Xiaojun LIU Han WANG

1Department of Mathematics,University of Shanghai for Science and Technology,Shanghai 200093,China.E-mail: xiaojunliu2007@hotmail.com 1356938164@qq.com

Abstract In this paper,the authors discuss a generalization of Lappan’s theorem to higher dimensional complex projective space and get the following result: Let f be a holomorphic mapping of ∆into Pn(C),and let H1,···,Hq be hyperplanes in general position in Pn(C).Assume that

if q≥2n2+3,then f is normal.

Keywords Holomorphic mapping,Normal family,Hyperplanes

1 Introduction

In the theory of normal families,perhaps the following criterion of Montel [1]is the most celebrated theorem.

Theorem 1.1LetFbe a family of meromorphic functions in a domain D⊂C,and let a,b,c be three distinct points in.Assume that all functions inFomit three points a,b,c in D.ThenFis a normal family in D.

In 1957,Lehto and Virtanen [2]proved the following well-known result,which says that a functionf(z) meromorphic in the unit disc ∆:={z∈C:|z|<1} is normal if and only if

In 1972,Pommerenke [3]posed an open question: For a given positive numberM >0,does there exist a finite subsetE⊂such that iffis a meromorphic function in ∆,then the condition that (1−|z|2)f♯(z)≤Mfor eachz∈f−1(E) implies thatfis a normal function?

Latter,Lappan [4]answered the above question and proved the following well-known result named Lappan’s theorem.

Theorem 1.2Let E be any set consisting of five complex numbers,finite or infinite.If f is a meromorphic function in∆such that

then f is a normal function.

In 2020,Tan [5]generalized the above theorem to thendimensional complex projective space,and proved the following theorem.

Theorem 1.3Let f be a holomorphic mapping of∆intoPn(C),and let H1,···,Hqbe hyperplanes in general position inPn(C).Assume that

if q≥n(2n+1)+2,then f is normal.

Inspried by the method of the proof of the main theorem in Chen and Yan [6],we reduce the number of hyperplanes in Theorem 1.3 and obtain the following main result.

Theorem 1.4Let f be a holomorphic mapping of∆intoPn(C),and let H1,···,Hqbe hyperplanes in general position inPn(C).Assume that

if q≥2n2+3,then f is normal.

2 Notations and Preliminaries

In this section,we introduce some notations and preliminaries related to this paper.For more details see [7].

Letf=[f0:···:fn]be a holomorphic mapping from a domain in C to Pn(C) given by homogeneous coordinate functionfj(j=0,1,···,n) which are holomorphic without common zeros.In this paper,we also need the following formula named Fubini-Study derivativef♯off(for details,see [8]),

Definition 2.1(see [7])Let ν be an effective divisor onC.For each positive integer(or+∞)p,we define the counting function of ν(where multiplicities are truncated by p)by

whereFor brevity,we will omit the character[p]in the countingfunction if p=+∞.

For a meromorphic functionϕon C (ϕ0,ϕ∞),we denote by (ϕ)0the divisor of zeros ofϕ.We have the following Jensen’s formula for the counting function:

Definition 2.2(see [7])We define the proximity function of ϕ by

wherelog+x=max{0,logx}for x>0.

Ifϕis nonconstant,then=o(Tϕ(r)) asr→∞,outside a set of finite Lebesgue measure (Nevanlinna’s lemma on the logarithmic derivative).

Nevanlinna’s first main theorem forϕstates that,for anya∈,

Definition 2.3(see [7])Let f be a holomorphic mapping ofCintoPn(C)with a reduced representation(f0,···,fn).The characteristic function Tf(r)of f is defined by

where

LetH=｛(ω0:···:ωn)∈Pn(C):=0｝ be a hyperplane in Pn(C)such thatf(C)/⊂H.Denote by(H(f))0the divisor of zeros ofand put(r,H):=N[p](r,(H(f))0).

Definition 2.4Let q,k be two positive integers,satisfying q≥k≥n and let H1,···,Hqbe q hyperplanes inPn(C).These hyperplanes are said to be in k-subgenerral position iffor all1≤j0<···

Definition 2.5(see [7])Let f be a holomorphic mapping ofCintoPn(C).If there exists a hyperplance H inPn(C),such that f(C) ⊂H,then we call that f is a linearly degenerate holomorphic mapping,otherwise f is linearly non-degenerate.

Definition 2.6(see [3])Let f: ∆→be a meromorphic function,and letF={f◦ϕ|ϕ:∆→∆be a conformal mapping}.IfFis normal in∆,then f is called a normal function.

Similarly,we can give the following definition for the normal curve.

Definition 2.7Let f: ∆→Pn(C)be a holomorphic mapping,and letF={f◦ϕ|ϕ:∆→∆be a conformal mapping}.IfFis normal in∆,then f is called a normal curve.

Nochka’s Second Main Theorem(see [9]) Letfbe a linearly non-degenerate holomorphic mapping of C into Pn(C),and letH1,···,Hqbeqhyperplanes ink-subgeneral position in Pn(C) (k≥nandq≥2k−n+1).Then

where “‖” means the estimate holds for all largeroutside a set of finite Lebesgue measure.

3 Preliminary Lemmas

Before we give the proof of our main theorem,we need the following version of Zalcman’s lemma for holomorphic mappings from the domain Ω ⊆C to Pn(C).

Lemma 3.1(see [10])LetFbe a family of holomorphic mappings of a hyperbolic domainΩinCintoPn(C).The familyFis not normal onΩif and only if there exist sequences{fk}⊂F,{zk}⊂Ωwith zk→z0∈Ω,{rk}with rk>0and rk→0,such that

converges uniformly on compact subsets ofCto a nonconstant holomorphic map g ofCintoPn(C).

Lemma 3.2(see [7])Let f: C→Pn(C)be a holomorphic mapping,and H1,···,Hqbe(q≥2n+1)hyperplanes inPn(C)in general position.If for each j=1,···,q,either f(C)is contained in Hj,or f(C)omits Hj,then f must be a constant.

The following lemma plays an important role in the proof of Theorem 1.3.

Lemma 3.3(see [5])Let f be a linearly non-degenerate holomorphic mapping ofCintoPn(C),and let H1,···,Hqbe q hyperplanes in k-subgeneral position inPn(C),where k≥n and q≥2k−n+1.Assume that f(z)∈Hj⇒f♯(z)=0,j=1,···,q.Then q≤2k(n+1)−n+1.

In this paper,inspired by the method of Chen and Yan [6],we improve the above lemma and get the following lemma,which plays a key role in the proof of our main theorem.

Lemma 3.4Let f be a linearly non-degenerate holomorphic mapping ofCintoPn(C),and let H1,···,Hqbe q hyperplanes in k-subgeneral position inPn(C),where k≥n and q≥2k−n+1.Assume that f(z)∈Hj⇒f♯(z)=0,j=1,···,q.Then q <2k(n+1)−2n+3.

ProofWe pick up a reduced presentation (f0,···,fn) off.We also write it asf=(f0,···,fn) and letf′=(f′0,···,f′n).For eachwe define

Since Azis a vector subspace of dimensionnof Cn+1and sinceis at most countable,it follows that there exists a vector

LetLbe a hyperplane in Pn(C),whereL(ω) is defined by the equation

By our choice,we have

Moreover,we can also choose suchL(ω) to satisfy that for alli∈{1,2,···,q},(ω)constant.

Set Φi=wherei=1,2,···,q.

If there exists somei0∈{1,2,···,q},such that Φi0≡C,thenHi0(f)≡CL(f),and we have 〈f,αi0〉≡〈f,Cℓ〉,which implies that 〈f,αi0−Cℓ〉≡0.Sincewe haveαi0−Cℓ≠0,which means thatfis linearly degenerate,a contradiction.

Then,for everyi∈{1,2,···,q},Φiconstant.and then

For anyz0∈C,Φ′i(z0)=0,we divided into two cases.

Case 1f(z0) ∈Hi,then 〈f(z0),αi〉=0.And ifz0∈{z|ν〈f,Hi〉(z)≥2},we have〈f′(z0),αi〉=0.Then,z0is a zero of Φ′i,and

We note that

Ifz0∈{z|ν〈f,Hi〉(z)=l}(={z|ν〈f′,Hi〉(z)=l−1}),2≤l≤n,then

Ifz0∈{z|ν〈f,Hi〉(z)>n},thenz0∈{z|ν〈f′,Hi〉(z)≥n},

Case 2f(z0) ∈Hj,wherej∈{1,···,q} andj≠i.By the condition of this lemma,we have

and

ThenHi(f)(z0)=λ(z0)Hi(f′)(z0)=λ(z0)(Hi(f))′(z0) andL(f)(z0)=λ(z0)L(f′)(z0)=λ(z0)(L(f)′)(z0),whereλ(z0) is a constant.Thus,Φ′i(z0)=0.So,we have any zero of〈f(z0),αj〉 is also a zero of Φ′i.

SinceH1,···,Hqare ink-subgeneral position in Pn(C),combining with the discussion above,we have

By the first main theorem and the logarithmic derivative lemma of Nevanlinna theory for meromorphic function,we can easily get

By (3.1)–(3.2) and [7,p.162],we have

Take summation of (3.3) over 1≤i≤q,we have

By Nochka’s second main theorem,it follows that

Comparing the coefficients ofTf(r) in the both sides of above inequality,we have

then

Since,

By calculation,

Thus,this lemma is proved.

4 Proof of Theorem 1.4

If not the case,we may assume thatThen there exist a sequencezk,|zk|<1,such that

Let

Since |zk+(1−|zk|)z|≤|zk|+(1−|zk|)|z|<|zk|+1−|zk|=1,we havefkis well-defined.By calculation,

then

By Lemma 3.1,there exist pointsz∗k∈∆,positive numbersρkwithρk→0+such that

wheregis a nonconstant holomorphic mapping of C into Pn(C).

If for eachj∈{1,···,q},g(C) is contained inHj,org(C) omitsHj.By Lemma 3.2,gis a constant,a contradiction.So there exist somej∈{1,···,q} andξ0∈C,such that〈g(ξ0),αj〉=0 but 〈g(ξ),αj〉0.

We now prove thatg♯(ξ0)=0. By Hurwitz’s theorem,there exist pointsξkwithξk→ξ0(ask→∞),such thatgk(ξk)∈Hj,and hencefk(z∗k+ρkξk)∈Hj.Thenf(zk+(1−|zk|)(z∗k+ρkξk))∈Hj.

Denoting

we have,by the condition of this theorem,there is a positive constantMsuch that

for allksufficiently large.

By calculation,

By the proof of Lemma 3.1,we havethen

ask→∞.

So,we have

Letk→∞,g♯(ξ0)=0.

Without loss of the generality,we may assume that there exists some integerq0with 1≤q0≤q,such that for anyj∈{1,···,q0},g(C)/⊂Hj,andj∈{q0+1,···,q},g(C) ⊂Hj.Denote the smallest subspace of Pn(C) containingg(C) by P.Thenp:=dim P≥1,andgis a linearly non-degenerate entire curve in P.SinceH1,···,Hqare in general position,we haveq−q0+p≤n,furthermore,:=H1∩P,···,:=Hq0∩P are hyperplanes inn−(q−q0)subgeneral position in P.

For eachj=1,···,q0and for allξ0∈C,such thatg(ξ0) ∈Hj,we haveg(ξ0) ∈andg♯(ξ0)=0.

Sinceq≥2n2+3>2n+1,we haveq0>q0−(q−q0)−(q−2n−1)−p=2[n−(q−q0)]−p+1.

Applying Lemma 3.4,we have

Then

Therefore,

This contradicts to the assumption thatq≥2n2+3.Thus,we have

The proof of Theorem 1.4 is finished.

AcknowledgementThe authors would like to thank the referees for their important comments and advices.