The main purpose of this paper is to investigate the characteristic functions and Borel exceptional values of E-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.
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 of E 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.
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, 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).
If f(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.
If f(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.
Let B(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.
ρ(x) is continuous and nondecreasing for x≥x0(x0>0) and tends to +∞ as x→+∞.
The function U(x)=xρ(x)(x≥x0) satisfies the following:
(1.12)limx→+∞logU(X)logU(x)=1,X=x+xlogU(x).
limsupx→+∞(logB(x)/logU(x))=1.
Lemma 1.1 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.
An E-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 that f(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.
Let f(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.
Let f(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).
If R=+∞, 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.
If 0<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 set J is independent of the choice of a[k].
2. Characteristic Function of E-Valued Meromorphic Functions in the Unit Disc ℂ1
In [4], Xuan and Wu proved the following.
Theorem A.
Let f(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.
Let f(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 nonconstant E-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 for E-valued meromorphic function in the unit disc ℂ1.
Lemma 2.1.
Let f(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.
Let f(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.
Lemmas 2.1 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.
Let f(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).
Let f(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]).
Let f(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.
Let f(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.
Denote R1=(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.
Let f(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 Lemma 2.5, 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 nonconstant E-valued meromorphic function f(z)(z∈ℂ1) of order λ(f)<+∞, one has λ(f)=λ(f'),μ(f)=μ(f').
Theorem 2.8 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.
If f(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.
Let h(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.
Let f(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.
Let f(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.
Let f(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
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);
E-valued evB for f for distinct zeros if ρ¯(a,f)<λ(f);
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.
Let f(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 function f 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.
Let f(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 Theorem 1.4, 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 Theorem 3.4, we can get the following.
Corollary 3.5.
Let f(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:
if all of a[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),
if one of a[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.
If R=+∞, 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.
Let f(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.
Let f(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
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;
E-valued evB for f for distinct zeros if
(3.22)limsupr→+∞log+[V(a,f)+N¯(r,a)]logU(r)<1;
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.
Let f(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 Corollary 3.5, 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.
Acknowledgments
The first author is supported in part by the Science Foundation of Educational Commission of Hubei Province (Grant nos. T201009, Q20112807). The second author is supported in part by the Science and Technology Research Program of Beijing Municipal Commission of Education (KM201211417011).
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