# Characteristic Functions and Borel Exceptional Values ofE-Valued Meromorphic Functions

Authors: Zhaojun Wu; Zuxing Xuan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The main purpose of this paper is to investigate the characteristic functions and Borel exceptional values ofE-valued meromorphic functions from the ℂR={z:|z|<R},0<R≤+∞ to an infinite-dimensional complex Banach space E with a Schauder basis. Results obtained extend the relative results by Xuan, Wu and Yang, Bhoosnurmath, and Pujari.


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## Body

## 1. Introduction and Preliminaries

In 1980s, Ziegler [1] succeeded in extending the Nevanlinna theory of meromorphic functions to the vector-valued meromorphic functions in finite dimensional spaces. Later, Hu and Yang [2] established the Nevanlinna theory of meromorphic mappings with the range in an infinite-dimensional Hilbert spaces. In 2006, C.-G. Hu and Q. Hu [3] established the Nevanlinna's first and second main theorems of meromorphic mappings with the range in an infinite-dimensional Banach spaces E with a Schauder basis. Recently, Xuan and Wu [4] established the Nevanlinna's first and second main theorems for an E-valued meromorphic mapping from a generic domain D⊆ℂ to an infinite-dimensional Banach spaces E with a Schauder basis.In [4], Xuan and Wu also proved Chuang's inequality (see, e.g., [5]) of E-valued meromorphic mapping f(z) in the whole complex plane, which compares the relationship between T(r,f) and T(r,f'), and also obtained that the order and the lower order of E-valued meromorphic mapping f(z) and those of its derivative f'(z) are the same. In Section 2, we shall prove that Chuang's inequality is valid for E-valued meromorphic mapping f(z) in the unit disc and prove that for any infinite-order E-valued meromorphic function f(z) defined in the unit disc has the same Xiong's proximate order as its derivative f'(z).In [5], Yang obtained much stronger results than those of Gopalakrishna and Bhoosnurmath [6] for the Borel exceptional values of meromorphic functions dealing with multiple values. In Section 3, we shall extend Le Yang's result to E-valued meromorphic functions of finite and infinite orders in
(1.1)ℂR:={z:|z|<R},0<R≤+∞.In the following, we introduce the definitions, notations, and results of [3, 4] which will be used in this paper.Let(E,∥•∥) be an infinite dimension complex Banach space with Schauder basis {ej} and the norm ∥•∥. Thus, an E-valued meromorphic function f(z) defined in ℂR, 0<R≤+∞ can be written as
(1.2)f(z)=(f1(z),f2(z),…,fk(z),…).
Let En be an n-dimensional projective space of E with a basis {ej}1n. The projective operator Pn:E→En is a realization of En associated with basis.The elements ofE are called vectors and are usually denoted by letters from the alphabet: a,b,c,…. 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 mappings is called holomorphic (meromorphic) if all fj(z) are holomorphic (some of fj(z) are meromorphic). The jth derivative j=1,2,… of f(z) is defined by
(1.3)f(j)(z)=(f1(j)(z),f2(j)(z),…,fk(j)(z),…).
A point z0∈ℂr is called a “pole" (or ∞^ point) of
(1.4)f(z)=(f1(z),f2(z),…,fk(z),…)
if z0 is a pole (or ∞ point) of at least one of the component functions fk(z)(k=1,2,…). A point z0∈ℂr is called a “zero" of f(z)=(f1(z),f2(z),…,fk(z),…) if z0 is a zero of all the component functions fk(z)(k=1,2,…). A point z0∈ℂr is called a pole or an ∞^-point of f(z) of multiplicity q∈ℕ+, meaning that in such a point z0 at least one of the meromorphic component functions fj(z) has a pole of this multiplicity in the ordinary sense of function theory. A point z0∈ℂr is called a zero of f(z) of multiplicity q∈ℕ+, meaning that in such a point z0 all component functions fj(z) vanish, each with at least this multiplicity.Letn(r,f) or n(r,∞^) denote the number of poles of f(z) in |z|≤r  and let n(r,a,f) denote the number of a-points of f(z) in |z|≤r, counting with multiplicities. Define the volume function associated with E-valued meromorphic function f(z) by
(1.5)V(r,∞^,f)=V(r,f)=12π∫Cr‍log|rξ|Δlog∥f(ξ)∥dx∧dy,ξ=x+iy,V(r,a,f)=12π∫Cr‍log|rξ|Δlog∥f(ξ)-a∥dx∧dy,ξ=x+iy,
and the counting function of finite or infinite a-points by
(1.6)N(r,f)=n(0,f)logr+∫0r‍n(t,f)-n(0,f)tdt,(1.7)N(r,∞^)=n(0,∞^)logr+∫0r‍n(t,∞^)-n(0,∞^)tdt,(1.8)N(r,a,f)=n(0,a,f)logr+∫0r‍n(t,a,f)-n(0,a,f)tdt,
respectively. Next, we define
(1.9)m(r,f)=m(r,∞^,f)=12π∫02π‍log+∥f(reiθ)∥dθ,m(r,a)=m(r,a,f)=12π∫02π‍log+1∥f(reiθ)-a∥dθ,T(r,f)=m(r,f)+N(r,f).
Let n¯(r,f) or n¯(r,∞^) denote the number of poles of f(z) in |z|≤r, and let n¯(r,a,f) denote the number of a-points of f(z) in |z|≤r, ignoring multiplicities. Similarly, we can define the counting functions N¯(r,f),  N¯(r,∞^), and N¯(r,a,f) of n¯(r,f), n¯(r,∞^), and n¯(r,a,f).Iff(z) is an E-valued meromorphic function in the whole complex plane, then the order and the lower order of f(z) are defined by
(1.10)λ(f)=limsupr→+∞log+T(r,f)logr,μ(f)=liminfr→+∞log+T(r,f)logr.Iff(z) is an E-valued meromorphic function in ℂR,0<R<+∞, then the order and the lower order of f(z) are defined by
(1.11)λ(f)=limsupr→R-logT(r,f)log+(1/(R-r)),μ(f)=liminfr→R-logT(r,f)log+(1/(R-r)).Lemma 1.1.
LetB(x) be a positive and continuous function in [0,+∞) which satisfies limsupx→+∞(logB(x)/logx)=∞. Then there exists a continuously differentiable function ρ(x), which satisfies the following conditions.(i)
ρ(x) is continuous and nondecreasing for x≥x0(x0>0) and tends to +∞ as x→+∞.(ii)
The functionU(x)=xρ(x)(x≥x0) satisfies the following:
(1.12)limx→+∞logU(X)logU(x)=1,X=x+xlogU(x).(iii)
limsupx→+∞(logB(x)/logU(x))=1.Lemma1.1 is due to K. L. Hiong (also Qinglai Xiong) and ρ(x) is called the proximate order of Hiong. A simple proof of the existence of ρ(r) was given by Chuang [7]. Suppose that f(z) is an E-valued meromorphic function of infinite order in the unit disk ℂ1. Let x=1/(1-r) and X=1/(1-R). From (ii) and (iii) in Lemma 1.1, we have
(1.13)limr→1-logU(1/(1-R))logU(1/(1-r))=1,R=rlogU(1/(1-r))+1logU(1/(1-r))+1,limsupx→1-logT(r,f)logU(1/(1-r))=1.
Here, the functions ρ(1/(1-r)) and U(1/(1-r)) are called the proximate order and type function of f(z), respectively.Definition 1.2.
AnE-valued meromorphic function f(z) in ℂR,0<R≤+∞ is of compact projection, if for any given ε>0, ∥Pn(f(z))-f(z)∥<ε has sufficiently larg n in any fixed compact subset D⊂CR.
Throughout this paper, we say thatf(z) is an E-valued meromorphic function meaning that f(z) is of compact projection. C.-G.  Hu and Q. Hu [3] established the following Nevanlinna's first and second main theorems of E-valued meromorphic functions.Theorem 1.3.
Letf(z) be a nonconstant E-valued meromorphic function in ℂR,0<R≤+∞. Then for 0<r<R,a∈E,f(z)≢a,
(1.14)T(r,f)=V(r,a)+N(r,a)+m(r,a)+log+∥cq(a)∥+ε(r,a).
Here, ε(r,a) is a function satisfying that
(1.15)|ε(r,a)|≤log+∥a∥+log2,ε(r,0)≡0,
and cq(a)∈E is the coefficient of the first term in the Laurent series at the point a.Theorem 1.4.
Letf(z) be a nonconstant E-valued meromorphic function in ℂR,0<R≤+∞ and a[k]∈E∪{∞^}(k=1,2,…,q) be q≥3 distinct points. Then for 0<r<R,
(1.16)(q-2)T(r,f)≤∑k=1q‍[V(r,a[k])+N¯(r,a[k])]+S(r,f).IfR=+∞, then(1.17)S(r,f)=O(logT(r,f)+logr)
holds as r→+∞ without exception if f(z) has finite order and otherwise as r→+∞ outside a set J of exceptional intervals of finite measure ∫J‍dr<+∞. If the order of f(z) is infinite and ρ(r) is the proximate order of f(z), then
(1.18)S(r,f)=O(logU(r))
holds as r→+∞ without exception.If0<R<+∞, then
(1.19)S(r,f)=O(logT(r,f)+log1R-r)
holds as r→R without exception if f(z) has finite order and otherwise as r→R outside a set J of exceptional intervals of finite measure ∫J‍d((r/(R-r))<+∞.In all cases, the exceptional setJ is independent of the choice of a[k].


## 2. Characteristic Function ofE-Valued Meromorphic Functions in the Unit Disc ℂ1

In [4], Xuan and Wu proved the following.Theorem A.
Letf(z)(z∈ℂ) be a nonconstant E-valued meromorphic function and f(0)≠∞^. Then for τ>1 and 0<r<R, one has
(2.1)T(r,f)<CτT(τr,f′)+log+τr+4+log+∥f(0)∥,
where Cτ is a positive constant.Theorem B.
Letf(z)(z∈ℂ) be a nonconstant E-valued meromorphic function. Then we have
(2.2)T(r,f′)<2T(r,f)+O(logr+log+T(r,f)).Theorem C.
For a nonconstantE-valued meromorphic function f(z)(z∈ℂ) of order λ(f)<+∞, one has λ(f)=λ(f′),μ(f)=μ(f').In this section, we shall prove that Theorems A, B, and C are valid forE-valued meromorphic function in the unit disc ℂ1.Lemma 2.1.
Letf(z) be an E-valued meromorphic function defined in the unit disc, and f(0)≠∞^. If 0<R<R'<1, then there exists a θ0∈[0,2π), such that for any 0≤r≤R, one has
(2.3)log+∥f(reiθ0)∥≤R′+RR′-Rm(R′,f)+n(R′,f)log4+N(R′,f).Lemma 2.2.
Letf(z) be an E-valued meromorphic function defined in the unit disc, and let 0<R<R'<R′′<1. Then there exists a positive number R≤ρ≤R', such that for |z|=ρ, one has
(2.4)log+∥f(reiθ0)∥≤R′′+R′R′′-R′m(R′′,f)+n(R′′,f)log8eR′′R'-R.Lemmas2.1 and 2.2 are due to Xuan and Wu [4] for the E-valued meromorphic function defined in the whole complex plane. From the proof of Xuan and Wu   [4], we know that Lemmas 2.1 and 2.2 are also valid for the E-valued meromorphic function defined in the unit disc ℂ1.Lemma 2.3.
Letf(z)(z∈ℂ1) be a nonconstant E-valued meromorphic function and f(0)≠∞^. Suppose that h(r)≥1,R=(1+rh(r))/(1+h(r)), then when r sufficiently tends to 1, one has
(2.5)n(r,f)≤6h(r)1-rN(R,f).Proof.
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Letf(z)(z∈ℂR,0<R≤+∞) be a nonconstant E-valued meromorphic function and f(0)≠∞^, and L a curve from the origin along the segment argz=θ0 to ρeiθ0, and along {|z|=ρ<r} turn a rotation to ρeiθ0. Then for any {|z|=r≤ρ}, one has
(2.7)log+∥f(z)∥≤log+M+O(1),
where M=max{∥f′(z)∥,z∈L}.Lemma 2.5 (see [3]).
Letf(z) be a nonconstant E-valued meromorphic function in ℂ1. Then for 0<r<1,
(2.8)12π∫02π‍log+∥f′(reiθ)∥∥f(reiθ)∥dθ<K(logT(r,f)+log11-r),
where K is a sufficiently large constant.We are now in the position to establish the main results of this section.Theorem 2.6.
Letf(z)(z∈ℂ1) be a nonconstant E-valued meromorphic function and f(0)≠∞^. Then for ε>1 and any real function h(x)≥1, when r sufficiently tend to 1, one has
(2.9)T(r,f)<ch1+ε(r)(1-r)1+εT(R,f′),R=1+rh(r)1+h(r).Proof.
DenoteR1=(R+2r)/3,R2=(r+2R)/3, we can get
(2.10)r<R1<R2<R,R1-r=R2-R1=R-R2=R-r3,R=1-3R2h(r)1+3h(r),R2+R1=r+R<2,1-R2=(1-r)(1+3h(r))3(1+h(r))≥1-r2;R-r=1-r1+h(r)≥1-r2h(r).
Applying Lemma 2.1 to f'(z) and combining Lemma 2.3, we can find a real number θ0∈[0,2π) such that for any 0≤t≤R1, one has(2.11)log+∥f′(teiθ0)∥≤R2+R1R2-R1m(R2,f′)+n(R2,f′)log4+N(R2,f′)≤(6R-r+6h(r)1-R2log4+1)T(R2,f′)≤(6+6h(r)1-r+12h(r)1-rlog4+1-r1-r)T(R,f′)≤6+6h(r)+24h(r)+1-r1-rT(R,f′)≤40h(r)1-rT(R,f′).
In view of  Lemma 2.2, there is a ρ∈[r,R1] such that for any z∈{|z|=ρ}, one has
(2.12)log+∥f′(z)∥≤R2+R1R2-R1m(R2,f′)+n(R2,f′)log8eR2R1-R≤(6R-r+6h(r)1-R2log48eh(r)1-r)T(R2,f′)≤(6+6h(r)1-r+12h(r)1-rlog144h(r)1-r)T(R,f′)≤(12h(r)1-r(9+logh(r)1-r))T(R,f′)≤(12h(r)1-r(9+(h(r)1-r)ε))T(R,f′)≤120(h(r)1-r)1+εT(R,f′).
From the origin along the segment argz=θ0 to ρeiθ0 and along {|z|=ρ}, turn a rotation to ρeiθ0. We denote this curve by L. In virtue of Lemma 2.4, we have
(2.13)log+∥f(z)∥≤log+M+O(1)
holds for any {|z|=r≤ρ}, where M=max{∥f'(z)∥,z∈L}. In virtue of (2.11), (2.12), and (2.13), we have
(2.14)m(r,f)≤m(ρ,f)≤m(ρ,f′)≤12π∫02π‍log+Mdθ≤121(h(r)1-r)1+εT(R,f′).
Hence,
(2.15)T(r,f)=m(r,f)+N(r,f)≤m(r,f)+2N(r,f′)≤123(h(r)1-r)1+εT(R,f′).Theorem 2.7.
Letf(z)(z∈ℂ1) be a nonconstant E-valued meromorphic function and f(0)≠0,∞^. Then for any 0<r<R<1, one has
(2.16)T(r,f′)<2T(r,f)+O(log+11-r+log+T(r,f)).Proof.
By Lemma2.5, we have
(2.17)T(r,f′)=m(r,f′)+N(r,f′)≤m(r,f)+m(r,f′f)+2N(r,f)≤2T(r,f)+m(r,f'f)≤2T(r,f)+O(log+11-r+log+T(r,f)).Theorem 2.8.
For a nonconstantE-valued meromorphic function f(z)(z∈ℂ1) of order λ(f)<+∞, one has λ(f)=λ(f'),μ(f)=μ(f').Theorem2.8 only discussed the E-valued meromorphic function of finite order. In fact, for any E-valued meromorphic function of infinite order, we have the following.Theorem 2.9.
Iff(z)(z∈ℂ1) is a nonconstant E-valued meromorphic function of order λ(f)=+∞, then the proximate orders of f(z) and f'(z) are the same.Proof.
Leth(r)=logU(1/(1-r)), in view of Theorems 2.6 and 2.7, we can easily derive Theorem 2.9.


## 3.E-Valued Borel Exceptional Values of Meromorphic Functions in ℂℝ

Some definitions in this section can be found in [8].Definition 3.1.
Letf(z)(z∈ℂR,0<R≤+∞) be an E-valued meromorphic function and a∈E∪{∞^}, if k is a positive integer, let n¯k(r,f) or n¯k(r,∞^) denote the number of distinct poles of f(z) of order ≤k in |z|≤r, and let n¯k(r,a) denote the number of distinct a-points of f(z) of order ≤k in |z|≤r. Similarly, we can define the counting functions N¯k(r,f), N¯k(r,∞^), and N¯k(r,a) of n¯k(r,f), n¯k(r,∞^), and n¯k(r,a).Definition 3.2.
Letf(z)(z∈ℂR,0<R≤+∞) be an E-valued meromorphic function and a∈E∪{∞^}. If R=+∞, we define
(3.1)ρ¯k(a,f)=limsupr→+∞log+[V(a,f)+N¯k(r,a)]logr,ρ¯(a,f)=limsupr→+∞log+[V(a,f)+N¯(r,a)]logr,ρ(a,f)=limsupr→+∞log+[V(a,f)+N(r,a)]logr.
If R<+∞, we define
(3.2)ρ¯k(a,f)=limsupr→R-log+[V(a,f)+N¯k(r,a)]log(1/(R-r)),ρ¯(a,f)=limsupr→R-log+[V(a,f)+N¯(r,a)]log(1/(R-r)),ρ(a,f)=limsupr→R-log+[V(a,f)+N(r,a)]log(1/(R-r)).Definition 3.3.
Letf(z)(z∈ℂR,0<R≤+∞) be an E-valued meromorphic function and a∈E∪{∞^} and k is a positive integer, we say that a is an (i)
E-valued evB (exceptional value in the sense of Borel) for f for distinct zeros of order ≤k if ρ¯k(a,f)<λ(f);(ii)
E-valued evB for f for distinct zeros if ρ¯(a,f)<λ(f);(iii)
E-valued evB for f (for the whole aggregate of zeros) if ρ(a,f)<λ(f).
In [5], Yang proved the following result.Theorem D.
Letf(z)(z∈ℂR,R=+∞) be a meromorphic function with finite order λ>0 and kj(j=1,2,…,q) be q positive integers. a is called a pseudo-Borel exceptional value of f(z) of order k if
(3.3)limsupr→+∞log+n¯k(r,a)logr<λ(f).
If f(z) has q distinct pseudo-Borel exceptional values aj of order kj(j=1,2,…,q), then
(3.4)∑j=1q‍(1-1kj+1)≤2.It is natural to consider whether there exists a similar result, if meromorphic functionf is replaced by E-valued meromorphic function f. In this section, we extend the above theorem to E-valued meromorphic function in ℂR,0<R≤+∞.Theorem 3.4.
Letf(z)(z∈ℂR,0<R≤+∞) be an E-valued meromorphic function with finite order λ>0, a[j](j=1,2,…,q) any system of distinct elements in E∪{∞^}, and kj(j=1,2,…,q) any system such that kj is a positive integer or +∞. If a[j] is an E-valued evB for f for distinct zeros of order ≤kj(j=1,2,…,q), then
(3.5)∑j=1q‍(1-1kj+1)≤2.Proof.
By Theorem1.4, we have
(3.6)(q-2)T(r,f)≤∑j=1q‍[V(r,a[j])+N¯(r,a[j])]+S(r,f)
holds for 0<r<R. For any j=1,2,…,q, we have
(3.7)N¯(r,a[j])≤1kj+1{kjN¯kj(r,a[j])+N(r,a[j])},N(r,a[j])≤T(r,f)-V(r,a[j])+O(1).
Using (3.7) and (7) in (3.6), we get
(3.8)(q-2)T(r,f)≤∑j=1q‍(V(r,a[j])+1kj+1{kjN¯kj(r,a[j])+N(r,a[j])})+S(r,f)=∑j=1q‍(V(r,a[j])+kjkj+1N¯kj(r,a[j])+1kj+1N(r,a[j]))+S(r,f)≤∑j=1q‍kjkj+1(V(r,a[j])+N¯kj(r,a[j]))+∑j=1q‍1kj+1T(r,f)+S(r,f).
Therefore, we have
(3.9)[∑j=1q‍(1-1kj+1)-2]T(r,f)≤∑j=1q‍kjkj+1(V(r,a[j])+N¯kj(r,a[j]))+S(r,f).
By hypothesis, we have
(3.10)ρ¯kj(a[j],f)<λ,j=1,2,…,q.
If R=+∞, then there is a positive number ρ<λ, such that for j=1,2,…,q, we can get
(3.11)V(r,a[j])+N¯kj(r,a[j]≤rρ).
Using (3.11) to (3.9), we have
(3.12)[∑j=1q‍(1-1kj+1)-2]T(r,f)≤∑j=1q‍kjkj+1rρ+S(r,f).
If∑j=1q(1-(1/(kj+1)))>2, then by Theorem 1.4 and (3.12), we can get a contradiction λ≤ρ. So
(3.13)∑j=1q‍(1-1kj+1)≤2.
If R<+∞, then there is a positive number ρ<λ, such that for j=1,2,…,q, we can get
(3.14)V(r,a[j])+N¯kj(r,a[j])≤(1R-r)ρ.
Using (3.14) to (3.9), we have
(3.15)[∑j=1q‍(1-1kj+1)-2]T(r,f)≤∑j=1q‍kjkj+1(1R-r)ρ+S(r,f).
If∑j=1q(1-(1/(kj+1)))>2, then by Theorem 1.4 and (3.15), we can get a contradiction λ≤ρ. So
(3.16)∑j=1q‍(1-1kj+1)≤2.From the proof of Theorem3.4, we can get the following.Corollary 3.5.
Letf(z)(z∈ℂR,0<R≤+∞) be a nonconstant E-valued meromorphic function. Then for any system a[j](j=1,2,…t) of distinct elements in E∪{∞^} and any system kj(j=1,2,…,t) such that kj is a positive integer or +∞, we have the following:  (1)
if all ofa[j](j=1,2,…,q) in E, then
(3.17)(q-∑j=1q1kj+1-2)T(r,f)≤∑j=1q‍kjkj+1(V(r,a[j],f)+N¯kj(r,a[j],f))+S(r,f),(2)
if one ofa[j](j=1,2,…,q) is ∞^, say a[q]=∞^. Then,
(3.18)(q-∑j=1q1kj+1-2)T(r,f)≤∑j=1q-1‍kjkj+1(V(r,a[j],f)+N¯kj(r,a[j],f))+kqkq+1N¯kq(r,f)+S(r,f).Remark 3.6.
IfR=+∞, let q=r+t+s and kj≡k(j=1,2,…,r), kj≡l(j=r+1,…,r+t) and kj≡m(j=r+t+1,…,r+t+s) in Theorem 3.4. We can get the following result by Bhoosnurmath and Pujari [8].Theorem E.
Letf(z)(z∈ℂR,0<R≤+∞) be an E-valued meromorphic function of order λ(f),0<λ(f)≤+∞. If there exist distinct elements
(3.19)a[1],a[2],…,a[r];b[1],b[2],…,b[t];c[1],c[2],…,c[s]
in E∪{∞^} such that a[1],a[2],…,a[r] are E-valued evB for f for distinct zeros of order ≤k, b[1],b[2],…,b[t] are E-valued evB for f for distinct zeros of order ≤l, c[1],c[2],…,c[s] are E-valued evB for f for distinct zeros of order ≤m, where k, l, and m are positive integers, then
(3.20)rkk+1+tll+1+smm+1≤2.Bhoosnurmath and Pujari [8] pointed out that Theorem E is valid for 0≤λ(f)≤+∞. In fact, Definition 3.3 is not well in the case of λ(f)=0. In the case of λ(f)=+∞,a is an E-valued evB for f if and only if ρ¯k(a,f) is finite. When ρ¯k(a,f) is infinite, we shall give the following definitions.Definition 3.7.
Letf(z)(z∈ℂ) be an E-valued meromorphic function of infinite order and ρ(r) is a proximate order of f and a∈E∪{∞^}. We say that a is an (i)
E-valued evB (exceptional value in the sense of Borel) for f for distinct zeros of order ≤k if
(3.21)limsupr→+∞log+[V(a,f)+N¯k(r,a)]logU(r)<1;(ii)
E-valued evB for f for distinct zeros if
(3.22)limsupr→+∞log+[V(a,f)+N¯(r,a)]logU(r)<1;(iii)
E-valued evB for f (for the whole aggregate of zeros) if
(3.23)limsupr→+∞log+[V(a,f)+N(r,a)]logU(r)<1.Theorem 3.8.
Letf(z)(z∈ℂ) be an E-valued meromorphic function of infinite order and ρ(r) is a proximate order of f, a[j](j=1,2,…,q) any system of distinct elements in E∪{∞^}, and kj(j=1,2,…,q) any system such that kj is a positive integer or +∞. If a[j] is an E-valued evB for f for distinct zeros of order ≤kj(j=1,2…,q), then
(3.24)∑j=1q‍(1-1kj+1)≤2.Proof.
By Corollary3.5, we have
(3.25)(q-∑j=1q‍1kj+1-2)T(r,f)≤∑j=1q‍kjkj+1(V(r,a[j])+N¯kj(r,a[j]))+S(r,f).
By hypothesis, there exists a positive number η<1 such that
(3.26)V(r,a[j])+N¯kj(r,a[j])<Uη(r),j=1,2,…,q.
Using (3.25) to (3.26), we have
(3.27)[∑j=1q‍(1-1kj+1)-2]T(r,f)≤∑j=1q‍kjkj+1Uη(r)+S(r,f).
If ∑j=1q(1-(1/(kj+1)))>2, then by Theorem 1.4 and (3.27), we can get a contradiction. So
(3.28)∑j=1q‍(1-1kj+1)≤2.



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*Source: 260506-2012-11-06.xml*