An Inequality of Meromorphic Vector Functions and Its Application

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 function 𝑓 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).

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 by ℂ^𝑛 the usual 𝑛 dimensional complex Euclidean space with the coordinates 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛), the Hermitian scalar product⟨𝑣,𝑤⟩=𝑣₁𝑤₁+𝑣₂𝑤₂+⋯+𝑣_𝑛𝑤_𝑛,𝑣,𝑤∈ℂ^𝑛, (1.1) and the distance‖𝑣−𝑤‖=+⟨𝑣−𝑤,𝑣−𝑤⟩^1/2. (1.2)

Let𝑤₁=𝑓₁(𝑧),𝑤₂=𝑓₂(𝑧),…,𝑤_𝑛=𝑓_𝑛(𝑧) (1.3) 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 discℂ_𝑅={|𝑧|<𝑅}⊂ℂ,0<𝑅<+∞. (1.4) Thus in ℂ_𝑅,0<𝑅≤+∞ (we put ℂ_+∞=ℂ), a vector-valued meromorphic function𝑓(𝑧)=𝑓₁(𝑧),𝑓₂(𝑧),…,𝑓_𝑛(𝑧) (1.5) 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 by𝑓^(𝑗)(𝑧)=𝑓₁^(𝑗)(𝑧),𝑓₂^(𝑗)(𝑧),…,𝑓_𝑛^(𝑗)(𝑧). (1.6)

For such a function, the notations “zero,” “pole,” and “multiplicity” are defined as in the scalar case 𝑛=1 of only one meromorphic function 𝑓₁(𝑧). More explicitly, in the punctured vicinity of each point 𝑧₀∈ℂ_𝑅, the vector function 𝑤=𝑓(𝑧) can developed into a Laurent series𝑓(𝑧)=𝑐_𝑘0𝑧−𝑧₀^𝑘0+𝑐_𝑘0+1𝑧−𝑧₀^𝑘0+1+⋯, (1.7) where the coefficients are vectors𝑐_𝑘=𝑐¹_𝑘,𝑐²_𝑘,…,𝑐^𝑛_𝑘∈ℂ^𝑛,𝑐_𝑘0≠(0,0,…,0). (1.8)

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 ℂ^𝑛 (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 𝑓_𝑗(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 𝑓_𝑗(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 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),𝑉()=1𝑟,∞=𝑉𝑟,𝑓2𝜋_𝐶𝑟𝑟log𝜉𝑓(𝜉𝑉()1Δlog)𝑑𝑥∧𝑑𝑦,𝜉=𝑥+𝑖𝑦𝑟,𝑎=𝑉𝑟,=1𝑓−𝑎2𝜋_𝐶𝑟𝑟log𝜉Δlog𝑓(𝜉)−𝑎𝑑𝑥∧𝑑𝑦,𝜉=𝑥+𝑖𝑦 (1.9) and the counting function of finite or infinite 𝑎-points by𝑁𝑟,𝑓=𝑛0,𝑓log𝑟+^𝑟₀𝑛𝑡,𝑓−𝑛0,𝑓𝑡𝑁()()𝑑𝑡,𝑟,∞=𝑛0,∞log𝑟+^𝑟₀𝑛(𝑡,∞)−𝑛(0,∞)𝑡𝑁()()𝑑𝑡,𝑟,𝑎=𝑛0,𝑎log𝑟+^𝑟₀𝑛(𝑡,𝑎)−𝑛(0,𝑎)𝑡𝑑𝑡, (1.10) respectively. Next, we define𝑚()=1𝑟,∞=𝑚𝑟,𝑓2𝜋₀^2𝜋log⁺𝑓𝑟𝑒^𝑖𝜃𝑚()=1𝑑𝜃,𝑟,𝑎2𝜋₀^2𝜋log⁺1𝑓𝑟𝑒^𝑖𝜃𝑇.−𝑎𝑑𝜃,𝑟,𝑓=𝑚𝑟,𝑓+𝑁𝑟,𝑓 (1.11) 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 by𝑁𝑟,𝑓=𝑛0,𝑓log𝑟+^𝑟₀𝑛𝑡,𝑓−𝑛0,𝑓𝑡𝑑𝑡,𝑁(𝑟,∞)=𝑛(0,∞)log𝑟+^𝑟₀𝑛(𝑡,∞)−𝑛(0,∞)𝑡𝑑𝑡,𝑁(𝑟,𝑎)=𝑛(0,𝑎)log𝑟+^𝑟₀𝑛(𝑡,𝑎)−𝑛(0,𝑎)𝑡𝑑𝑡, (1.12) respectively.

If 𝑓(𝑧) is a vector-valued meromorphic function in the whole complex plane, then the order and the lower order of 𝑓(𝑧) are defined by𝜆𝑓=limsup_𝑟→∞log𝑇𝑟,𝑓,𝜇𝑓log𝑟=liminf_𝑟→∞log𝑇𝑟,𝑓.log𝑟 (1.13)

We call the vector-valued meromorphic function 𝑓 admissible iflimsup_𝑟→+∞𝑇𝑟,𝑓log𝑟=+∞. (1.14)

Definition 1.1. For a meromorphic function 𝑓(𝑧) (vector-valued or scalar-valued), we denote by 𝑆(𝑟,𝑓) any quantity such that 𝑆𝑟,𝑓=𝑜𝑇𝑟,𝑓,𝑟⟶+∞ (1.15) without restriction if 𝑓(𝑧) is of finite order and otherwise except possibly for a set of values of 𝑟 of finite linear measure.

Definition 1.1 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. Let 𝑓(𝑧)=(𝑓₁(𝑧),𝑓₂(𝑧),…,𝑓_𝑛(𝑧)) be a meromorphic vector function in ℂ_𝑅. Then for 0<𝑟<𝑅≤+∞,𝑎∈ℂ^𝑛, 𝑓(𝑧)≢𝑎, then 𝑇𝑟,𝑓=𝑉(𝑟,𝑎)+𝑁(𝑟,𝑎)+𝑚(𝑟,𝑎)+𝑂(1). (1.16)

Theorem B. Let 𝑓(𝑧)=(𝑓₁(𝑧),𝑓₂(𝑧),…,𝑓_𝑛(𝑧)) be a nonconstant meromorphic vector function in ℂ. Then 12𝜋₀^2𝜋log⁺𝑓′𝑟𝑒^𝑖𝜃𝑓𝑟𝑒^𝑖𝜃−𝑎𝑑𝜃=𝑆𝑟,𝑓,𝑎∈ℂ^𝑛. (1.17)

By the second main theorem, Ziegler [1] studies the following deficiency theorem for meromorphic vector function. For any vector 𝑎∈ℂ^𝑛, we define the number 𝛿(𝑎)=𝛿(𝑎,𝑓) by putting𝛿(𝑎)=𝛿𝑎,𝑓=liminf_𝑟→+∞𝑚(𝑟,𝑎)𝑇𝑟,𝑓=1−limsup_𝑟→+∞𝑉(𝑟,𝑎)+𝑁(𝑟,𝑎)𝑇,𝛿(∞)𝑟,𝑓=𝛿∞,𝑓=liminf_𝑟→+∞𝑚𝑟,𝑓𝑇𝑟,𝑓=1−limsup_𝑟→+∞𝑁𝑟,𝑓𝑇,𝑟,𝑓 (1.18) and Θ(𝑎)=Θ(𝑎,𝑓) by puttingΘ(𝑎)=Θ𝑎,𝑓=1−limsup_𝑟→+∞𝑉(𝑟,𝑎)+𝑁(𝑟,𝑎)𝑇,Θ(∞)𝑟,𝑓=Θ∞,𝑓=1−limsup_𝑟→+∞𝑁𝑟,𝑓𝑇,𝑟,𝑓 (1.19)

Theorem C. Let 𝑓(𝑧)=(𝑓₁(𝑧),𝑓₂(𝑧),…,𝑓_𝑛(𝑧)) be an admissible meromorphic vector function in ℂ. Then the set {𝑎∈ℂ^𝑛∪{∞},𝛿(𝑎)>0} is at most countable and summing over all such points we have_𝑎𝛿(𝑎)≤_𝑎Θ(𝑎)≤2. (1.20)

2. A Fundamental Inequality of Meromorphic Vector Function

For meromorphic scalar-valued function 𝑓(𝑧), Milloux [5] has proved the following theorem.

Theorem D. If 𝑓(𝑧) is a nonconstant meromorphic scalar-valued function in Gaussian complex plane ℂ and if 𝑎_𝑖,𝑖=1,2,…,𝑞, are distinct elements of ℂ (where 𝑞 is any positive integer), then 𝑞𝑇𝑟,𝑓≤𝑇𝑟,𝑓′+^𝑞_𝑖=1𝑁𝑟,𝑎_𝑖+𝑆𝑟,𝑓. (2.1)

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. Let 𝑓(𝑧)=(𝑓₁(𝑧),𝑓₂(𝑧),…,𝑓_𝑛(𝑧)) be an admissible meromorphic vector function in ℂ and if 𝑎^[𝑗],𝑗=1,2,…,𝑞, are distinct elements of ℂ^𝑛 (where 𝑞 is any positive integer), then 𝑞𝑇𝑟,𝑓≤𝑇𝑟,𝑓′+^𝑞_𝑗=1𝑁𝑟,𝑎^[𝑗]+𝑉𝑟,𝑎^[𝑗]+𝑆𝑟,𝑓. (2.2)

Proof. Put 𝐹(𝑧)=^𝑞_𝑗=11𝑓(𝑧)−𝑎^[𝑗]. (2.3) We can get 12𝜋₀^2𝜋log⁺𝐹𝑟𝑒^𝑖𝜃+1𝑑𝜃≤𝑚𝑟,0,𝑓′2𝜋₀^2𝜋log⁺𝐹𝑟𝑒^𝑖𝜃𝑓^′𝑟𝑒^𝑖𝜃𝑑𝜃. (2.4) Put 𝛿=min_𝑖≠𝑗𝑎^[𝑖]−𝑎^[𝑗]. (2.5) Let for the moment 𝜇∈{1,2,…,𝑞} be fixed. Then we get in every point where 𝑓(𝑧)−𝑎^[𝜇]<𝛿≤𝛿2𝑞4, (2.6) the inequality 𝑓(𝑧)−𝑎^[𝜈]≥𝑎^[𝜇]−𝑎^[𝜈]−𝑓(𝑧)−𝑎^[𝜇]≥3𝛿4, (2.7) for 𝜇≠𝜈. Therefore, the set of points on 𝜕ℂ_𝑟 which is determined by (2.6) is either empty or any two such sets for different 𝜇 have empty intersection. In any case, 12𝜋₀^2𝜋log⁺𝐹𝑟𝑒^𝑖𝜃1𝑑𝜃≥2𝜋^𝑞_{𝜇=1𝑓(𝑧)−𝑎}^[𝜇]_{|𝑧|<𝛿/2𝑞,=𝑟}log⁺𝐹𝑟𝑒^𝑖𝜃≥1𝑑𝜃2𝜋^𝑞_{𝜇=1𝑓(𝑧)−𝑎}^[𝜇]_{|𝑧|<𝛿/2𝑞,=𝑟}log⁺1𝑓𝑟𝑒^𝑖𝜃−𝑎^[𝜇]𝑑𝜃. (2.8) Because of 12𝜋_{𝑓(𝑧)−𝑎}^[𝜇]_{|𝑧|<𝛿/2𝑞,=𝑟}log⁺1𝑓𝑟𝑒^𝑖𝜃−𝑎^[𝜇]𝑑𝜃=𝑚𝑟,𝑎^[𝜇]−12𝜋_{𝑓(𝑧)−𝑎}^[𝜇]_{|𝑧|≥𝛿/2𝑞,=𝑟}log⁺1𝑓𝑟𝑒^𝑖𝜃−𝑎^[𝜇]𝑑𝜃≥𝑚𝑟,𝑎^[𝜇]−log⁺2𝑞𝛿, (2.9) it follows that 12𝜋₀^2𝜋log⁺𝐹𝑟𝑒^𝑖𝜃𝑑𝜃≥^𝑞_𝜇=1𝑚𝑟,𝑎^[𝜇]−log⁺2𝑞𝛿, (2.10) so that by (2.4) ^𝑞_𝜇=1𝑚𝑟,𝑎^[𝜇]+1≤𝑚𝑟,0,𝑓′2𝜋₀^2𝜋log⁺𝐹𝑟𝑒^𝑖𝜃𝑓^′𝑟𝑒^𝑖𝜃𝑑𝜃+log⁺2𝑞𝛿. (2.11) Thus by Theorem B, we have ^𝑞_𝜇=1𝑚𝑟,𝑎^[𝜇]≤𝑚𝑟,0,𝑓′+𝑆𝑟,𝑓. (2.12) It follows from Theorem A that 𝑚𝑟,0,𝑓′+𝑁𝑟,0,𝑓′+𝑉𝑟,0,𝑓′=𝑇𝑟,𝑓′+𝑂(1). (2.13) Thus from (2.12) and (2.13), we deduce ^𝑞_𝜇=1𝑚𝑟,𝑎^[𝜇]≤𝑇𝑟,𝑓′−𝑁𝑟,0,𝑓′+𝑆𝑟,𝑓. (2.14) Adding ∑^𝑞_𝜇=1𝑁(𝑟,𝑎^[𝜇]) to both sides, ^𝑞_𝜇=1𝑇1𝑟,𝑓−𝑎^[𝜇]≤𝑇𝑟,𝑓′+^𝑞_𝜇=1𝑁𝑟,𝑎^[𝜇]+−𝑁𝑟,0,𝑓′+𝑆𝑟,𝑓=𝑇𝑟,𝑓′^𝑞_𝜇=1𝑁𝑟,𝑎^[𝜇]−𝑁₀𝑟,0,𝑓′+𝑆𝑟,𝑓, (2.15) 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 ^𝑞_𝜇=1𝑇1𝑟,𝑓−𝑎^[𝜇]≤𝑇𝑟,𝑓′+^𝑞_𝜇=1𝑁𝑟,𝑎^[𝜇]+𝑆𝑟,𝑓. (2.16) Since 𝑇1𝑟,𝑓−𝑎^[𝜇]+𝑉𝑟,𝑎^[𝜇]=𝑇𝑟,𝑓+𝑂(1), (2.17) it follows that 𝑞𝑇𝑟,𝑓≤𝑇𝑟,𝑓′+^𝑞_𝑗=1𝑁𝑟,𝑎^[𝑗]+𝑉𝑟,𝑎^[𝑗]+𝑆𝑟,𝑓. (2.18)

3. Characteristic Function of Derivative of Meromorphic Vector Function

Let 𝑓(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. Let 𝑓(𝑧) be a transcendental meromorphic scalar-valued function of finite order and assume ∑_𝑎∈ℂ𝛿(𝑎)=𝜂≥1 and 𝛿(∞)=2−𝜂. Then 𝑇𝑟,𝑓′∼𝜂𝑇𝑟,𝑓,𝑟⟶+∞. (3.1)

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. Let 𝑓(𝑧) be a transcendental meromorphic scalar-valued function of finite order and assume ∑_𝑎Θ(𝑎)=2. Then lim_𝑟→+∞𝑇𝑟,𝑓′𝑇(∞),𝑟,𝑓=2−Θlim_𝑟→+∞𝑁(𝑟,𝑎)𝑇(𝑎)𝑟,𝑓=1−Θ (3.2) for every 𝑎∈ℂ∪{∞}.

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 purpose is to extend the above theorems to vector-valued meromorphic function by referring the method of [1, 7].

Theorem 3.1. Let 𝑓(𝑧)=(𝑓₁(𝑧),𝑓₂(𝑧),…,𝑓_𝑛(𝑧)) be an admissible meromorphic vector function of finite order in ℂ and assume ∑_𝑎Θ(𝑎)=2. Then lim_𝑟→+∞𝑇𝑟,𝑓′𝑇(∞),𝑟,𝑓=2−Θlim_𝑟→+∞𝑁𝑟,𝑓𝑇(∞)𝑟,𝑓=1−Θ,lim_𝑟→+∞𝑁(𝑟,𝑎)+𝑉(𝑟,𝑎)𝑇(𝑎)𝑟,𝑓=1−Θ (3.3) for every 𝑎∈ℂ^𝑛.

Proof. Now, basic estimates in vector-valued Nevanlinna theory [1] or [4] yields 𝑇𝑟,𝑓′=𝑚𝑟,𝑓′+𝑁𝑟,𝑓′=𝑚𝑟,𝑓𝑓′𝑓+𝑁𝑟,𝑓′≤𝑚𝑟,𝑓′𝑓+𝑚𝑟,𝑓+𝑁𝑟,𝑓+𝑁+𝑟,𝑓≤𝑇𝑟,𝑓𝑁𝑟,𝑓+𝑚𝑟,𝑓′𝑓. (3.4) By Theorem B and the above inequality, we have limsup_𝑟→+∞𝑇𝑟,𝑓′𝑇(∞)𝑟,𝑓≤2−Θ. (3.5)
Let {𝑎^[𝑗]} be a sequence of distinct vector in ℂ^𝑛 containing all the vector of 𝛿(𝑎^[𝑗])>0. From Theorem 2.1, for any positive integer 𝑞, we have 𝑞𝑇𝑟,𝑓≤𝑇𝑟,𝑓′+^𝑞_𝑗=1𝑁𝑟,𝑎^[𝑗]+𝑉𝑟,𝑎^[𝑗]+𝑆𝑟,𝑓. (3.6) Hence 𝑞≤liminf_𝑟→+∞𝑇𝑟,𝑓′𝑇+𝑟,𝑓^𝑞_𝑗=1limsup_𝑟→+∞𝑁𝑟,𝑎^[𝑗]+𝑉𝑟,𝑎^[𝑗]𝑇𝑟,𝑓+limsup_𝑟→+∞𝑆𝑟,𝑓𝑇𝑟,𝑓=liminf_𝑟→+∞𝑇𝑟,𝑓′𝑇+𝑟,𝑓^𝑞_𝑗=11−Θ𝑎^[𝑗]+limsup_𝑟→+∞𝑆𝑟,𝑓𝑇.𝑟,𝑓 (3.7) Thus liminf_𝑟→+∞𝑇𝑟,𝑓′𝑇≥𝑟,𝑓^𝑞_𝑗=1Θ𝑎^[𝑗]. (3.8) Since 𝑞 was arbitrary, we have 2−Θ(∞)=_{𝑎∈ℂ𝑛}Θ(𝑎)≤liminf_𝑟→+∞𝑇𝑟,𝑓′𝑇𝑟,𝑓. (3.9) This and (3.5) yield lim_𝑟→+∞𝑇𝑟,𝑓′𝑇(∞)𝑟,𝑓=2−Θ. (3.10)
Let 𝑎∈ℂ^𝑛∪{∞} and {𝑎^[𝑖]}^+∞_𝑖=1 an infinite sequence of distinct elements of ℂ^𝑛∪{∞} which includes every 𝑏∈ℂ^𝑛∪{∞} satisfying 𝑏≠𝑎 and Θ(𝑏)>0. Then ^+∞_𝑖=1Θ𝑎^[𝑖]=_{𝑏∈ℂ𝑛∪{∞},𝑏≠𝑎}Θ(𝑏)=2−Θ(𝑎). (3.11)
Let 𝑞 be any integer ≥3. From Generalized Second Main Theorem (see [1], Page 126), we have 𝑞−2𝑇𝑟,𝑓=^𝑞−1_𝑖=1𝑁𝑟,𝑎^[𝑖]+𝑉𝑟,𝑎^[𝑖]+𝑁𝑟,𝑓+𝑆𝑟,𝑓. (3.12) Hence 𝑞−2≤^𝑞−1_𝑖=11−Θ𝑎^[𝑖]+liminf_𝑟→+∞𝑁𝑟,𝑓𝑇𝑟,𝑓. (3.13) Thus ^𝑞−1_𝑖=1Θ𝑎^[𝑖]−1≤liminf_𝑟→+∞𝑁𝑟,𝑓𝑇𝑟,𝑓. (3.14) Since this holds for all 𝑞≥3, letting 𝑞→+∞ and combining (3.11), we get 1−Θ(∞)=^+∞_𝑖=1Θ𝑎^[𝑖]−1≤liminf_𝑟→+∞𝑁𝑟,𝑓𝑇𝑟,𝑓≤limsup_𝑟→+∞𝑁𝑟,𝑓𝑇(∞)𝑟,𝑓=1−Θ. (3.15) So lim_𝑟→+∞𝑁𝑟,𝑓𝑇(∞)𝑟,𝑓=1−Θ. (3.16)
For every 𝑎∈ℂ^𝑛, Let 𝑞 be any integer ≥3. From Generalized Second Main Theorem (see [1], Page 126), we have 𝑞−2𝑇𝑟,𝑓=^𝑞−2_𝑖=1𝑁𝑟,𝑎^[𝑖]+𝑉𝑟,𝑎^[𝑖]+𝑁(𝑟,𝑎)+𝑉(𝑟,𝑎)+𝑁𝑟,𝑓+𝑆𝑟,𝑓. (3.17) Hence 𝑞−2≤^𝑞−2_𝑖=11−Θ𝑎^[𝑖]+(1−Θ(∞))+liminf_𝑟→+∞𝑁(𝑟,𝑎)+𝑉(𝑟,𝑎)𝑇𝑟,𝑓. (3.18) Thus ^𝑞−2_𝑖=1Θ𝑎^[𝑖]+Θ(∞)−1≤liminf_𝑟→+∞𝑁(𝑟,𝑎)+𝑉(𝑟,𝑎)𝑇𝑟,𝑓. (3.19) Since this holds for all 𝑞≥3, letting 𝑞→+∞ and combining (3.11), we get 1−Θ(𝑎)=^+∞_𝑖=1Θ𝑎^[𝑖]−1≤liminf_𝑟→+∞𝑁(𝑟,𝑎)+𝑉(𝑟,𝑎)𝑇𝑟,𝑓≤limsup_𝑟→+∞𝑁(𝑟,𝑎)+𝑉(𝑟,𝑎)𝑇(𝑎).𝑟,𝑓=1−Θ (3.20) So lim_𝑟→+∞𝑁(𝑟,𝑎)+𝑉(𝑟,𝑎)𝑇(𝑎)𝑟,𝑓=1−Θ. (3.21)

From Theorem 3.1, we have the following corollary

Corollary 3.2. Let 𝑓(𝑧)=(𝑓₁(𝑧),𝑓₂(𝑧),…,𝑓_𝑛(𝑧)) be an admissible meromorphic vector function of finite order in ℂ and assume ∑_𝑎∈ℂ^𝑛Θ(𝑎)=2. Then 𝑇𝑟,𝑓^′∼2𝑇𝑟,𝑓,𝑟⟶+∞. (3.22)

Corollary 3.3. Let 𝑓(𝑧)=(𝑓₁(𝑧),𝑓₂(𝑧),…,𝑓_𝑛(𝑧)) be an admissible meromorphic vector function of finite order in ℂ and assume ∑_𝑎∈ℂ^𝑛Θ(𝑎)=𝜂≥1 and 𝛿(∞)=2−𝜂. Then 𝑇𝑟,𝑓^′∼𝜂𝑇𝑟,𝑓,𝑟⟶+∞. (3.23)

Corollary 3.4. Let 𝑓(𝑧)=(𝑓₁(𝑧),𝑓₂(𝑧),…,𝑓_𝑛(𝑧)) be an admissible meromorphic vector function of finite order in ℂ and assume ∑_𝑎𝛿(𝑎)=2. Then lim_𝑟→+∞𝑇𝑟,𝑓′𝑇(∞),𝑟,𝑓=2−𝛿lim_𝑟→+∞𝑁𝑟,𝑓𝑇(∞)𝑟,𝑓=1−𝛿,lim_𝑟→+∞𝑁(𝑟,𝑎)+𝑉(𝑟,𝑎)𝑇(𝑎)𝑟,𝑓=1−𝛿 (3.24) for every 𝑎∈ℂ^𝑛.

Proof. Since 𝛿(𝑎)≤Θ(𝑎) for every 𝑎∈ℂ^𝑛∪{∞} and Theorem C, it follows that, if ∑_𝑎𝛿(𝑎)=2, then ∑_𝑎Θ(𝑎)=2 and 𝛿(𝑎)=Θ(𝑎) for every 𝑎∈ℂ^𝑛∪{∞}. Hence lim_𝑟→+∞𝑇𝑟,𝑓′𝑇(∞)𝑟,𝑓=2−𝛿 (3.25) follows by Theorem 3.1.
Now, for every 𝑎∈ℂ^𝑛, lim_𝑟→+∞𝑁(𝑟,𝑎)+𝑉(𝑟,𝑎)𝑇(𝑎)(𝑎)𝑟,𝑓=1−Θ=1−𝛿. (3.26) Further 𝑁(𝑟,𝑎)≤𝑁(𝑟,𝑎). (3.27) Hence 1−𝛿(𝑎)≤lim_𝑟→+∞𝑁(𝑟,𝑎)+𝑉(𝑟,𝑎)𝑇𝑟,𝑓≤liminf_𝑟→+∞𝑁(𝑟,𝑎)+𝑉(𝑟,𝑎)𝑇𝑟,𝑓≤limsup_𝑟→+∞𝑁(𝑟,𝑎)+𝑉(𝑟,𝑎)𝑇(𝑎).𝑟,𝑓=1−𝛿 (3.28)
Similarly, lim_𝑟→+∞𝑁𝑟,𝑓𝑇(∞)(∞)𝑟,𝑓=1−Θ=1−𝛿. (3.29) Further 𝑁(𝑟,∞)≤𝑁(𝑟,∞). (3.30) Hence 1−𝛿(∞)≤lim_𝑟→+∞𝑁(𝑟,∞)𝑇𝑟,𝑓≤liminf_𝑟→+∞𝑁(𝑟,∞)𝑇𝑟,𝑓≤limsup_𝑟→+∞𝑁(𝑟,∞)𝑇(∞)𝑟,𝑓=1−𝛿. (3.31)

From Corollary 3.4, we have the following corollary.

Corollary 3.5. Let 𝑓(𝑧)=(𝑓₁(𝑧),𝑓₂(𝑧),…,𝑓_𝑛(𝑧)) be an admissible meromorphic vector function of finite order in ℂ and assume ∑_𝑎∈ℂ^𝑛𝛿(𝑎)=2. Then 𝑇𝑟,𝑓^′∼2𝑇𝑟,𝑓,𝑟⟶+∞. (3.32)

Corollary 3.6. Let 𝑓(𝑧)=(𝑓₁(𝑧),𝑓₂(𝑧),…,𝑓_𝑛(𝑧)) be an admissible meromorphic vector function of finite order in ℂ and assume ∑_𝑎∈ℂ^𝑛𝛿(𝑎)=𝜂≥1 and 𝛿(∞)=2−𝜂. Then 𝑇𝑟,𝑓^′∼𝜂𝑇𝑟,𝑓,𝑟⟶+∞. (3.33)

Acknowledgments

This research was partially supported by the NSF of Jiangxi Province (Grant 2010GZC0187), by NSF of Educational Department of the Hubei Province (Grant T201009, Q20112807), and by Grant PY1002 of Xianning University.