicities are truncated by3 ) by N [ p ] ( r, ν ) := Z r n [ p ] ν t dt (1 < r < ∞ )where n [ p ] ν ( t ) = P | z | t min { ν ( z ) , p } . For brevity we will omit the character[ p ] in the counting function if p = + ∞ . For a meromorphic function ϕ on C ( ϕ , ϕ 6≡ ∞ ), we denote by ( ϕ ) the divisor of zeros of ϕ . We have the following Jensen’s formula for thecounting function: N ( r, ( ϕ ) ) − N ( r, (cid:18) ϕ (cid:19) ) = 12 π Z π log (cid:12)(cid:12) ϕ ( re iθ ) (cid:12)(cid:12) dθ + O (1) . We define the proximity function of ϕ by m ( r, ϕ ) = 12 π Z π log + (cid:12)(cid:12) ϕ ( re iθ ) (cid:12)(cid:12) dθ, where log + x = max { , log x } for x ≥ . If ϕ is nonconstant then m ( r, ϕ ′ ϕ ) = o ( T ϕ ( r )) as r → ∞ , outside a set offinite Lebesgue measure (Nevanlinna’s lemma on the logarithmic derivative).Nevanlinna’s first main theorem for ϕ states that T ϕ ( r ) = N ϕ ( r ) + m ( r, ϕ ) + O (1) . Let f be a holomorphic mapping of C into P n ( C ) with a reduced repre-sentation ( f , . . . , f n ) . The characteristic function T f ( r ) of f is defined by T f ( r ) := 12 π Z π log k f ( re iθ ) k dθ − π Z π log k f ( e iθ ) k dθ, r > , where k f k = max i =0 ,...,n | f i | . Let H = { ( ω : · · · : ω n ) ∈ P n ( C ) : a ω + · · · + a n ω n = 0 } be ahyperplane in P n ( C ) such that f ( C ) H. Denote by ( H ( f )) the divisor ofzeros of a f + · · · + a n f n , and put N [ p ] f ( r, H ) := N [ p ] ( r, ( H ( f )) ) . Let q, κ be positive integers, q ≥ κ ≥ n and let H , . . . , H q be hyper-planes in P n ( C ) . These hyperplanes are said to be in κ -subgeneral positionif ∩ κi =0 H j i = ∅ , for all 1 j < · · · < j κ q. ochka’s second main theorem. Let f be a linearly nondegenerate holo-morphic mapping of C into P n ( C ) , and let H , . . . , H q be hyperplanes in κ -subgeneral position in P n ( C ) ( κ ≥ n and q ≥ κ − n + 1) . Then( q − κ + n − T f ( r ) q X j =1 N [ n ] f ( r, H j ) + o ( T f ( r )) , for all r ∈ (1 , + ∞ ) excluding a subset of finite Lebesgue measure. Let D be a domain in C m , and let f and H be two holomorphic mappingsof D into P n ( C ) . For each z ∈ D , we take reduced representations b f =( f , . . . , f n ) of f and b H = ( a , . . . , a n ) of H in a neighbourhood U of z andset (cid:10) b f , b H (cid:11) := a f + · · · + a n f n . Denote by (cid:10) b f , b H (cid:11) the zero divisor of theholomorphic function (cid:10) b f , b H (cid:11) . The divisor ( H ( f )) := (cid:10) b f , b H (cid:11) is determinedindependently of a choice of reduced representations, and hence is well definedon the totality of D. Put f − ( H ) := { z ∈ D : ( H ( f )) ( z ) > } . For n + 1 points a , . . . , a n in P n ( C ) , we denote by d F S ( a , . . . , a n ) theminimum of the Fubini-Study distances from each point to the subspacegenerated by these n other points.We shall prove the following normality criterion. Theorem 3.1. Let F be a family of holomorphic mappings of a domain D ⊂ C m into P n ( C ) . For each f ∈ F , we consider n + 1 holomorphic mappings H f , . . . , H (2 n +1) f of D into P n ( C ) satisfying the following condition:For each compact subset K of D, there is a positive constant δ K such that d F S ( H j f ( z ) , . . . , H j n f ( z )) ≥ δ K for all subsets { j , . . . , j n } ⊂ { , . . . n + 1 } and all z ∈ K, f ∈ F . Assume that f − ( H jf ) = ∅ for all f ∈ F and j ∈ { , . . . , n + 1 } . Then F is normal on D. Lemma 3.2 (Zalcman lemma, [1], Lemma 3.1) . Let F be a family of holo-morphic mappings of a domain D ⊂ C m into P n ( C ) . If F is not nor-mal then there exist sequences { z k } ⊂ D with z k → z ∈ D, { f k } ⊂ F , ρ k } ⊂ R with ρ k → + , and Euclidean unit vectors { u k } ⊂ C m , such that g k ( ζ ) := f k ( z k + ρ k u k ζ ) , where ζ ∈ C satisfies z k + ρ k u k ζ ∈ D, convergesuniformly on compact subsets of C to a nonconstant holomorphic mapping g of C into P n ( C ) . Lemma 3.3 ([4], Corollary 14) . Let P = ( ω : · · · : ω n ) , . . . , P n = ( ω n : · · · : ω nn ) be n + 1 points in P n ( C ) . Then d nF S ( P , . . . , P n ) | det( P , . . . , P n ) |k P k · · · k P n k d F S ( P , . . . , P n ) , where k P j k = ( | ω j | + · · · + | ω jn | ) and det( P , . . . , P n ) := det( ω ji ) i,j n . In fact, in ([4], Corollary 14), the points P , . . . , P n are projectively indepen-dent, however, if they are projectively dependent, thendet( P , . . . , P n ) = 0 = d F S ( P , . . . , P n ) . In the case n = m = 1, the following lemma is due to Grahl and Nevo [5]. Lemma 3.4. Let { H α } α ∈A , . . . , { H qα } α ∈A be q ( q ≥ n + 1) families of holo-morphic mappings of D ⊂ C m into P n ( C ) . Assume that for each compactsubset K of D , there is a positive constant δ K such that d F S ( H j α ( z ) , . . . , H j n α ( z )) ≥ δ K , for all z ∈ K , α ∈ A , and j < j < · · · < j n q .Then { H α } α ∈A , . . . , { H qα } α ∈A are normal families on D .Proof. Suppose that there is an index j ∈ { , . . . , q } such that { H jα } α ∈A isnot normal on D, say j = 1 . By induction, we prove the following claim:For each s ∈ { , . . . , q } , there exist sequences { α k } ∞ k =1 ⊂ A , { z k } ⊂ D with z k → a ∈ D, { ρ k } ⊂ R with ρ k → + , and Euclidean unit vectors { u k } ⊂ C m , such that for all j ∈ { , . . . , s } , H j,k ( ζ ) := H jα k ( z k + ρ k u k ζ ) , where ζ ∈ C satisfies z k + ρ k u k ζ ∈ D, converges uniformly on compact subsets of C to aholomorphic mapping L j of C into P n ( C ) , where at least one of L , . . . , L s is nonconstant.The case s = 1 is just Lemma 3.2.Assume that the claim is true for some s ∈ { , . . . , q − } , we provethat it holds for s + 1 . By the induction hypothesis, there exist sequences6 α k } ∞ k =1 ⊂ A , { z ′ k } ⊂ D with z ′ k → a ∈ D, { ρ ′ k } ⊂ R with ρ ′ k → + , and Euclidean unit vectors { u k } ⊂ C m , such that for all j ∈ { , . . . , s } ,H j,k ( ζ ) := H jα k ( z ′ k + ρ ′ k u k ζ ) , where ζ ∈ C satisfies z ′ k + ρ ′ k u k ζ ∈ D, convergesuniformly on compact subsets of C to a holomorphic mapping L j of C into P n ( C ) , where at least one of L , . . . , L s is nonconstant.We consider the sequence H s +1 ,k ( ζ ) := H ( s +1) α k ( z ′ k + ρ ′ k u k ζ ) , where ζ ∈ C satisfies z ′ k + ρ ′ k u k ζ ∈ D. If { H s +1 ,k } ∞ k =1 is normal on C , then by replacing by an appropriate subse-quence, without loss of generality, we assume that H s +1 ,k converges uniformlyon compact subsets of C to a holomorphic mapping L s +1 of C into P n ( C ) . Hence, in this case, combining with the induction hypothesis, we get thatthe claim is also true for p + 1 . If { H s +1 ,k } ∞ k =1 is not normal on C , then by Lemma 3.2, there exist asubsequence of { H s +1 ,k } ∞ k =1 which without loss of generality we also denote by { H s +1 ,k } ∞ k =1 and sequences { ξ ′ k } ⊂ C with ξ ′ k → ξ ∈ C , { t k } ⊂ R with t k → + , such that h s +1 ,k ( ζ ) := H s +1 ,k ( ξ ′ k + t k ζ ) = H ( s +1) α k ( z ′ k + ρ ′ k ξ ′ k u k + ρ ′ k t k u k ζ )converges uniformly on compact subsets of C to a nonconstant holomorphicmapping L s +1 of C into P n ( C ) . Set z k := z ′ k + ρ ′ k ξ ′ k u k , ρ k := ρ ′ k t k . Then z k → a, ρ k → + , h s +1 ,k ( ζ ) = H ( s +1) α k ( z k + ρ k u k ζ ) converges uniformly oncompact subsets of C to a nonconstant holomorphic mapping L s +1 of C into P n ( C ), and h j,k ( ζ ) := H j,k ( ξ ′ k + t k ζ ) = H jα k ( z k + ρ k u k ζ ) converges uniformlyon compact subsets of C to the point L j ( ξ ) , for all j ∈ { , . . . , s } (note that H j,k → L j and ξ ′ k → ξ ). Therefore, the claim is true for p + 1 . By induction, we get the claim.In our claim, without loss of generality, we assume that L is nonconstant.Take a ball B ( a, r ) := { z : k z − a k r } ⊂ D, for some r > a ∈ D ). By the assumption, there is a constant δ > d F S ( H α ( z ) , . . . , H ( n +1) α ( z )) ≥ δ for all z ∈ K, α ∈ A . For each ζ ∈ C , it is clear that z k + ρ k u k ζ ∈ B ( a, r ) for all k sufficiently large.Hence, by our above claim and by Lemma 3.3, we have7 det ( L ( ζ ) , . . . , L n +1 ( ζ )) |k L ( ζ ) k · · · k L n +1 ( ζ ) k = lim k →∞ (cid:12)(cid:12) det (cid:0) H ,k ( ζ ) , . . . , H ( n +1) ,k ( ζ ) (cid:1)(cid:12)(cid:12) k H ,k ( ζ ) k · · · k H ( n +1) ,k ( ζ ) k = lim k →∞ (cid:12)(cid:12) det (cid:0) H α k ( z k + ρ k u k ζ ) , . . . , H ( n +1) α k ( z k + ρ k u k ζ ) (cid:1)(cid:12)(cid:12) k H α k ( z k + ρ k u k ζ ) k · · · k H ( n +1) α k ( z k + ρ k u k ζ ) k≥ lim k →∞ (cid:2) d F S ( H α k ( z k + ρ k u k ζ ) , . . . , H ( n +1) α k ( z k + ρ k u k ζ ) (cid:3) n ≥ δ n . This implies that det ( L ( ζ ) , . . . , L n +1 ( ζ )) is nowhere vanishing, andlog | det ( L ( ζ ) , . . . , L n +1 ( ζ )) | ≥ n +1 X i =1 log k L i ( ζ ) k + n log δ. (3.1)Applying integration on both sides of (3.1) and using Jensen’s Lemma, weget 0 = N ( r, (det ( L , . . . , L n +1 )) ) ≥ n +1 X i =1 T L i ( r ) − O (1) . for all r > . This contradicts to the assumption that L is nonconstant.The proof of Theorem 3.1 is based on the Zalcman lemma, Lemma 3.4,and a notice given by Green in [6]. In [13], we also given some applicationsof Lemma 3.4 in the normal problem concerning the condition of uniformboundedness of tangent mappings. Proof of Theorem 3.1. Suppose that F is not normal, then by Lemma 3.2there exist sequences { z k } ⊂ D with z k → z ∈ D, { f k } ⊂ F , { ρ k } ⊂ R with ρ k → + , and Euclidean unit vectors u k ⊂ C m , such that g k ( ζ ) := f k ( z k + ρ k u k ζ ) , where ζ ∈ C satisfies z k + ρ k u k ζ ∈ D, converges uniformlyon compact subsets of C to a nonconstant holomorphic mapping g of C into P n ( C ) . By Lemma 3.4, { H f k } ∞ k =1 , . . . , { H (2 n +1) f k } ∞ k =1 are normal families on D. By replacing by subsequences, without loss of generality, we assume that { H jf k } ∞ k =1 (1 j n + 1) converges uniformly on compact subsets of D toa nonconstant holomorphic mapping h j of D into P n ( C ) . 8e take reduced representations b h j = ( a j , . . . , a jn ) of h j , b f k = ( f k , . . . , f kn )of f k and d H jf k = ( a jk , . . . , a jkn ) of H jf k ( j = 1 , , . . . , n + 1) in a neighbour-hood V z of z such that { a jki } ∞ k =1 converges uniformly on compact subsetsof V z to a ji ( i = 0 , . . . , n ).We consider hyperplanes H j : a j ( z ) ω + · · · + a jn ( z ) ω n = 0 ( j =1 , . . . , n + 1) in P n ( C ) . Take a closed ball B ( z , R ) = { z : k z − z k R } ⊂ D. By the assumptionand by Lemma 3.3, there is a positive constant δ B ( z ,R ) such that for allsubsets { j , . . . , j n } ⊂ { , . . . , n + 1 } we have | det( a j s i ( z )) s,i n ) |k H j k · · · k H j n k = lim k →∞ | det ( H j f k ( z ) , . . . , H j n f k ( z )) |k H j f k ( z ) k · · · k H j n f k ( z ) k≥ [ d F S ( H j f k ( z ) , . . . , H j n f k ( z ))] n ≥ δ nB ( z ,R ) > . Hence, det( a j s i ( z )) s,i n ) = 0 , for all subsets { j , . . . , j n } ⊂ { , . . . , n + 1 } .Therefore, H , . . . , H n +1 are in general position.For each j ∈ { , . . . , n + 1 } , by Hurwitz’s theorem g ( C ) ⊂ H j or g ( C ) ∩ H j = ∅ ; this is impossible, by the notice given by Green ([6], p. 112), thereare no non-constant holomorphic maps of C into ( H i ∩ · · · ∩ H i p ) \ ( H i p +1 ∪· · · ∪ H i n +1 ), where ( i , . . . , i n +1 ) is a permutation of (1 , . . . , n + 1).We have completed the proof of Theorem 3.1. (cid:3) Let f = ( f : · · · : f n ) be a holomorphic map from a domain in C to P n ( C )given by homogeneous coordinate functions f j which are holomorphic withoutcommon zeros. We have the following formula for the Fubini-Study derivative f of f (for details, see [3])( f ) := ∂ ∂z∂z log n X i =0 | f i | = P s Let f be a holomorphic mapping of C into P n ( C ) , and let H , H , . . . , H q be hyperplanes in general position in P n ( C ) . Assume that sup { f ( z ) : z ∈ ∪ qj =1 f − ( H j ) } < ∞ . If q ≥ n (2 n + 1) + 2 , then f is upper bounded on C . Lemma 4.3 ([7], Theorem 4) . The holomorphic mapping f : △ → P n ( C ) isnot normal if and only if there exist sequences { z k } ⊂ △ , { r k } ⊂ R , r k > , with lim k →∞ r k −| z k | = 0 such that g k ( ξ ) := f ( z k + r k ξ ) converges uniformlyon compact subsets of C to a non-constant holomorphic mapping g of C into P n ( C ) . Lemma 4.4. Let f be a linearly non-degenerate holomorphic mapping of C into P n ( C ) . Let H , . . . , H q be q hyperplanes in κ -subgeneral position in P n ( C ) , where κ ≥ n and q ≥ κ − n +1 . Assume that f = 0 on ∪ qj =1 f − ( H j ) .Then q κ ( n + 1) − n + 1 . Proof. Let ( f , . . . , f n ) be a reduced presentation of f . For each a ∈ ∪ qj =1 f − ( H j ),we define C a := { ( c , . . . , c n ) ∈ C n +1 : c f ( a ) + · · · + c n f n ( a ) = 0 } . Since C a is a vector subspace of dimension n of C n +1 and since ∪ qj =1 f − ( H j )is at most countable, it follows that there exists( c , . . . , c n ) ∈ C n +1 \ ( ∪ a ∈∪ qj =1 f − ( H j ) C a ) . Let L , . . . , L n be n + 1 hyperplanes in general position in P n ( C ) , where L is defined by the equation: c ω + · · · + c n ω n = 0 . By our choice for ( c , . . . , c n ) f − ( L ) ∩ ( ∪ qj =1 f − ( H j )) = ∅ . (4.2)10et F := ( L ( f ) : · · · : L n ( f )) : C → P n ( C ) . Then, F is linearly nondegener-ate and T F ( r ) = T f ( r ) + O (1) . Since f vanishes on ∪ qj =1 f − ( H j ), we have( f : · · · : f n ) = ( f ′ : · · · : f ′ n ) on ∪ qj =1 f − ( H j ) . Hence,( L ( f ) : · · · : L n ( f )) = (( L ( f )) ′ : · · · : ( L n ( f )) ′ ) on ∪ qj =1 f − ( H j ) . (4.3)Since F is linearly nondegenerate, the Wronskian of F is not identically equalto zero. Therefore, there exists t ∈ { , . . . , n } such thatdet (cid:18) L ( f ) L t ( f )( L ( f )) ′ ( L t ( f )) ′ (cid:19) , hence, (cid:18) L t ( f ) L ( f ) (cid:19) ′ . By (4.2) and (4.3), we have (cid:18) L t ( f ) L ( f ) (cid:19) ′ = 0 on ∪ qj =1 f − ( H j ) . (4.4)From the first main theorem and the lemma on logarithmic derivative ofNevanlinna theory for meromorphic functions, we get easily that T (cid:16) Lt ( f ) L f ) (cid:17) ′ ( r ) T (cid:16) Lt ( f ) L f ) (cid:17) ( r ) + o (cid:18) T (cid:16) Lt ( f ) L f ) (cid:17) ( r ) (cid:19) . On the other hand, for each a ∈ C , since H , . . . , H q are in κ -subgeneralposition in P n ( C ), it follows that there are at most κ of them passing through f ( a ) . Hence, by (4.2) and (4.4), we have q X j =1 N [1] f ( r, H j ) κN (cid:16) Lt ( f ) L f ) (cid:17) ′ ( r ) κT (cid:16) Lt ( f ) L f ) (cid:17) ′ ( r ) + O (1) κT Lt ( f ) L f ) ( r ) + o (cid:18) T Lt ( f ) L f ) ( r ) (cid:19) κT F ( r ) + o ( T F ( r ))= 2 κT f ( r ) + o ( T f ( r )) . (cid:13)(cid:13)(cid:13) κT f ( r ) + o ( T f ( r )) ≥ q X j =1 N [1] f ( r, H j ) ≥ n q X j =1 N [ n ] f ( r, H j ) ≥ q − κ + n − n T f ( r ) − o ( T f ( r )) . Hence, q κ ( n + 1) − n + 1 . Proof of Theorem 4.1. Suppose that f is not normal, then by Lemma4.3, there exist sequences { z k } ⊂ △ , { r k } , r k > , with lim k →∞ r k −| z k | = 0such that g k ( ξ ) := f ( z k + r k ξ ) converges uniformly on compact subsets of C to a non-constant holomorphic mapping g of C into P n ( C ) . Without loss of the generality, we may assume that g ( C ) H j for all j ∈{ , . . . , q } and g ( C ) ⊂ H j for all j ∈ { q + 1 , . . . , q } , for some q q. Denoteby P the smallest subspace of P n ( C ) containing g ( C ). Then p := dim P ≥ g is a linearly non-degenerate entire curve in P . Since H , . . . , H q are ingeneral position, we have q − q + p n, furthermore, H ′ := H ∩ P , . . . , H ′ q := H q ∩ P are hyperplanes in n − ( q − q )-subgeneral position in P . Since q ≥ n (2 n +1)+2 > n +1, we have q > q − ( q − q ) − ( q − n − − p =2[ n − ( q − q )] − p + 1 . We now prove that g ( ξ ) = 0 for all ξ ∈ ∪ q j =1 g − ( H j ) = ∪ q j =1 g − ( H ′ j ) . To do this, we consider an arbitrary point ξ ∈ ∪ q j =1 g − ( H ′ j ) . Take an index j ∈ { , . . . , q } such that ξ ∈ g − ( H ′ j ) = 0 . By Hurwitz’s Theorem thereare values { ξ k } (for all k sufficiently large), ξ k → ξ such that ξ k ∈ g − k ( H j ) , and hence, z k + r k ξ k ∈ f − ( H j ) . Therefore, by the assumption, there is apositive constant M such that(1 − | z k + r k ξ k | ) f ( z k + r k ξ k ) < M for all k sufficiently large.We have g ( ξ ) = lim k →∞ g k ( ξ k ) = lim r k f ( z k + r k ξ k )= lim k →∞ r k ( −| z k | − | z k −| z k | + r k −| z k | ξ k | ) − (1 − | z k | )(1 + | z k + r k ξ k | ) (1 − | z k + r k ξ k | ) f ( z k + r k ξ k )= 0 , k →∞ r k −| z k | = 0 and −| z k | − | z k −| z k | + r k −| z k | ξ k | ≥ ( −| z k | − | z k | −| z k | ) − r k −| z k | | ξ k | > , for all k sufficiently large).Hence, g = 0 in ∪ q j =1 g − ( H ′ j ) . Applying Lemma 4.4, we have q n − ( q − q ))( p + 1) − p + 1 . Then q + 2( p + 1)( q − q ) n ( p + 1) − p + 1 . Therefore, q = q + ( q − q ) q + 2( p + 1)( q − q ) n ( p + 1) − p + 1 n (2 n + 1) + 1 . This contradicts to the assumption that q ≥ n (2 n + 1) + 2 . We have completed the proof of Theorem 4.1. (cid:3) References [1] G. Aladro and S. G. Krantz A criterion for normality in C n , J. Math.Anal. Appl. 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