# An Inequality of Meromorphic Vector Functions and Its Application

Authors: Wu Zhaojun; Chen Yuxian
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Firstly, an inequality for vector-valued meromorphic functions is established which extend a corresponding inequality of Milloux for meromorphic scalar-valued function (1946).  As an application, the relationship between the characteristic function of a vector-valued meromorphic functionf and its derivative f′ is studied, results are obtained to extend some related results for meromorphic scalar-valued function of Weitsman  (1969)  and Singh and Gopalakrishna  (1971).


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## Body

## 1. Introduction of Vector-Valued Meromorphic Function

In 1980s, Ziegler [1] established Nevanlinna’s theory for the vector-valued meromorphic function in finite dimensional spaces. After Ziegler some works related to vector-valued meromorphic function were done in 1990s [2–4]. In this section, we shall introduce the following fundamental notations and results of vector-valued Nevanlinna theory which were quoted from Ziegler [1].We denote byℂn the usual n dimensional complex Euclidean space with the coordinates w=(w1,w2,…,wn), the Hermitian scalar product(1.1)〈v,w〉=v1w¯1+v2w¯2+⋯+vnw¯n,(v,w∈Cn),
and the distance(1.2)‖v-w‖=+〈v-w,v-w〉1/2.Let(1.3)w1=f1(z),w2=f2(z),…,wn=fn(z)
be n≥1 complex valued functions of the complex variable z, which are meromorphic and not all constant in the Gaussian plane ℂ1=ℂ, or in a finite disc(1.4)CR={|z|<R}⊂C,0<R<+∞.
Thus in ℂR,0<R≤+∞ (we put ℂ+∞=ℂ), a vector-valued meromorphic function(1.5)f(z)=(f1(z),f2(z),…,fn(z))
is given, which does not reduce to the constant zero vector 0=(0,0,…,0). The jth derivative j=1,2,… of f(z) are defined by(1.6)f(j)(z)=(f1(j)(z),f2(j)(z),…,fn(j)(z)).For such a function, the notations “zero,” “pole,” and “multiplicity” are defined as in the scalar casen=1 of only one meromorphic function f1(z). More explicitly, in the punctured vicinity of each point z0∈ℂR, the vector function w=f(z) can developed into a Laurent series(1.7)f(z)=ck0(z-z0)k0+ck0+1(z-z0)k0+1+⋯,
where the coefficients are vectors(1.8)ck=(ck1,ck2,…,ckn)∈Cn,ck0≠(0,0,…,0).In order to introduce the Nevanlinna theory of vector-valued meromorphic function, we will denote by "∞" the ideal element of the Aleksandrov one-point compactification of ℂn (the two real infinities will be denoted by +∞ and -∞, resp.). Now, if k0≤0 in the above Laurent expansion, then z0 will be called a pole or an ∞-point of f(z) of multiplicity -k0; 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, so that in z0 itself f(z) is not defined. If k0>0 in Laurent expansion, then z0 is called a zero of f(z) of multiplicity k0; 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 n(r,a) denote the number of a-points of f(z) in |z|≤r, counting with multiplicities. Define the volume function associated with vector-valued meromorphic function f(z),(1.9)V(r,∞)=V(r,f)=12π∫Crlog⁡|rξ|Δlog⁡‖f(ξ)‖dx∧dy,ξ=x+iyV(r,a)=V(r,1f-a)=12π∫Crlog⁡|rξ|Δlog⁡‖f(ξ)-a‖dx∧dy,ξ=x+iy
and the counting function of finite or infinite a-points by(1.10)N(r,f)=n(0,f)log⁡r+∫0rn(t,f)-n(0,f)tdt,N(r,∞)=n(0,∞)log⁡r+∫0rn(t,∞)-n(0,∞)tdt,N(r,a)=n(0,a)log⁡r+∫0rn(t,a)-n(0,a)tdt,
respectively. Next, we define(1.11)m(r,∞)=m(r,f)=12π∫02πlog⁡+‖f(reiθ)‖dθ,m(r,a)=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 n¯(r,a) denote the number of a-points of f(z) in |z|≤r, ignoring multiplicities. Define the counting function of finite or infinite a-points by(1.12)N¯(r,f)=n¯(0,f)log⁡r+∫0rn¯(t,f)-n¯(0,f)tdt,N¯(r,∞)=n¯(0,∞)log⁡r+∫0rn¯(t,∞)-n¯(0,∞)tdt,N¯(r,a)=n¯(0,a)log⁡r+∫0rn¯(t,a)-n¯(0,a)tdt,
respectively.Iff(z) is a vector-valued meromorphic function in the whole complex plane, then the order and the lower order of f(z) are defined by(1.13)λ(f)=limsup⁡r→∞log⁡T(r,f)log⁡r,μ(f)=liminf⁡r→∞log⁡T(r,f)log⁡r.We call the vector-valued meromorphic functionf admissible if(1.14)limsup⁡r→+∞T(r,f)log⁡r=+∞.Definition 1.1.
For a meromorphic functionf(z) (vector-valued or scalar-valued), we denote by S(r,f) any quantity such that
(1.15)S(r,f)=o(T(r,f)),r⟶+∞
without restriction if f(z) is of finite order and otherwise except possibly for a set of values of r of finite linear measure.Definition1.1 quoted from [2]. In [1], Ziegler established the following first main theorem, logarithmic derivative lemma, and deficient values theorem for meromorphic vector function.Theorem A.
Letf(z)=(f1(z),f2(z),…,fn(z)) be a meromorphic vector function in ℂR. Then for 0<r<R≤+∞,a∈ℂn, f(z)≢a, then
(1.16)T(r,f)=V(r,a)+N(r,a)+m(r,a)+O(1).Theorem B.
Letf(z)=(f1(z),f2(z),…,fn(z)) be a nonconstant meromorphic vector function in ℂ. Then
(1.17)12π∫02πlog⁡+‖f′(reiθ)‖‖f(reiθ)-a‖dθ=S(r,f),a∈Cn.By the second main theorem, Ziegler [1] studies the following deficiency theorem for meromorphic vector function. For any vector a∈ℂn, we define the number δ(a)=δ(a,f) by putting(1.18)δ(a)=δ(a,f)=liminf⁡r→+∞m(r,a)T(r,f)=1-limsup⁡r→+∞V(r,a)+N(r,a)T(r,f),δ(∞)=δ(∞,f)=liminf⁡r→+∞m(r,f)T(r,f)=1-limsup⁡r→+∞N(r,f)T(r,f),
and Θ(a)=Θ(a,f) by putting(1.19)Θ(a)=Θ(a,f)=1-limsup⁡r→+∞V(r,a)+N¯(r,a)T(r,f),Θ(∞)=Θ(∞,f)=1-limsup⁡r→+∞N¯(r,f)T(r,f),Theorem C.
Letf(z)=(f1(z),f2(z),…,fn(z)) be an admissible meromorphic vector function in ℂ. Then the set {a∈ℂn∪{∞},δ(a)>0} is at most countable and summing over all such points we have(1.20)∑aδ(a)≤∑aΘ(a)≤2.


## 2. A Fundamental Inequality of Meromorphic Vector Function

For meromorphic scalar-valued functionf(z), Milloux [5] has proved the following theorem.Theorem D.
Iff(z) is a nonconstant meromorphic scalar-valued function in Gaussian complex plane ℂ and if ai,i=1,2,…,q, are distinct elements of ℂ (where q is any positive integer), then
(2.1)qT(r,f)≤T(r,f′)+∑i=1qN¯(r,ai)+S(r,f).For some alternative proofs of Theorem D, see [6] or [7]. It is natural to consider whether there exists a similar results if meromorphic scalar-valued function f(z) is replaced by meromorphic vector-valued function f(z). In this section, the main contribution is to extend Theorem D to vector-valued meromorphic function by referring the method of [1, 7].Theorem 2.1.
Letf(z)=(f1(z),f2(z),…,fn(z)) be an admissible meromorphic vector function in ℂ and if a[j],j=1,2,…,q, are distinct elements of ℂn (where q is any positive integer), then
(2.2)qT(r,f)≤T(r,f′)+∑j=1q(N¯(r,a[j])+V(r,a[j]))+S(r,f).Proof.
Put(2.3)F(z)=∑j=1q1‖f(z)-a[j]‖.
We can get
(2.4)12π∫02πlog⁡+F(reiθ)dθ≤m(r,0,f′)+12π∫02πlog⁡+{F(reiθ)‖f′(reiθ)‖}dθ.
Put
(2.5)δ=min⁡i≠j‖a[i]-a[j]‖.
Let for the moment μ∈{1,2,…,q} be fixed. Then we get in every point where
(2.6)‖f(z)-a[μ]‖<δ2q≤δ4,
the inequality
(2.7)‖f(z)-a[ν]‖≥‖a[μ]-a[ν]‖-‖f(z)-a[μ]‖≥3δ4,
for μ≠ν. Therefore, the set of points on ∂ℂr which is determined by (2.6) is either empty or any two such sets for different μ have empty intersection. In any case,
(2.8)12π∫02πlog⁡+F(reiθ)dθ≥12π∑μ=1q∫‖f(z)-a[μ]‖<δ/2q,|z|=rlog⁡+F(reiθ)dθ≥12π∑μ=1q∫‖f(z)-a[μ]‖<δ/2q,|z|=rlog⁡+1‖f(reiθ)-a[μ]‖dθ.
Because of
(2.9)12π∫‖f(z)-a[μ]‖<δ/2q,|z|=rlog⁡+1‖f(reiθ)-a[μ]‖dθ=m(r,a[μ])-12π∫‖f(z)-a[μ]‖≥δ/2q,|z|=rlog⁡+1‖f(reiθ)-a[μ]‖dθ≥m(r,a[μ])-log⁡+2qδ,
it follows that
(2.10)12π∫02πlog⁡+F(reiθ)dθ≥∑μ=1qm(r,a[μ])-log⁡+2qδ,
so that by (2.4)
(2.11)∑μ=1qm(r,a[μ])≤m(r,0,f′)+12π∫02πlog⁡+{F(reiθ)‖f′(reiθ)‖}dθ+log⁡+2qδ.
Thus by Theorem B, we have
(2.12)∑μ=1qm(r,a[μ])≤m(r,0,f′)+S(r,f).
It follows from Theorem A that
(2.13)m(r,0,f′)+N(r,0,f′)+V(r,0,f′)=T(r,f′)+O(1).
Thus from (2.12) and (2.13), we deduce
(2.14)∑μ=1qm(r,a[μ])≤T(r,f′)-N(r,0,f′)+S(r,f).
Adding ∑μ=1qN(r,a[μ]) to both sides,
(2.15)∑μ=1qT(r,1f-a[μ])≤T(r,f′)+∑μ=1qN(r,a[μ])-N(r,0,f′)+S(r,f)=T(r,f′)+∑μ=1qN¯(r,a[μ])-N0(r,0,f′)+S(r,f),
where N0(r,0,f′) is formed with the zeros of f′ which are not zeros of any of f-a[μ],(i=1,2,…,q). Since N0(r,0,f′)≥0, we have
(2.16)∑μ=1qT(r,1f-a[μ])≤T(r,f′)+∑μ=1qN¯(r,a[μ])+S(r,f).
Since
(2.17)T(r,1f-a[μ])+V(r,a[μ])=T(r,f)+O(1),
it follows that
(2.18)qT(r,f)≤T(r,f′)+∑j=1q(N¯(r,a[j])+V(r,a[j]))+S(r,f).


## 3. Characteristic Function of Derivative of Meromorphic Vector Function

Letf(z) be a meromorphic scalar-valued function in ℂ.  The characteristic function of derivative of f(z) with ∑aδ(a)=2 has been studied by Edrei [8], Shan and Singh [9], Singh and Gopalakrishna [7], Singh and Kulkarni [10] and Weitsman [11]. For example, Edrei [8] and Weitsman [11] have proved the following theorem.Theorem E.
Letf(z) be a transcendental meromorphic scalar-valued function of finite order and assume ∑a∈ℂδ(a)=η≥1 and δ(∞)=2-η. Then
(3.1)T(r,f′)~ηT(r,f),r⟶+∞.If∑aδ(a)=2 is replaced by ∑aΘ(a)=2, Singh and Gopalakrishna [7] and Singh and Kulkarni [10] have proved the following theorem.Theorem F.
Letf(z) be a transcendental meromorphic scalar-valued function of finite order and assume ∑aΘ(a)=2. Then
(3.2)lim⁡r→+∞T(r,f′)T(r,f)=2-Θ(∞),lim⁡r→+∞N¯(r,a)T(r,f)=1-Θ(a)
for every a∈ℂ∪{∞}.It is natural to consider whether there exists a similar results if meromorphic scalar-valued functionf(z) is replaced by meromorphic vector-valued function f(z). In this section, the main purpose is to extend the above theorems to vector-valued meromorphic function by referring the method of [1, 7].Theorem 3.1.
Letf(z)=(f1(z),f2(z),…,fn(z)) be an admissible meromorphic vector function of finite order in ℂ and assume ∑aΘ(a)=2. Then
(3.3)lim⁡r→+∞T(r,f′)T(r,f)=2-Θ(∞),lim⁡r→+∞N¯(r,f)T(r,f)=1-Θ(∞),lim⁡r→+∞N¯(r,a)+V(r,a)T(r,f)=1-Θ(a)
for every a∈ℂn.Proof.
Now, basic estimates in vector-valued Nevanlinna theory [1] or [4] yields
(3.4)T(r,f′)=m(r,f′)+N(r,f′)=m(r,ff′f)+N(r,f′)≤m(r,f′f)+m(r,f)+N(r,f)+N¯(r,f)≤T(r,f)+N¯(r,f)+m(r,f′f).
By Theorem B and the above inequality, we have
(3.5)limsup⁡r→+∞T(r,f′)T(r,f)≤2-Θ(∞).
Let{a[j]} be a sequence of distinct vector in ℂn containing all the vector of δ(a[j])>0. From Theorem 2.1, for any positive integer q, we have
(3.6)qT(r,f)≤T(r,f′)+∑j=1q(N¯(r,a[j])+V(r,a[j]))+S(r,f).
Hence
(3.7)q≤liminf⁡r→+∞T(r,f′)T(r,f)+∑j=1qlimsup⁡r→+∞N¯(r,a[j])+V(r,a[j])T(r,f)+limsup⁡r→+∞S(r,f)T(r,f)=liminf⁡r→+∞T(r,f′)T(r,f)+∑j=1q{1-Θ(a[j])}+limsup⁡r→+∞S(r,f)T(r,f).
Thus
(3.8)liminf⁡r→+∞T(r,f′)T(r,f)≥∑j=1qΘ(a[j]).
Since q was arbitrary, we have
(3.9)2-Θ(∞)=∑a∈CnΘ(a)≤liminf⁡r→+∞T(r,f′)T(r,f).
This and (3.5) yield
(3.10)lim⁡r→+∞T(r,f′)T(r,f)=2-Θ(∞).
Leta∈ℂn∪{∞} and {a[i]}i=1+∞ an infinite sequence of distinct elements of ℂn∪{∞} which includes every b∈ℂn∪{∞} satisfying b≠a and Θ(b)>0. Then
(3.11)∑i=1+∞Θ(a[i])=∑b∈Cn∪{∞},b≠aΘ(b)=2-Θ(a).
Letq be any integer ≥3. From Generalized Second Main Theorem (see [1], Page 126), we have
(3.12)(q-2)T(r,f)=∑i=1q-1(N¯(r,a[i])+V(r,a[i]))+N¯(r,f)+S(r,f).
Hence
(3.13)q-2≤∑i=1q-1{1-Θ(a[i])}+liminf⁡r→+∞N¯(r,f)T(r,f).
Thus
(3.14)∑i=1q-1Θ(a[i])-1≤liminf⁡r→+∞N¯(r,f)T(r,f).
Since this holds for all q≥3, letting q→+∞ and combining (3.11), we get
(3.15)1-Θ(∞)=∑i=1+∞Θ(a[i])-1≤liminf⁡r→+∞N¯(r,f)T(r,f)≤limsup⁡r→+∞N¯(r,f)T(r,f)=1-Θ(∞).
So
(3.16)lim⁡r→+∞N¯(r,f)T(r,f)=1-Θ(∞).
For everya∈ℂn, Let q be any integer ≥3. From Generalized Second Main Theorem (see [1], Page 126), we have
(3.17)(q-2)T(r,f)=∑i=1q-2(N¯(r,a[i])+V(r,a[i]))+(N¯(r,a)+V(r,a))+N¯(r,f)+S(r,f).
Hence
(3.18)q-2≤∑i=1q-2{1-Θ(a[i])}+(1-Θ(∞))+liminf⁡r→+∞N¯(r,a)+V(r,a)T(r,f).
Thus
(3.19)∑i=1q-2Θ(a[i])+Θ(∞)-1≤liminf⁡r→+∞N¯(r,a)+V(r,a)T(r,f).
Since this holds for all q≥3, letting q→+∞ and combining (3.11), we get
(3.20)1-Θ(a)=∑i=1+∞Θ(a[i])-1≤liminf⁡r→+∞N¯(r,a)+V(r,a)T(r,f)≤limsup⁡r→+∞N¯(r,a)+V(r,a)T(r,f)=1-Θ(a).
So
(3.21)lim⁡r→+∞N¯(r,a)+V(r,a)T(r,f)=1-Θ(a).From Theorem3.1, we have the following corollaryCorollary 3.2.
Letf(z)=(f1(z),f2(z),…,fn(z)) be an admissible meromorphic vector function of finite order in ℂ and assume ∑a∈ℂnΘ(a)=2. Then
(3.22)T(r,f′)~2T(r,f),r⟶+∞.Corollary 3.3.
Letf(z)=(f1(z),f2(z),…,fn(z)) be an admissible meromorphic vector function of finite order in ℂ and assume ∑a∈ℂnΘ(a)=η≥1 and δ(∞)=2-η. Then
(3.23)T(r,f′)~ηT(r,f),r⟶+∞.Corollary 3.4.
Letf(z)=(f1(z),f2(z),…,fn(z)) be an admissible meromorphic vector function of finite order in ℂ and assume ∑aδ(a)=2. Then
(3.24)lim⁡r→+∞T(r,f′)T(r,f)=2-δ(∞),lim⁡r→+∞N(r,f)T(r,f)=1-δ(∞),lim⁡r→+∞N(r,a)+V(r,a)T(r,f)=1-δ(a)
for every a∈ℂn.Proof.
Sinceδ(a)≤Θ(a) for every a∈ℂn∪{∞} and Theorem C, it follows that, if ∑aδ(a)=2, then ∑aΘ(a)=2 and δ(a)=Θ(a) for every a∈ℂn∪{∞}. Hence
(3.25)lim⁡r→+∞T(r,f′)T(r,f)=2-δ(∞)
follows by Theorem 3.1.
Now, for everya∈ℂn,
(3.26)lim⁡r→+∞N¯(r,a)+V(r,a)T(r,f)=1-Θ(a)=1-δ(a).
Further
(3.27)N¯(r,a)≤N(r,a).
Hence
(3.28)1-δ(a)≤lim⁡r→+∞N¯(r,a)+V(r,a)T(r,f)≤liminf⁡r→+∞N(r,a)+V(r,a)T(r,f)≤limsup⁡r→+∞N(r,a)+V(r,a)T(r,f)=1-δ(a).
Similarly,(3.29)lim⁡r→+∞N¯(r,f)T(r,f)=1-Θ(∞)=1-δ(∞).
Further
(3.30)N¯(r,∞)≤N(r,∞).
Hence
(3.31)1-δ(∞)≤lim⁡r→+∞N¯(r,∞)T(r,f)≤liminf⁡r→+∞N(r,∞)T(r,f)≤limsup⁡r→+∞N(r,∞)T(r,f)=1-δ(∞).From Corollary3.4, we have the following corollary.Corollary 3.5.
Letf(z)=(f1(z),f2(z),…,fn(z)) be an admissible meromorphic vector function of finite order in ℂ and assume ∑a∈ℂnδ(a)=2. Then
(3.32)T(r,f′)~2T(r,f),r⟶+∞.Corollary 3.6.
Letf(z)=(f1(z),f2(z),…,fn(z)) be an admissible meromorphic vector function of finite order in ℂ and assume ∑a∈ℂnδ(a)=η≥1 and δ(∞)=2-η. Then
(3.33)T(r,f′)~ηT(r,f),r⟶+∞.



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