Second main theorem for holomorphic mappings from the discs into the projective spaces

JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2015-0027 Mathematical and Physical Sci., 2015, Vol. 60, No. 7, pp. 21-29 This paper is available online at SECONDMAIN THEOREM FOR HOLOMORPHICMAPPINGS FROM THE DISCS INTO THE PROJECTIVE SPACES Nguyen Van An1 and Nguyen Thi Nhung2 1Division of Mathematics, Banking Academy 2Department of Mathematics and Informatics, Thang Long University Abstract. In this paper, we establish a second main theorem for holomorphic mappings from a disc ∆(R) into Pn(C) and families of hyperplanes in subgeneral position. Our result is an extension the classical second main theorem of Cartan-Nochka and the second main theorem of Fujimoto. Keywords: Second main theorem, holomorphic mapping, subgeneral position. 1. Introduction Let f be a holomorphic mapping from C into Pn(C) with a reduced presentation f = (f0 : · · · : fn) in a fixed homogeneous coordinate system (ω0 : · · · : ωn) of Pn(C). Let H be a hyperplane in Pn(C) defined by H : {(ω0 : · · · : ωn) : a0ω0 + · · ·+ anωn = 0}, where ai ∈ C, 0 ≤ i ≤ n. If there is no confusion, we will also denote by H the linear form H(ω0, . . . , ωn) = a0ω0 + · · ·+ anωn. We set ||ω|| = (|ω0|2 + · · ·+ |ωn|2)1/2, ||f || = (|f0|2 + · · · + |fn|2)1/2, ||H|| = (|a0|2+ · · ·+ |an|2)1/2 andH(f) := a0f0+ · · ·+anfn. Then the functionH(f) depends on the choice of the reduce representation of f and representation ofH . However its divisor of the zeros νH(f) does not depend on these choices. Here, we may consider νH(f) as a function whose value at a point z0 is the multiple intersection of the image of f and H at the point f(z0). The classical second main theorem for holomorphic mappings into projective spaces of H. Cartan, which is the most important second main theorem in Nevanlinna theory, is stated as follows: Theorem A (see [4]). Let f be linearly nondegenerate holomorphic mapping of C into Pn(C) and {Hj}qj=1 be hyperplanes of Pn(C) in general position, where q ≥ n+ 2. Then (q − n− 1)Tf (r) ≤ q∑ j=1 Nn(r, νHj (f)) + o(Tf (r)). Received November 16, 2015. Accepted December 10, 2015. Contact Nguyen Thi Nhung, e-mail address: hoangnhung227@gmail.com 21 Nguyen Van An and Nguyen Thi Nhung Here, Tf (r) denotes the characteristic function of f andNn(r, νHj (f)) denotes the counting function of the divisor νHj(f) with multiplicity truncated to level n. These notions are defined in the next section. In 1994, H. Fujimoto [1] generalized the above result to the case of meromorphic mappings from the balls in Cm into Pn(C). We state here that result for the case of holomorphic mappings from the discs ∆(R) = {z ∈ C : |z| < R} ⊂ C as follows: Theorem B (see [1]). Let Hj (1 ≤ j ≤ q) be q (≥ n + 2) hyperplanes in general position and let f be a linear meromorphic mapping from ∆(R) ⊂ C into Pn(C) satisfying the condition f(0) /∈ ⋃qj=1Hj . Then it holds that, for any r (0 < r < R), (q − n− 1)Tf (r) ≤ q∑ j=1 Nn(r, νHj (f)) + S(r), where, for any given positive ε and ρ (r < ρ < R), S(r) is evaluated as follows: S(r) ≤K0 +K1 log+ ρ+K2 log+ 1 ρ− r +K3 log + 1 r +K4 q∑ j=1 log+ ∣∣∣ log |Hj(f)(0)|||f(0)|| ∣∣∣+ q∑ j=1 |Hj(f)(0)| ||f(0)|| + εTf (ρ)−W ∗ f with constants K0 depending only on ε, Hj (1 ≤ j ≤ q) and with constants Kκ(1 ≤ κ ≤ 4) depending only on n. Here, the quantity W ∗f is defined by W ∗f = 1 ||f(0)||n+1 · ∣∣∣∣∣∣∣∣∣ f0(0) f1(0) · · · fn(0) f ′0(0) f ′ 1(0) · · · f ′n(0) ... ... · · · ... f (n) 0 (0) f (n) 1 (0) · · · f (n)n (0) ∣∣∣∣∣∣∣∣∣ . Let H1, ...,Hq be q hyperplanes in Pn(C). Let Q = {1, . . . , q} be an index set. For the subset R ⊂ Q, |R| stands for its cardinality. Definition 1.1. Suppose N ≥ n and q ≥ N + 1. The family {Hj}qj=1 is said to be in N− subgeneral position if for any arbitrary R ⊂ Qwith |R| = N+1 then⋂j∈RHj = ∅. In particular, if N = n then we will say that H1, . . . ,Hq are in general position. In this paper, we will extend the above mentioned results by establishing a second main theorem for holomorphic mappings from a disc ∆(R) into Pn(C) and families of hyperplanes in N− subgeneral position, where N ≥ n. Our main result is stated as follows: Theorem 1.1. Let f : ∆(R) → Pn(C) be a linearly nondegenerate holomorphic mapping and {Hj}qj=1 be hyperplanes of Pn(C) in N−subgeneral position, N ≥ n, q ≥ 2N − n + 1. Then for any r (0 < r < R), we have (q − 2N + n− 1)Tf (r) ≤ q∑ j=1 Nn(r, νHj (f)) + S(r), 22 Second main theorem for holomorphic mappings from the discs into the projective spaces where for any ρ (0 < r < ρ < R), S(r) is evaluated as follows: S(r) ≤K0 +K1 log+ ρ+K2 log+ 1 ρ− r +K3 log + 1 r +K4 q∑ j=1 log+ ∣∣∣ log |Hj(f)(0)|||f(0)|| ∣∣∣+K5 log+ Tf (ρ) +O(1). with Kκ(0 ≤ κ ≤ 5) are constants depending only onHi (0 ≤ i ≤ n), N and n. 2. Basic notions and auxiliary results from Nevanlinna theory In this section, we will recall some basic notions and auxiliary results in Nevanlinna theory due to [1] and [4]. For z ∈ C we set ∆(R) := {z ∈ C : |z| < R}, S(R) := {z ∈ C : |z| = R} (0 < R ≤ +∞). Let f be a not identically zero holomorphic function on ∆(R). For a point a ∈ ∆(R) we expand f as a compactly convergent series f(u+ a) = ∞∑ k=0 Pk(u), on a neighborhood of a, where Pk is either identically zero or a homogeneous polynomial of degree k. The number νf (a) := min{k : Pk(u) 6≡ 0} is said to be the zero multiplicity of f at a. By definition, a divisor on ∆(R) is an integer-valued function ν on∆(R) such that for every a ∈ ∆(R) there are holomorphic functions g(z)(6≡ 0) and h(z)(6≡ 0) on a neighborhood U of a with ν(z) = νg(z) − νf (z) on U . The support of a divisor ν is defined as an analytic set |ν| := {z : ν(z) 6= 0} in ∆(R). Let us consider next a meromorphic mapping f of ∆(R) into Pn(C). For any a ∈ ∆(R), f has a representation f(z) = ( f0(z) : · · · : fn(z) ) on some neighborhood U of a with fixed homogeneous coordinates (w0 : . . . : wn) on Pn(C) and holomorphic functions fi (0 ≤ i ≤ n) on Pn(C). Let H be a hyperplane in Pn(C) given by H = {a0ω0 + . . . + anωn = 0}, where a := (a0, ..., an) 6= (0, ..., 0). For every a ∈ ∆(R), taking an admissible representation f(z) = ( f0(z) : · · · : fn(z) ) on a neighborhood U of a, we consider a holomorphic function F := a0f0 + . . .+ anfn. Then, the divisor νH(f)(z) := νF (z) (z ∈ U) is determined independently of a choice of admissible representations and hence is well-defined on the totality of ∆(R). A meromorphic function ϕ on G induces a meromorphic map ϕ∗ of G into Pn(C) defined by ϕ∗(z) = (f0(z) : f1(z)) on a connected open set U if ϕ = f0(z)/f1(z) for holomorphic functions f0, f1(6≡ 0) on U . In this case, ν0ϕ := ν(ϕ∗,H0) and ν ∞ ϕ := ν(ϕ∗,H1) are nothing but the divisors of zeros of ϕ and of poles of ϕ respectively, where Hi = {ωi = 0}, (i = 0, 1). 23 Nguyen Van An and Nguyen Thi Nhung Let ν be a non-negative divisor on disc∆(R) (0 < R ≤ +∞). Suppose that 0 < m ≤ +∞ and 0 < s < r < R. We define Vm(r, ν) = ∑ |z|≤r min(ν(z),m), and Nm(r, s, ν) = r∫ s Vm(r, ν) t dt. We put V (r, ν) = V∞(r, ν), N(r, s, ν) = N∞(r, s, ν) and Nm(r, ν) := lims→+0Nm(r, s, ν) for the case 0 /∈ |ν|. Since B(R) is a Cousin-II domain, any given divisor ν on B(R) can be written ν = νg − νh with holomorphic functions g (6≡ 0) and h (6≡ 0). Let f be a holomorphic mapping of ∆(R) into Pn(C) with a reduce representation f = (f0 : . . . : fn) on ∆(R). The characteristic function is define by Tf (r) := 1 2π ∫ 2π 0 log ||f(reiθ)||dθ − log ||f(0)||. LetH be a hyperplane in Pn(C) such that f(0) 6∈ H . The proximity function of f with respect to H is defined as follows: mf (r,H) = 1 2π ∫ 2π 0 log ||f(reiθ)|| |H(f)(reiθ)|dθ − log ||f(0)|| |H(f)(0)| . The first main theorem states that Tf (r) = N(r, νH(f)) +mf (r,H). Let ϕ be a nonzero meromorphic function on ∆(R). We define m(r, ϕ) := 1 2π ∫ 2π 0 log+ |ϕ(reiθ)|dθ, where log+ x = max(log x, 0) for any x ≥ 0. If 0 /∈ |ν0ϕ| ∪ |ν∞ϕ | for a meromorphic function ϕ(z) on ∆(R), then we define T (r, ϕ) = N(r, ν∞ϕ ) +m(r, ϕ) − log+ |ϕ(0)|. If we consider ϕ is a holomorphic mapping into P1(C), then T (r, ϕ) = Tϕ(r). Let f0, . . . , fn be n + 1 holomorphic functions on ∆(R). We define the Wronskian of the family f0, . . . , fn as follows: W (f0, . . . , fn) := ∣∣∣∣∣∣∣∣∣ f0 f1 · · · fn f ′0 f ′ 1 · · · f ′n ... ... · · · ... f (n) 0 f (n) 1 · · · f (n)n ∣∣∣∣∣∣∣∣∣ , and if f is a holomorphic mapping of ∆(R) into Pn(C) with a reduce representation f = (f0 : . . . : fn), we define the WronskianW (f) of f asW (f) =W (f0, . . . , fn). It is easy to see that 24 Second main theorem for holomorphic mappings from the discs into the projective spaces • W (hf0, . . . , hfn) = hn+1W (f0, . . . , fn) for any meromorphic function h (6≡ 0) on ∆(R). • For n+ 1 hyperplanes H0, . . . ,Hn in general position such that Hi(f) 6≡ 0, ∀i = 0, . . . , n W (H0(f), . . . ,Hn(f)) = det(aij)1≤i,j≤n.W (f0, . . . , fn), where Hj : ∑n k=0 ajkωk = 0. Let Q = {1, . . . , q} and for the subset R ⊂ Q, we denote by V (R) the vector subspace of C n+1 spanned by vectors (ajk)0≤k≤n, j ∈ R and rk(R) = dimV (R). Lemma 2.1 (see [4]). Let q > 2N − n + 1 and let {Hj}qj=1 be hyperplanes of Pn(C) in N− subgeneral position. Then there are rational constants ω(j), j ∈ Q, which are called Nochka’s weights, satisfying the following conditions: i 0 < ω(j) ≤ 1,∀j ∈ Q. ii Put ω˜ = maxj∈Q ω(j), then q∑ j=1 ω(j) = ω˜(q − 2N + n− 1) + n+ 1, ω˜ is called Nochka’s constant. iii n+ 1 2N − n+ 1 ≤ ω˜ ≤ n N . iv If R ⊂ Q and 0 < |R| ≤ N + 1, then∑j∈R ω(j) ≤ rk(R). Lemma 2.2 (see [4]). Let q > 2N − n + 1 and let {Hj}qj=1 be hyperplanes of Pn(C) in N− subgeneral position. Let {ω(j)}j∈Q be Nochka’s weights for {Hj}qj=1. Let Ej ≥ 1, j ∈ Q be arbitrary constants. Then for every subset R ⊂ Q with 0 < |R| ≤ N + 1, there are distinct j1, . . . , jrk(R) ∈ R such that rk({jl}rk(R)l=1 ) = rk(R) and ∏ j∈R E ω(j) j ≤ rk(R)∏ l=1 Ejl . Lemma 2.3 (Jensen’s formula, see [1]). Let ϕ (6≡ 0) be a meromorphic function on∆(R). Assume that 0 /∈ |ν0ϕ| ∪ |ν∞ϕ |. Then we have N(r, ν0ϕ)−N(r, ν∞ϕ ) = 1 2π ∫ 2π 0 log |ϕ(reiθ)|dθ − log |ϕ(0)|, where 0 < s < r < R. Lemma 2.4 (Logarithmic derivative lemma, see [1]). Let ϕ(z) be a meromorphic function on ∆(R) with ϕ(0) 6= 0,∞. For any r, ρ (0 < r < ρ < R) and a positive integer l, there are some constants K ′κ(0 ≤ κ ≤ 5) depending only on l such that m ( r, dl−1 dzl−1 (ϕ′(z) ϕ(z) )) ≤K ′0 +K ′1 log+ ρ+K ′2 log+ 1ρ− r +K ′3 log+ 1r +K ′4 log + | log |ϕ(0)|| +K ′5 log+ Tϕ(ρ). 25 Nguyen Van An and Nguyen Thi Nhung 3. Proof of main theorem Lemma 3.1. Let ϕ(z) be a meromorphic function on ∆(R) with ϕ(0) 6= 0,∞. We have m ( r, dk dzk (ϕ(z)) ϕ(z) ) ≤ K k−1∑ j=0 m ( r, dj dzj (ϕ′(z) ϕ(z) )) +K ′, where K,K ′ are positive numbers depending only on k. Proof. By simple computation, we get dk dzk (ϕ(z)) ϕ(z) = P (ϕ′ ϕ , . . . , dk−1 dzk−1 (ϕ′(z) ϕ(z) )) where P (z0, . . . , zk−1) is a polynomial in k variables with integer coefficients not depending on ϕ. Applying properties of proximity function, we have m ( r, dk dzk (ϕ(z)) ϕ(z) ) ≤ K k−1∑ j=0 m ( r, dj dzj (ϕ′(z) ϕ(z) )) +K ′ for positive numbers K and K ′ depending only on k. Lemma 3.2. Let f be meromorphic of ∆(R) into Pn(C) and let H0, . . . ,Hn be (n + 1) hyperplanes in general position of Pn(C) such that Hi(f) 6≡ 0, i = 0, . . . , n. Then for any r and ρ (0 < r < ρ < R), we have m ( r, W (H0(f), . . . ,Hn(f))∏n i=0Hi(f) ) ≤K ′′0 +K ′′1 log+ ρ+K ′′2 log+ 1 ρ− r +K ′′ 3 log + 1 r +K ′′4 n∑ i=0 log+ ∣∣∣ log |Hi(f)(0)|||f(0)|| ∣∣∣+K ′′5 log+ Tf (ρ). where K ′′κ(0 ≤ κ ≤ 5) are constants depending only on Hi (0 ≤ i ≤ n) and n. Proof. By Lemma 3.1 and Lemma 2.4, for any r and ρ (0 < r < ρ < R), we have m ( r, W ( H0(f), . . . ,Hn(f) )∏n i=0Hi(f) ) =m ( r, W (H0(f) H0(f) , . . . , Hn(f)H0(f) ) H0(f) H0(f) . . . Hn(f)H0(f) ) ≤K1 ∑ 0≤k,i≤n m ( r, dk dzk (Hi(f) H0(f) ) / Hi(f) H0(f) ) +K ′1 ≤K2 ∑ 0≤i,j≤n m ( r, dj dzj (( Hi(f) H0(f) )′ / Hi(f) H0(f) )) +K ′2 ≤K ′0 +K ′′1 log+ ρ+K ′′2 log+ 1 ρ− r +K ′′ 3 log + 1 r +K ′4 n∑ i=0 log+ ∣∣∣ log |Hi(f)(0)||H0(f)(0)| ∣∣∣+K ′5 n∑ i=0 log+ THi(f) H0(f) (ρ). 26 Second main theorem for holomorphic mappings from the discs into the projective spaces where K1,K ′1,K2,K ′ 2,K ′ 0,K ′′ 1 ,K ′′ 2 ,K ′′ 3 ,K ′ 4,K ′ 5 are constants depending only on Hi (0 ≤ i ≤ n) and n. Since log |Hi(f)(0)| |H0(f)(0)| ≤ log |Hi(f)(0)| ||f(0)|| +O(1) and THi(f)H0(f) (ρ) ≤ Tf (ρ), where O(1) depends only onHi,H0. Then we get m ( r, W ( H0(f), . . . ,Hn(f) )∏n i=0Hi(f) ) ≤K ′′0 +K ′′1 log+ ρ+K ′′2 log+ 1 ρ− r +K ′′ 3 log + 1 r +K ′′4 n∑ i=0 log+ ∣∣∣ log |Hi(f)(0)|||f(0)|| ∣∣∣+K ′′5 log+ Tf (ρ). for any r and ρ (0 < r < ρ < R) and K ′′κ(0 ≤ κ ≤ 5) are constants depending only on Hi (0 ≤ i ≤ n) and n. Proof of the Theorem 1.1 We take a homogeneous coordinate system (ω0 : . . . : ωn) of Pn(C) and let f = (f0 : . . . : fn) be the presentation of f . We assume that Hj is defined by Hj : n∑ k=0 ajkωk = 0. Put Q = {1, 2, . . . , q} and denote by ω(j), ω˜ the Nochka’s weights and the Nochka’s constant respectively. Set Φ(ω) = ∑ |S|=q−N−1 ∏ j∈S ( |Hj(ω)| ||ω|| )ω(j) . Then Φ is a nonzezo continuous function on Pn(C). Therefore there is a constant C > 0 such that C−1 ≤ Φ(ω) ≤ C,∀ω ∈ Pn(C). For R ⊂ Q, |R| = N + 1, we denote R0 ⊂ R by the indices satifying lemma 2.2, we get Φ(ω) = ∑ |R|=N+1 q∏ j=1 ( |Hj(ω)| ||ω|| )ω(j) ×∏ j∈R ( 1 ||Hj||ω(j) ) × ∏ j∈R ( ||ω||.||Hj || |Hj(ω)| )ω(j) ≤ q∏ j=1 ( |Hj(ω)| ||ω|| )ω(j) × ∑ |R|=N+1 (∏ j∈R ( 1 ||Hj ||ω(j) )× ∏ j∈R0 ( ||ω||.||Hj || |Hj(ω)| )) Therefore q∏ j=1 ( ||ω|| |Hj(ω)| )ω(j) ≤ C × ∑ |R|=N+1 ((∏ j∈R 1 ||Hj||ω(j) )× ∏ j∈R0 ( ||ω||.||Hj || |Hj(ω)| )) 27 Nguyen Van An and Nguyen Thi Nhung Setting R0 = {j1, . . . , jrk(R)}, then for z /∈ (⋃ j∈Q f −1(Hj), we have q∏ j=1 ( ||f || |Hj(f)| )ω(j) × |W (f)(z)| ≤C||f(z)||n+1 × ∑ |R|=N+1 ∏ j∈R ( 1 ||Hj||ω(j) )ω(j) × ( |W (f)(z)|.∏j∈R0 ||Hj ||∏ j∈R0 |Hj(f)(z)| ) =CR||f(z)||n+1 ∑ |R|=N+1 (W (Hj1(f), . . . ,Hjrk(R)(f))(z)∏ j∈R0 |Hj(f)(z)| ) , where CR is a positive constant depending only on R. It follows that q∑ j=1 ω(j) log ( ||f ||(z) |Hj(f(z))| ) + log |W (f)(z)| ≤ log ||f(z)||n+1 + ∑ |R|=N+1 ∑ R0⊂R log+ (W (Hj1(f), . . . ,Hjrk(R)(f))(z)∏ j∈R0 |Hj(f)(z)| ) + C0, where C0 is a positive constant not depending on f . Integrating in [0, 2π] both sides of the above inequality and applying lemma 3.2, we deduce q∑ j=1 ω(j)Tf (r) +N(r, νW (f))− q∑ j=1 ω(j)N(r, νHj (f)) ≤(n+ 1)Tf (r) + ∑ |R|=N+1 ∑ R0⊂R m ( r, W (Hj1(f), . . . ,Hjrk(R)(f))(z)∏ j∈R0 |Hj(f)(z)| ) +O(1) ≤(n+ 1)Tf (r) + S(r) (3.1) for any ρ (0 < r < ρ < R), where O(1) depends only on {Hi}qi=1 and S(r) is evaluated as follows: S(r) ≤K0 +K1 log+ ρ+K2 log+ 1 ρ− r +K3 log + 1 r +K4 q∑ j=1 log+ ∣∣∣ log |Hj(f)(0)|||f(0)|| ∣∣∣+K5 log+ Tf (ρ) +O(1). with Kκ(0 ≤ κ ≤ 5) are constants depending only on Hi (0 ≤ i ≤ n), N and n. Since ∑q j=1 ω(j) = ω˜(q − 2N + n− 1) + n+ 1, we obtain (q − 2N + n− 1)Tf (r) ≤ 1 ω˜ ( q∑ j=1 ω(j)N(r, νHj (f))−N(r, νW (f) ) . To continue, we prove the following inequality q∑ j=1 ω(j)νHj (f)(a)− νW (f)(a) ≤ q∑ j=1 ω(j)min{νHj(f)(a), n}. (3.2) Indeed, for each point a, there are at most N indices j ∈ Q such that νHj(f)(a) = 0. Without loss of generality, we may assume that νH1(f)(a) ≥ . . . ≥ νHN (f)(a) ≥ 0 = νHN+1(f)(a) = . . . = νHq(f)(a). 28 Second main theorem for holomorphic mappings from the discs into the projective spaces Putting R = {1, . . . , N + 1}, we have q∑ j=1 ω(j)νHj(f)(a)− νW (f)(a) = ∑ j∈R ω(j)νHj(f)(a)− νWR0(H(f))(a) ≤ ∑ j∈R ω(j)νHj(f)(a)− ∑ j∈R0 max{νHj(f)(a)− n, 0} ≤ ∑ j∈R ω(j)νHj(f)(a)− ∑ j∈R ω(j)max{νHj(f)(a)− n, 0} ≤ ∑ j∈R ω(j)min{νHj(f)(a), n}. According to (3.2), it follows that q∑ j=1 ω(j)N(r, νHj (f))−N(r, νW (f) ≤ q∑ j=1 ω(j)Nn(r, νHj (f)). (3.3) Combining (3.1) and (3.3), we get (q − 2N + n− 1)Tf (r) ≤ q∑ j=1 ω(j) ω˜ Nn(r, νHj (f)) + S(r) ≤ q∑ j=1 Nn(r, νHj (f)) + S(r) The theorem is proved.  REFERENCES [1] H. Fujimoto, 1974. 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