The purpose of this paper is to establish the first and second fundamental theorems for an E-valued meromorphic mapping from a generic domain D⊂ℂ to an infinite dimensional complex Banach space E with a Schauder basis. It is a continuation of the work of C. Hu and Q. Hu. For f(z) defined in the disk, we will prove Chuang's inequality, which is to compare the relationship between T(r,f) and T(r,f′). Consequently, we obtain that the order and the lower order of f(z) and its derivative f′(z) are the same.

1. Introduction

In 1980s, Ziegler [1] established Nevanlinna's theory for the vector-valued meromorphic functions in finite dimensional spaces. After Ziegler some works in finite dimensional spaces were done in 1990s [2–4]. In 2006, C. Hu and Q. Hu [5] considered the case of infinite dimensional spaces and they investigated the E-valued meromorphic mappings defined in the disk Cr={z:|z|<r}. In this article, by using Green function technique, we will consider this theory defined in generic domain D⊆ℂ (see Section 2). In Section 3, motivated by the work of [6–8], we will prove Chuang's inequality, which is to compare the relationship between T(r,f) and T(r,f'). Consequently, we obtain that the order and the lower order of f(z) and its derivative f'(z) are the same. This is an extension of an important result for meromorphic functions.

2. First and Second Fundamental Theorem in Generic Domains

Let (E,∥·∥) be a complex Banach space with a Schauder basis {ei} and the norm ∥·∥. Thus an E-valued meromorphic mapping f(z) defined in a domain D⊆ℂ can be written as f(z)=(f1(z),f2(z),…,fk(z),…). The elements of E are called vectors and are usually denoted by letters from the alphabet: a,b,c,d,…. The symbol 0 denotes the zero vector of E. We denote vector infinity, complex number infinity, and the norm infinity by ∞̂, ∞, and +∞, respectively. A vector-valued mapping is called holomorphic(meromorphic) if all fj(z) are holomorphic (meromorphic). The jth derivative (j=1,2,…) and the integration of f(z) are defined by f(j)(z)=(f1(j)(z),f2(j)(z),…,fk(j)(z),…),∫zf(ζ)dζ=(∫zf1(ζ)dζ,∫zf2(ζ)dζ,…,∫zfk(ζ)dζ,…),
respectively. We assume that f(0)(z)=f(z). A point z0∈D is called a “pole” or “∞̂-point” of f(z)=(f1(z),…,fk(z),…) if z0 is a pole of at least one of the component functions fk(z)(k=1,2,…). We define ∥f(z0)∥=+∞ when z0 is a pole. A point z0∈D is called “zero” of f(z) if all the component functions fk(z)(k=1,2,…) have zeros at z0.

Remark 2.1.

The integrals are well defined because the set of singularities making ∞̂-∞̂ meaningless is zero measurable.

In order to make our statement clear, we first recall some knowledge of Green functions.

Definition 2.2.

Let D be a domain surrounded by finitely many piecewise analytic curves. Then for any a∈D, there exists a Green function, denoted by GD(z,a), for D with singularity at a∈D which is uniquely determined by the following:

GD(z,a) is harmonic in D∖{a};

in a neighborhood of a, GD(z,a)=-log|z-a|+w(z,a) for some function w(z,a) harmonic in D;

GD(z,a)≡0, on the boundary of D.

By ∂D we denote the boundary of D and n⃗ the inner normal of ∂D with respect to D. Using Green function we can establish the following general Poisson formula for the E-valued meromorphic mapping, which is similar with [5, Lemma 2.2] (see [9, Theorem 2.1], or [10, Theorem 2.1.1]). We do not give the details here.

Theorem 2.3.

Let f:D¯(⊂ℂ)→E be an E-valued meromorphic mapping, which does not reduce to the constant zero element 0∈E. Then
log∥f(z)∥=12π∫∂Dlog∥f(ζ)∥∂GD(ζ,z)∂n⃗ds-∑am∈DGD(am,z)+∑bn∈DGD(bn,z)-12π∫DGD(ζ,z)Δlog∥f(ζ)∥dx∧dy,
where ζ=x+iy, {am} are the zeros of f(z) and {bn} are the poles of f(z) according to their multiplicities.

Remark 2.4.

A simple inspection to the ℂ2-valued case shows that log∥f(z)∥ is not harmonic for a holomorphic (or meromorphic) E-valued function. Therefore we have an additional term in formula (2.2).

Following Theorem 2.3, we introduce some notations.N(D,a,f)=∑bn∈DGD(bn,a),m(D,a,f)=12π∫∂Dlog+∥f(ζ)∥∂GD(ζ,a)∂n⃗ds,V(D,a,f)=12π∫DG(ζ,a)Δlog∥f(ζ)∥dx∧dy,
where a is a point in D and {bn} are the poles of f(z) in D appearing according to their multiplicities, log+x=logmax{x,1}. Define T(D,a,f)=m(D,a,f)+N(D,a,f).

T(D,a,f) is called the Nevanlinna characteristic function of f(z) with the center a∈D.

Next, we give the first (FFT) and the second (SFT) fundamental theorems for f(z).

Theorem 2.5 (FFT).

Let f(z) be an E-valued meromorphic mapping on D¯. Then for a fixed vector b∈E and for any a∈D such that f(a)≠b, one has
T(D,a,1f-b)=T(D,a,f)-V(D,a,f-b)-log∥f(a)-b∥+ɛ(b,D),
where
|ɛ(b,D)|≤{log+∥b∥+log2,b≠0,0,b=0.

Proof.

We can rewrite Theorem 2.3 as follows:
T(D,a,f)=T(D,a,1f)+V(D,a,f)+log∥f(a)∥.
Applying this formula to the function f(z)-b, we can prove the theorem.

Theorem 2.6 (SFT).

Let f(z) be an E-valued meromorphic mapping on D¯, let a[j](j=1,2,…,q)∈E⋃{∞̂} be q distinct vectors, and let f(a)≠a[j]. Then,
(q-2)T(D,a,f)≤∑j=1q[N(D,a,f=a[j])+V(D,a,f-a[j])]-V(D,a,f′)-N1(D,a,f)+S(D,a,f),
where
S(D,a,f)=12π∫∂Dlog+[∑j=1q∥f′(ζ)∥∥f(ζ)-a[j]∥]∂GD(ζ,a)∂n⃗ds-log∥f′(a)∥+qlog+2qδ+∑i=1qlog∥f(a)-a[i]∥,N1(D,a,f)=2N(D,a,f)-N(D,a,f′)+N(D,a,1f′),δ=mini≠j∥a[i]-a[j]∥>0.
Furthermore, one has the following form:
(q-2)T(D,a,f)≤∑j=1q[N¯(D,a,1f-a[j])+V(D,a,f-a[j])]-V(D,a,f′)+S(D,a,f).

Proof.

Set
F(ζ)=∑j=1q1∥f(ζ)-a[j]∥.
According to the property of the logarithm function, we get
12π∫∂Dlog+F(ζ)∂GD(ζ,a)∂n⃗ds≤m(D,a,1f′)+12π∫∂Dlog+[F(ζ)∥f′(ζ)∥]∂GD(ζ,a)∂n⃗ds.
Denote δ=mini≠j∥a[i]-a[j]∥, and fix μ∈{1,2,…,q}. Then we obtain
∥f(z)-a[v]∥≥∥a[μ]-a[v]∥-∥a[μ]-f(z)∥>3δ4
for μ≠v by
∥f(z)-a[μ]∥<δ2q≤δ4.

So either the set of points on ∂D which is determined by (2.14) is empty or any two of some sets for different μ have intersection. In any case, on ∂D we have
12π∫∂Dlog+F(ζ)∂GD(ζ,a)∂n⃗ds≥12π∑μ=1q∫∥f-a[μ]∥<δ/2qlog+F(ζ)∂GD(ζ,a)∂n⃗ds≥12π∑μ=1q∫∥f-a[μ]∥<δ/2qlog+1∥f(ζ)-a[μ]∥∂GD(ζ,a)∂n⃗ds.
Since
12π∫∥f-a[μ]∥<δ/2qlog+1∥f(ζ)-a[μ]∥∂GD(ζ,a)∂n⃗ds=m(D,a,a[μ])-12π∫∥f-a[μ]∥>δ/2qlog+1∥f(ζ)-a[μ]∥∂GD(ζ,a)∂n⃗ds≥m(D,a,a[μ])-log+2qδ,
it follows that
12π∫∂Dlog+F(ζ)∂GD(ζ,a)∂n⃗ds≥∑μ=1qm(D,a,a[μ])-qlog+2qδ.
From (2.12), we get
m(D,a,1f′)≥∑μ=1qm(D,a,a[μ])-qlog+2qδ-12π∫∂Dlog+[F(ζ)∥f′(ζ)∥]∂GD(ζ,a)∂n⃗ds.
Since f(z) is nonconstant vector, f′(z) does not reduce to the constant zero element 0. Applying FFT to f′(z), we can obtain
T(D,a,f′)=N(D,a,1f′)+m(D,a,1f′)+V(D,a,f′)+log∥f′(a)∥.
Using this formula, we have
T(D,a,f')≥∑μ=1qm(D,a,a[μ])-qlog+2qδ-12π∫∂Dlog+[F(ζ)∥f′(ζ)∥]∂GD(ζ,a)∂n⃗ds+N(D,a,1f′)+V(D,a,f′)+log∥f′(a)∥.
On the other hand, we have
T(D,a,f′)=m(D,a,f′)+N(D,a,f′)≤m(D,a,f)+N(D,a,f′)+12π∫∂Dlog∥f′(ζ)∥∥f(ζ)∥∂GD(ζ,a)∂n⃗ds.
The two inequalities above give
∑μ=1qm(D,a,a[μ])+V(D,a,f′)≤m(D,a,f)+N(D,a,f′)-N(D,a,1f′)+12π∫∂Dlog+[F(ζ)∥f′(ζ)∥]∂GD(ζ,a)∂n⃗ds+12π∫∂Dlog∥f′(ζ)∥∥f(ζ)∥∂GD(ζ,a)∂n⃗ds-log∥f′(a)∥+qlog+2qδ.
That is to say,
∑μ=1qm(D,a,a[μ])+V(D,a,f′)≤m(D,a,f)+N(D,a,f′)-N(D,a,1f′)+S(D,a,f).
Adding ∑μ=1qN(D,a,f=a[μ]) to the above inequality and applying FFT, we can formulate
(q-1)T(D,a,f)<N(D,a,f)+∑j=1q[N(D,a,f=a[j])+V(D,a,f-a[j])]-N1(D,a,f)+S(D,a,f),
where
S(D,a,f)=12π∫∂Dlog+[∑j=0q∥f′(ζ)∥∥f(ζ)-a[j]∥]∂GD(ζ,a)∂n⃗ds-log∥f′(a)∥+qlog+2qδ+∑i=0qlog∥f(a)-a[i]∥,a[0]=0.
Since N(D,a,f)≤T(D,a,f), (2.24) can be written as
(q-2)T(D,a,f)<∑j=1q[N(D,a,f=a[j])+V(D,a,f-a[j])]-N1(D,a,f)+S(D,a,f).
If {a[j]} contains ∞̂, (2.26) also holds. Let a[q+1]=∞̂, and substitute q with q+1; then we have (2.26), where a[q]=∞̂, and
S(D,a,f)=12π∫∂Dlog+[∑j=0q-1∥f′(ζ)∥∥f(ζ)-a[j]∥]∂GD(ζ,a)∂n⃗ds-log∥f′(a)∥+qlog+2qδ+∑i=0q-1log∥f(a)-a[i]∥.

Next we establish Hiong King-Lai's inequality for f(z).

Theorem 2.7.

Let f(z) be an E-valued meromorphic mapping on D¯, l∈D, let a,b,c∈E be three finite vectors, and let b≠0,c≠0,b≠c, f(k)(l)≠0,b,c. Then one has
T(D,l,f)<N(D,l,f=a)+N(D,l,f(k)=b)+N(D,l,f(k)=c)+V(D,l,f(k))+V(D,l,f(k)-b)+V(D,l,f(k)-c)-N(D,l,1f(k+1))+S(D,l,f(k)).

Proof.

First, we have
12π∫∂Dlog+∥1f(ζ)-a∥∂GD(ζ,l)∂n⃗ds≤12π∫∂Dlog+∥1f(k)(ζ)∥∂GD(ζ,l)∂n⃗ds+12π∫∂Dlog+∥f(k)(ζ)f(ζ)-a∥∂GD(ζ,l)∂n⃗ds.
Applying FFT to f(z) and f(k)(z), respectively, we have
12π∫∂Dlog+∥1f(ζ)-a∥∂GD(ζ,l)∂n⃗ds=T(D,l,f)-N(D,l,f=a)-V(D,l,f-a)-log∥f(l)-a∥+ɛ(a,D),12π∫∂Dlog+∥1f(k)(ζ)∥∂GD(ζ,l)∂n⃗ds=T(D,l,f(k))-N(D,l,1f(k))-V(D,l,f(k))-log∥f(k)(l)∥.
Thus we have
T(D,l,f)≤T(D,l,f(k))+N(D,l,f=a)+V(D,l,f-a)-N(D,l,1f(k))-V(D,l,f(k))+log∥f(l)-a∥∥f(k)(l)∥-ɛ(a,D).
Applying SFT to f(k) with 0,b,c, we have
T(D,l,f(k))≤N¯(D,l,1f(k))+N¯(D,l,f(k)=b)+N¯(D,l,f(k)=c)-N(D,l,f(k+1))+V(D,l,f(k))+V(D,l,f(k)-b)+V(D,l,f(k)-c)-V(D,l,f(k+1))+S(D,l,f(k)).
Combining (2.31) with (2.32), we have
T(D,l,f)≤N(D,l,f=a)+N¯(D,l,f(k)=b)+N¯(D,l,f(k)=c)-N(D,l,f(k+1))+V(D,l,f-a)+V(D,l,f(k)-b)+V(D,l,f(k)-c)-V(D,l,f(k+1))+S(D,l,f(k)).

3. The Vector-Valued Mapping and Its Derivative

In this section, we will discuss the value distribution theory of f(z) defined in the disk Cr={z:|z|<r}. We will prove Chuang's inequality. According to (2.3), we have the following terms: m(r,f)=12π∫02πlog+∥f(reiθ)∥dθ,N(r,f)=∫0rn(t,f)-n(0,f)tdt+n(0,f)logr,V(r,f)=12π∫Crlog|rζ|Δlog∥f(ζ)∥dx∧dy,ζ=x+iy,T(r,f)=m(r,f)+N(r,f),

where n(r,f) denotes the number of poles of f(z) in {z:|z|<r}. The order and the lower order of an E-valued meromorphic mapping f(z) are defined by λ(f)=lim supr→∞logT(r,f)logr,μ(f)=liminfr→∞logT(r,f)logr.
The following lemma is well known.

Lemma 3.1 (see [<xref ref-type="bibr" rid="B9">11</xref>, Boutroux-Cartan Theorem]).

Let {aj}j=1n be n complex numbers. Then the set of the point z satisfying
∏j=1n|z-aj|<hn
can be contained in several disks, denoted by (γ); the total sum of its radius does not exceed 2eh.

The next lemma is a special case of Theorem 2.3.

Lemma 3.2 (see [<xref ref-type="bibr" rid="B2">5</xref>]).

Let f:Cr→E be an E-valued meromorphic mapping, which does not reduce to the constant zero element 0∈E. Then, for a z∈Cr, one has
log∥f(z)∥=12π∫02πlog∥f(reiθ)∥r2-t2r2-2rtcos(θ-ϕ)+t2dϕ-∑zj(0)∈Crlog|r2-zj(0)¯zr(z-zj(0))|+∑zj(∞̂)∈Crlog|r2-zj(∞̂)¯zr(z-zj(∞̂))|-12π∫Crlog|r2-ξ¯zr(z-ξ)|Δlog∥f(ξ)∥dx∧dy.
Here zj(0)andzj(∞̂) are all the zeros and poles counting their multiplies of f in D.

In order to obtain the relationship between T(r,f) and T(r,f'), we should first establish the following two lemmas.

Lemma 3.3.

Let f:ℂ→E be a nonzero E-valued meromorphic mapping, and f(0)≠∞̂. If R and R' are two positive numbers, and R<R', then there exists a θ0∈[0,2π), such that for any 0≤r≤R one has
log+∥f(reiθ0)∥≤R′+RR′-Rm(R′,f)+n(R′,f)log4+N(R′,f).

Proof.

For z=reiθ,0≤r≤R. By Lemma 3.2 we have
log∥f(z)∥≤12π∫02πlog∥f(R′eiθ)∥R'2-r2R'2-2R′rcos(θ-ϕ)+r2dϕ+∑j=1nlog|R'2-bj¯zR′(z-bj)|,
where {bj}j=1n are the poles of f(z) in |z|≤R′. Then
log+∥f(z)∥≤R′+rR′-rm(R′,f)+∑j=1nlog2R′|z-bj|≤R′+rR′-rm(R′,f)+log(2R′)n∏j=1n|z-bj|.
Writing bj=|bj|eiϕj, we have
|reiθ-|bj|eiϕj|≥|bj∥sin(θ-ϕj)|.
Thus
∏j=1n|z-bj|≥(∏j=1n|bj|)(∏j=1n|sin(θ-ϕj)|).
However,
∫0πlog|∏j=1n|sin(θ-ϕj)∣dθ=n∫0πlog|sinθ|dθ=-nπlog2.
Hence there exists a real number θ0 such that
|∏j=1nsin(θ0-ϕj)|>12n.
Combining (3.7) and (3.9) with (3.11), we have
log+∥f(reiθ0)∥≤R′+RR′-Rm(R′,f)+nlog4+∑j=1nlogR′|bj|≤R′+RR′-Rm(R',f)+nlog4+N(R′,f).

Lemma 3.4.

Let f:ℂ→E be a nonzero E-valued meromorphic mapping, and let R<R'<R′′ be three positive numbers. Then there exists a positive number R≤ρ≤R′, and for |z|=ρ, one has
log+∥f(z)∥≤R′′+R′R′′-R′m(R′′,f)+n(R′′,f)log8eR′′R′-R.

Proof.

Let {bj}j=1n be the poles of f(z) in |z|≤R′′. By Boutroux-Cartan Theorem, we have
∏j=1n|z-bj|≥(R′-R4e)n,
except for some points contained in a pack of disks whose radius does not exceed (R′-R)/2. Then there exists a circle {z:|z|=ρ} such that R≤ρ≤R′ and {|z|=ρ}⋂(γ)=∅. Thus (3.14) holds on {|z|=ρ}. For any z∈{z:|z|=ρ}, we have
log+∥f(reiθ0)∥≤R′′+ρR′′-ρm(R′′,f)+∑j=1nlog|R′′2-bj¯zR′′(z-bj)|≤R′′+R′R′′-R′m(R′′,f)+nlog8eR′′R′-R.

Now we are in the position to establish the following Chuang's inequality.

Theorem 3.5.

Let f:ℂ→E be a nonzero E-valued meromorphic mapping and f(0)≠∞̂. Then for τ>1 and 0<r<R, one has
T(r,f)<CτT(τr,f′)+log+τr+4+log+∥f(0)∥,
where Cτ is a positive constant.

Proof.

Take a σ such that σ3=τ and denote r1=σr,r2=σr1,r3=σr2. Applying Lemma 3.3 to f′(z), we can find a real number θ0 such that 0≤t≤r1, and we have
log+∥f′(teiθ0)∥≤r2+r1r2-r1m(r2,f′)+n(r2,f′)log4+N(r2,f′).
In view of Lemma 3.4, for a fixed ρ∈[r,r1] we have
log+∥f′(z)∥≤r2+r1r2-r1m(r2,f′)+n(r2,f′)log8er2r1-r,
on {z:|z|=ρ}.

From the origin along the segment argz=θ0 to ρeiθ0, and along {z:|z|=ρ} turn a rotation to ρeiθ0. We denote this curve by L, and its length is (2π+1)ρ.

We notice that φ(z)=∥f(z)∥ is continuous on L. As in [5], En is an n-dimensional projective space of E with a basis {ei}i=1n. The projection operator Pn:E→En is a realization of En associated to the basis and Pn(f(z))=(f1(z),f2(z),…,fn(z)). We have Pn(f′(z))=(Pn(f(z)))′=∑i=1nfi′(z)ei and Pn(f(z))=Pn(f(0))+∑i=1n(∫0zfi′(ζ)dζ)ei. Therefore, since En is finite dimensional, there exists K>0 (appearing in the inequality ∥·∥1≤K∥·∥2, where ∥·∥1 and ∥·∥2 are any two norms on En) such that
∥Pn(f(z))∥≤∥Pn(f(0))∥+∥∑i=1n(∫0zfi′(ξ)dξ)ei∥≤∥Pn(f(0))∥+1K(∑i=1n|∫0zfi′(ξ)dξ|2)1/2≤∥Pn(f(0))∥+1K(∑i=1nmaxξϵL|fi′(ξ)|2)1/2(2π+1)ρ≤∥Pn(f(0))∥+K′KMn(2π+1)ρ,
where Mn=maxz∈L∥Pn(f′(z))∥. Thus, we have
∥Pn(f(z))∥≤∥Pn(f(0))∥+Mn(2π+1)ρ+O(1),|z|=ρ.

In virtue of [6–8], every meromorphic mapping f(z) with values in a Banach space E with a Schauder basis and the projections Pn(f) are convergent in its natural topology; that is, they converge uniformly to f in any compact subset W of ℂ∖Pf (Pf being the set of poles the f in ℂ). Thus for n large enough, we have
∥Pn(f(z))∥=∥f(z)∥+O(1),foranyz∈W⊆ℂ∖Pf.
A similar argument to f'(z) implies that for n large enough
∥Pn(f′(z))∥=∥f′(z)∥+O(1),Mn≤M+O(1)foranyz∈W⊆ℂ∖Pf′,
where M=maxz∈L∥f′(z)∥.

Combining (3.20), (3.21), and (3.22) and the fact that the compact set {z:|z|=ρ}⊆L⊆ℂ∖Pf, we get
∥f(z)∥≤∥f(0)∥+M(2π+1)ρ+O(1).
Then
log+∥f(z)∥≤log+∥f(0)∥+log+M+log+ρ+log8eπ+O(1).
In virtue of (3.13) and (3.17), we have
log+M≤r2+r1r2-r1m(r2,f′)+n(r2,f′)log8er2r1-r+N(r2,f′)≤{log(8er2/(r1-r))log(r3/r2)+r2+r1r2-r1}T(r3,f′)=Cτ′T(r3,f′).
Therefore,
m(ρ,f)<Cτ′T(r3,f′)+log+(τr)+4+log+∥f(0)∥.
Thus we have
T(r,f)≤T(ρ,f)<(Cτ′+1)T(r3,f′)+log+(τr)+4+log+∥f(0)∥=CτT(τr,f′)+log+(τr)+4+log+∥f(0)∥.

The following result says that we can also control the T(r,f′) by T(r,f).

Theorem 3.6.

Let f(z)(z∈ℂ) be a nonconstant E-valued meromorphic mapping. Then one has
T(r,f′)≤2T(r,f)+O(logr+log+T(r,f)).

Proof.

One has
T(r,f')=m(r,f′)+N(r,f′)≤m(r,f)+N(r,f′)+12π∫02πlog+∥f′(reiϕ)∥∥f(reiϕ)∥dϕ=m(r,f)+N(r,f)+N¯(r,f)+12π∫02πlog+∥f'(reiϕ)∥∥f(reiϕ)∥dϕ≤2T(r,f)+12π∫02πlog+∥f′(reiϕ)∥∥f(reiϕ)∥dϕ=2T(r,f)+O(logr+log+T(r,f)).
From Theorems 3.5 and 3.6, we have the following.

Corollary 3.7.

For a nonconstant E-valued meromorphic mapping f(z)(z∈ℂ), One has λ(f)=λ(f′), μ(f)=μ(f′).

Acknowledgments

The authors would like to thank the referee for his/her many helpful comments and suggestions on an early version of the manuscript. The work is supported by NSF of China (Grant no. 10871108). The first author is also supported partially by NSF of China (Grant no. 10926049).

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