Characteristic Functions and Borel Exceptional Values of 𝐸-Valued Meromorphic Functions

Abstract

The main purpose of this paper is to investigate the characteristic functions and Borel exceptional values of 𝐸-valued meromorphic functions from the ℂ_𝑅={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 ℂ_𝑅∶={𝑧∶|𝑧|<𝑅},0<𝑅≤+∞. (1.1)

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 {𝑒_𝑗} and the norm ‖•‖. Thus, an 𝐸-valued meromorphic function 𝑓(𝑧) defined in ℂ_𝑅, 0<𝑅≤+∞ can be written as 𝑓(𝑧)=𝑓₁(𝑧),𝑓₂(𝑧),…,𝑓_𝑘(𝑧),…. (1.2) Let 𝐸_𝑛 be an 𝑛-dimensional projective space of 𝐸 with a basis {𝑒_𝑗}^𝑛₁. The projective operator 𝑃_𝑛∶𝐸→𝐸_𝑛 is a realization of 𝐸_𝑛 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 𝑓_𝑗(𝑧) are holomorphic (some of 𝑓_𝑗(𝑧) are meromorphic). The 𝑗th derivative 𝑗=1,2,… of 𝑓(𝑧) is defined by 𝑓^(𝑗)(𝑧)=𝑓₁^(𝑗)(𝑧),𝑓₂^(𝑗)(𝑧),…,𝑓_𝑘^(𝑗)(𝑧),…. (1.3) A point 𝑧₀∈ℂ_𝑟 is called a “pole" (or ∞ point) of 𝑓(𝑧)=𝑓₁(𝑧),𝑓₂(𝑧),…,𝑓_𝑘(𝑧),… (1.4) if 𝑧₀ is a pole (or ∞ point) of at least one of the component functions 𝑓_𝑘(𝑧)(𝑘=1,2,…). A point 𝑧₀∈ℂ_𝑟 is called a “zero" of 𝑓(𝑧)=(𝑓₁(𝑧),𝑓₂(𝑧),…,𝑓_𝑘(𝑧),…) if 𝑧₀ is a zero of all the component functions 𝑓_𝑘(𝑧)(𝑘=1,2,…). A point 𝑧₀∈ℂ_𝑟 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 𝑓_𝑗(𝑧) has a pole of this multiplicity in the ordinary sense of function theory. A point 𝑧₀∈ℂ_𝑟 is called a zero of 𝑓(𝑧) of multiplicity 𝑞∈ℕ⁺, meaning that 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 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 𝑉=1𝑟,∞,𝑓=𝑉𝑟,𝑓2𝜋_𝐶𝑟𝑟log𝜉𝑓(𝜉𝑉=1Δlog)𝑑𝑥∧𝑑𝑦,𝜉=𝑥+𝑖𝑦,𝑟,𝑎,𝑓2𝜋_𝐶𝑟𝑟log𝜉Δlog𝑓(𝜉)−𝑎𝑑𝑥∧𝑑𝑦,𝜉=𝑥+𝑖𝑦, (1.5) and the counting function of finite or infinite 𝑎-points by 𝑁𝑟,𝑓=𝑛0,𝑓log𝑟+^𝑟₀𝑛𝑡,𝑓−𝑛0,𝑓𝑡𝑑𝑡,𝑁(𝑟,∞)=𝑛(0,∞)log𝑟+^𝑟₀𝑛(𝑡,∞)−𝑛(0,∞)𝑡𝑑𝑡,𝑁𝑟,𝑎,𝑓=𝑛0,𝑎,𝑓log𝑟+^𝑟₀𝑛𝑡,𝑎,𝑓−𝑛0,𝑎,𝑓𝑡𝑑𝑡, (1.6)(1.7)(1.8) respectively. Next, we define 𝑚=1𝑟,𝑓=𝑚𝑟,∞,𝑓2𝜋₀^2𝜋log⁺𝑓𝑟𝑒^𝑖𝜃𝑚()=1𝑑𝜃,𝑟,𝑎=𝑚𝑟,𝑎,𝑓2𝜋₀^2𝜋log⁺1𝑓𝑟𝑒^𝑖𝜃𝑇.−𝑎𝑑𝜃,𝑟,𝑓=𝑚𝑟,𝑓+𝑁𝑟,𝑓 (1.9) 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 𝜆𝑓=limsup_𝑟→+∞log⁺𝑇𝑟,𝑓,𝜇𝑓log𝑟=liminf_𝑟→+∞log⁺𝑇𝑟,𝑓.log𝑟 (1.10)

If 𝑓(𝑧) is an 𝐸-valued meromorphic function in ℂ_𝑅,0<𝑅<+∞, then the order and the lower order of 𝑓(𝑧) are defined by 𝜆𝑓=limsup_{𝑟→𝑅−}log𝑇𝑟,𝑓log⁺((,𝜇𝑓1/𝑅−𝑟))=liminf_{𝑟→𝑅−}log𝑇𝑟,𝑓log⁺((.1/𝑅−𝑟)) (1.11)

Lemma 1.1. Let 𝐵(𝑥) be a positive and continuous function in [0,+∞) which satisfies limsup_𝑥→+∞(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 function 𝑈(𝑥)=𝑥^𝜌(𝑥)(𝑥≥𝑥₀) satisfies the following: lim_𝑥→+∞log𝑈(𝑋)(𝑥)𝑥log𝑈=1,𝑋=𝑥+(𝑥).log𝑈 (1.12)(iii)limsup_𝑥→+∞(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 lim_𝑟→1−log𝑈(1/(1−𝑅))((((log𝑈1/1−𝑟))=1,𝑅=𝑟log𝑈1/1−𝑟))+1((,log𝑈1/1−𝑟))+1limsup_𝑥→1−log𝑇𝑟,𝑓log𝑈(1/(1−𝑟))=1. (1.13) 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 ℂ_𝑅,0<𝑅≤+∞ is of compact projection, if for any given 𝜀>0, ‖𝑃_𝑛(𝑓(𝑧))−𝑓(𝑧)‖<𝜀 has sufficiently larg 𝑛 in any fixed compact subset 𝐷⊂𝐶_𝑅.
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 ℂ_𝑅,0<𝑅≤+∞. Then for 0<𝑟<𝑅,𝑎∈𝐸,𝑓(𝑧)≢𝑎, 𝑇𝑟,𝑓=𝑉(𝑟,𝑎)+𝑁(𝑟,𝑎)+𝑚(𝑟,𝑎)+log⁺𝑐_𝑞(𝑎)+𝜀(𝑟,𝑎). (1.14) Here, 𝜀(𝑟,𝑎) is a function satisfying that |𝜀(𝑟,𝑎)|≤log⁺‖𝑎‖+log2,𝜀(𝑟,0)≡0, (1.15) and 𝑐_𝑞(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 ℂ_𝑅,0<𝑅≤+∞ and 𝑎^[𝑘]∈𝐸∪{∞}(𝑘=1,2,…,𝑞) be 𝑞≥3 distinct points. Then for 0<𝑟<𝑅, 𝑞−2𝑇𝑟,𝑓≤^𝑞_𝑘=1𝑉𝑟,𝑎^[𝑘]+𝑁𝑟,𝑎^[𝑘]+𝑆𝑟,𝑓. (1.16)

If 𝑅=+∞, then𝑆𝑟,𝑓=𝑂log𝑇𝑟,𝑓+log𝑟 (1.17) 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 𝑆𝑟,𝑓=𝑂log𝑈(𝑟) (1.18) holds as 𝑟→+∞ without exception.

If 0<𝑅<+∞, then 𝑆1𝑟,𝑓=𝑂log𝑇𝑟,𝑓+log𝑅−𝑟 (1.19) 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 𝑇𝑟,𝑓<𝐶_𝜏𝑇𝜏𝑟,𝑓^′+log⁺𝜏𝑟+4+log⁺𝑓(0), (2.1) where 𝐶_𝜏 is a positive constant.

Theorem B. Let 𝑓(𝑧)(𝑧∈ℂ) be a nonconstant 𝐸-valued meromorphic function. Then we have 𝑇𝑟,𝑓^′<2𝑇𝑟,𝑓+𝑂log𝑟+log⁺𝑇𝑟,𝑓. (2.2)

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 log⁺𝑓𝑟𝑒^𝑖𝜃0≤𝑅^′+𝑅𝑅^′𝑚𝑅−𝑅^′,𝑓+𝑛𝑅^′,𝑓log4+𝑁𝑅^′,𝑓. (2.3)

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 log⁺𝑓𝑟𝑒^𝑖𝜃0≤𝑅^′′+𝑅^′𝑅^′′−𝑅^′𝑚𝑅^′′,𝑓+𝑛𝑅^′′,𝑓log8𝑒𝑅^′′𝑅^′.−𝑅 (2.4)

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 𝑛≤(𝑟)𝑟,𝑓6ℎ𝑁.1−𝑟𝑅,𝑓 (2.5)

Proof. 𝑁𝑅,𝑓=𝑛0,𝑓log𝑟+^𝑅₀𝑛𝑡,𝑓−𝑛0,𝑓𝑡𝑑𝑡=^𝑅₀𝑛𝑡,𝑓𝑡𝑑𝑡≥^𝑅_𝑟𝑛𝑡,𝑓𝑡𝑅𝑑𝑡≥𝑛𝑟,𝑓log𝑟=𝑛𝑟,𝑓log1+1−𝑟𝑟(1+ℎ(𝑟))≥𝑛𝑟,𝑓1−𝑟𝑟((𝑟−)((𝑟1+ℎ))((1−𝑟/𝑟+ℎ)))²2()((𝑟≥𝑛𝑟,𝑓1−𝑟/𝑟1+ℎ))2≥𝑛𝑟,𝑓1−𝑟(𝑟).6ℎ (2.6)

Lemma 2.4 (see [4]). Let 𝑓(𝑧)(𝑧∈ℂ_𝑅,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 𝜌𝑒^𝑖𝜃0, and along {|𝑧|=𝜌<𝑟} turn a rotation to 𝜌𝑒^𝑖𝜃0. Then for any {|𝑧|=𝑟≤𝜌}, one has log⁺𝑓(𝑧)≤log⁺𝑀+𝑂(1), (2.7) 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, 12𝜋₀^2𝜋log⁺𝑓^′𝑟𝑒^𝑖𝜃𝑓𝑟𝑒^𝑖𝜃1𝑑𝜃<𝐾log𝑇𝑟,𝑓+log,1−𝑟 (2.8) 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 𝑇𝑟,𝑓<𝑐ℎ^1+𝜀(𝑟)(1−𝑟)^1+𝜀𝑇𝑅,𝑓^′(𝑟),𝑅=1+𝑟ℎ(𝑟).1+ℎ (2.9)

Proof. Denote 𝑅₁=(𝑅+2𝑟)/3,𝑅₂=(𝑟+2𝑅)/3, we can get 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((𝑟≥1+ℎ))1−𝑟2;𝑅−𝑟=1−𝑟(𝑟)≥1+ℎ1−𝑟(𝑟).2ℎ (2.10) 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 haslog⁺𝑓^′𝑡𝑒^𝑖𝜃0≤𝑅₂+𝑅₁𝑅₂−𝑅₁𝑚𝑅₂,𝑓^′+𝑛𝑅₂,𝑓^′log4+𝑁𝑅₂,𝑓^′≤6+(𝑟)𝑅−𝑟6ℎ1−𝑅₂log4+1𝑇𝑅₂,𝑓^′≤6+6ℎ(𝑟)+(𝑟)1−𝑟12ℎ1−𝑟log4+1−𝑟𝑇1−𝑟𝑅,𝑓^′≤6+6ℎ(𝑟)+24ℎ(𝑟)+1−𝑟𝑇1−𝑟𝑅,𝑓^′≤40ℎ(𝑟)𝑇1−𝑟𝑅,𝑓^′. (2.11) In view of Lemma 2.2, there is a 𝜌∈[𝑟,𝑅₁] such that for any 𝑧∈{|𝑧|=𝜌}, one has log⁺𝑓^′(𝑧≤𝑅)²+𝑅₁𝑅₂−𝑅₁𝑚𝑅₂,𝑓^′+𝑛𝑅₂,𝑓^′log8𝑒𝑅₂𝑅₁≤6−𝑅+(𝑟)𝑅−𝑟6ℎ1−𝑅₂(𝑟)log48𝑒ℎ𝑇𝑅1−𝑟²,𝑓^′≤6+6ℎ(𝑟)+(𝑟)1−𝑟12ℎ(𝑟)1−𝑟log144ℎ𝑇1−𝑟𝑅,𝑓^′≤12ℎ(𝑟)ℎ(𝑟)1−𝑟9+log𝑇1−𝑟𝑅,𝑓^′≤12ℎ(𝑟)ℎ(𝑟)1−𝑟9+1−𝑟^𝜀𝑇𝑅,𝑓^′ℎ(𝑟)≤1201−𝑟^1+𝜀𝑇𝑅,𝑓^′. (2.12) From the origin along the segment arg𝑧=𝜃₀ to 𝜌𝑒^𝑖𝜃0 and along {|𝑧|=𝜌}, turn a rotation to 𝜌𝑒^𝑖𝜃0. We denote this curve by 𝐿. In virtue of Lemma 2.4, we have log⁺𝑓(𝑧)≤log⁺𝑀+𝑂(1) (2.13) 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 𝑚𝑟,𝑓≤𝑚𝜌,𝑓≤𝑚𝜌,𝑓^′≤12𝜋₀^2𝜋log⁺ℎ(𝑟)𝑀𝑑𝜃≤1211−𝑟^1+𝜀𝑇𝑅,𝑓^′. (2.14) Hence, 𝑇𝑟,𝑓=𝑚𝑟,𝑓+𝑁𝑟,𝑓≤𝑚𝑟,𝑓+2𝑁𝑟,𝑓^′ℎ(𝑟)≤1231−𝑟^1+𝜀𝑇𝑅,𝑓^′. (2.15)

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 𝑇𝑟,𝑓^′<2𝑇𝑟,𝑓+𝑂log⁺11−𝑟+log⁺𝑇𝑟,𝑓. (2.16)

Proof. By Lemma 2.5, we have 𝑇𝑟,𝑓^′=𝑚𝑟,𝑓^′+𝑁𝑟,𝑓^′𝑓≤𝑚𝑟,𝑓+𝑚𝑟,^′𝑓𝑓+2𝑁𝑟,𝑓≤2𝑇𝑟,𝑓+𝑚𝑟,^′𝑓≤2𝑇𝑟,𝑓+𝑂log⁺11−𝑟+log⁺𝑇𝑟,𝑓. (2.17)

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 ℂ_ℝ

Some definitions in this section can be found in [8].

Definition 3.1. Let 𝑓(𝑧)(𝑧∈ℂ_𝑅,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∈ℂ_𝑅,0<𝑅≤+∞) be an 𝐸-valued meromorphic function and 𝑎∈𝐸∪{∞}. If 𝑅=+∞, we define 𝜌_𝑘𝑎,𝑓=limsup_𝑟→+∞log⁺𝑉𝑎,𝑓+𝑁_𝑘(𝑟,𝑎),log𝑟𝜌𝑎,𝑓=limsup_𝑟→+∞log⁺𝑉𝑎,𝑓+𝑁(𝑟,𝑎),𝜌log𝑟𝑎,𝑓=limsup_𝑟→+∞log⁺𝑉𝑎,𝑓+𝑁(𝑟,𝑎).log𝑟 (3.1) If 𝑅<+∞, we define 𝜌_𝑘𝑎,𝑓=limsup_{𝑟→𝑅−}log⁺𝑉𝑎,𝑓+𝑁_𝑘(𝑟,𝑎)((,log1/𝑅−𝑟))𝜌𝑎,𝑓=limsup_{𝑟→𝑅−}log⁺𝑉𝑎,𝑓+𝑁(𝑟,𝑎)((,𝜌log1/𝑅−𝑟))𝑎,𝑓=limsup_{𝑟→𝑅−}log⁺𝑉𝑎,𝑓+𝑁(𝑟,𝑎)((.log1/𝑅−𝑟)) (3.2)

Definition 3.3. Let 𝑓(𝑧)(𝑧∈ℂ_𝑅,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. Let 𝑓(𝑧)(𝑧∈ℂ_𝑅,𝑅=+∞) be a meromorphic function with finite order 𝜆>0 and 𝑘_𝑗(𝑗=1,2,…,𝑞) be 𝑞 positive integers. 𝑎 is called a pseudo-Borel exceptional value of 𝑓(𝑧) of order 𝑘 if limsup_𝑟→+∞log⁺𝑛_𝑘(𝑟,𝑎)𝑓.log𝑟<𝜆 (3.3) If 𝑓(𝑧) has 𝑞 distinct pseudo-Borel exceptional values 𝑎_𝑗 of order 𝑘_𝑗(𝑗=1,2,…,𝑞), then ^𝑞_𝑗=111−𝑘_𝑗+1≤2. (3.4)

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 ℂ_𝑅,0<𝑅≤+∞.

Theorem 3.4. Let 𝑓(𝑧)(𝑧∈ℂ_𝑅,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 𝑘_𝑗(𝑗=1,2,…,𝑞) any system such that 𝑘_𝑗 is a positive integer or +∞. If 𝑎^[𝑗] is an 𝐸-valued evB for 𝑓 for distinct zeros of order ≤𝑘_𝑗(𝑗=1,2,…,𝑞), then ^𝑞_𝑗=111−𝑘_𝑗+1≤2. (3.5)

Proof. By Theorem 1.4, we have 𝑞−2𝑇𝑟,𝑓≤^𝑞_𝑗=1𝑉𝑟,𝑎^[𝑗]+𝑁𝑟,𝑎^[𝑗]+𝑆𝑟,𝑓 (3.6) holds for 0<𝑟<𝑅. For any 𝑗=1,2,…,𝑞, we have 𝑁𝑟,𝑎^[𝑗]≤1𝑘_𝑗𝑘+1^𝑗𝑁_𝑘𝑗𝑟,𝑎^[𝑗]+𝑁𝑟,𝑎^[𝑗],𝑁𝑟,𝑎^[𝑗]≤𝑇𝑟,𝑓−𝑉𝑟,𝑎^[𝑗]+𝑂(1). (3.7) Using (3.7) and (7) in (3.6), we get 𝑞−2𝑇𝑟,𝑓≤^𝑞_𝑗=1𝑉𝑟,𝑎^[𝑗]+1𝑘_𝑗𝑘+1^𝑗𝑁_𝑘𝑗𝑟,𝑎^[𝑗]+𝑁𝑟,𝑎^[𝑗]=+𝑆𝑟,𝑓_𝑞𝑗=1𝑉𝑟,𝑎^[𝑗]+𝑘_𝑗𝑘_𝑗+1𝑁_𝑘𝑗𝑟,𝑎^[𝑗]+1𝑘_𝑗𝑁+1𝑟,𝑎^[𝑗]≤+𝑆𝑟,𝑓_𝑞𝑗=1𝑘_𝑗𝑘_𝑗𝑉+1𝑟,𝑎^[𝑗]+𝑁_𝑘𝑗𝑟,𝑎^[𝑗]+^𝑞_𝑗=11𝑘_𝑗𝑇.+1𝑟,𝑓+𝑆𝑟,𝑓 (3.8) Therefore, we have ^𝑞_𝑗=111−𝑘_𝑗𝑇≤+1−2𝑟,𝑓^𝑞_𝑗=1𝑘_𝑗𝑘_𝑗𝑉+1𝑟,𝑎^[𝑗]+𝑁_𝑘𝑗𝑟,𝑎^[𝑗]+𝑆𝑟,𝑓. (3.9) By hypothesis, we have 𝜌_𝑘𝑗𝑎^[𝑗],𝑓<𝜆,𝑗=1,2,…,𝑞. (3.10) If 𝑅=+∞, then there is a positive number 𝜌<𝜆, such that for 𝑗=1,2,…,𝑞, we can get 𝑉𝑟,𝑎^[𝑗]+𝑁_𝑘𝑗𝑟,𝑎^[𝑗]≤𝑟^𝜌. (3.11) Using (3.11) to (3.9), we have ^𝑞_𝑗=111−𝑘_𝑗𝑇≤+1−2𝑟,𝑓^𝑞_𝑗=1𝑘_𝑗𝑘_𝑗𝑟+1^𝜌+𝑆𝑟,𝑓. (3.12) If∑^𝑞_𝑗=1(1−(1/(𝑘_𝑗+1)))>2, then by Theorem 1.4 and (3.12), we can get a contradiction 𝜆≤𝜌. So ^𝑞_𝑗=111−𝑘_𝑗+1≤2. (3.13) If 𝑅<+∞, then there is a positive number 𝜌<𝜆, such that for 𝑗=1,2,…,𝑞, we can get 𝑉𝑟,𝑎^[𝑗]+𝑁_𝑘𝑗𝑟,𝑎^[𝑗]≤1𝑅−𝑟^𝜌. (3.14) Using (3.14) to (3.9), we have ^𝑞_𝑗=111−𝑘_𝑗𝑇≤+1−2𝑟,𝑓^𝑞_𝑗=1𝑘_𝑗𝑘_𝑗1+1𝑅−𝑟^𝜌+𝑆𝑟,𝑓. (3.15) If∑^𝑞_𝑗=1(1−(1/(𝑘_𝑗+1)))>2, then by Theorem 1.4 and (3.15), we can get a contradiction 𝜆≤𝜌. So ^𝑞_𝑗=111−𝑘_𝑗+1≤2. (3.16)

From the proof of Theorem 3.4, we can get the following.

Corollary 3.5. Let 𝑓(𝑧)(𝑧∈ℂ_𝑅,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 𝑘_𝑗(𝑗=1,2,…,𝑡) such that 𝑘_𝑗 is a positive integer or +∞, we have the following: (1)if all of 𝑎^[𝑗](𝑗=1,2,…,𝑞) in 𝐸, then 𝑞−∑^𝑞_𝑗=11𝑘_𝑗𝑇≤+1−2𝑟,𝑓^𝑞_𝑗=1𝑘_𝑗𝑘_𝑗𝑉+1𝑟,𝑎^[𝑗],𝑓+𝑁_𝑘𝑗𝑟,𝑎^[𝑗],𝑓+𝑆𝑟,𝑓, (3.17)(2)if one of 𝑎^[𝑗](𝑗=1,2,…,𝑞) is ∞, say 𝑎^[𝑞]=∞. Then, 𝑞−∑^𝑞_𝑗=11𝑘_𝑗𝑇≤+1−2𝑟,𝑓^𝑞−1_𝑗=1𝑘_𝑗𝑘_𝑗𝑉+1𝑟,𝑎^[𝑗],𝑓+𝑁_𝑘𝑗𝑟,𝑎^[𝑗]+𝑘,𝑓_𝑞𝑘_𝑞+1𝑁_𝑘𝑞𝑟,𝑓+𝑆𝑟,𝑓. (3.18)

Remark 3.6. If 𝑅=+∞, let 𝑞=𝑟+𝑡+𝑠 and 𝑘_𝑗≡𝑘(𝑗=1,2,…,𝑟), 𝑘_𝑗≡𝑙(𝑗=𝑟+1,…,𝑟+𝑡) and 𝑘_𝑗≡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∈ℂ_𝑅,0<𝑅≤+∞) be an 𝐸-valued meromorphic function of order 𝜆(𝑓),0<𝜆(𝑓)≤+∞. If there exist distinct elements 𝑎^[1],𝑎^[2],…,𝑎^[𝑟];𝑏^[1],𝑏^[2],…,𝑏^[𝑡];𝑐^[1],𝑐^[2],…,𝑐^[𝑠] (3.19) 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 𝑟𝑘+𝑘+1𝑡𝑙+𝑙+1𝑠𝑚𝑚+1≤2. (3.20)

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 (i)𝐸-valued evB (exceptional value in the sense of Borel) for 𝑓 for distinct zeros of order ≤𝑘 if limsup_𝑟→+∞log⁺𝑉𝑎,𝑓+𝑁_𝑘(𝑟,𝑎)log𝑈(𝑟)<1; (3.21)(ii)𝐸-valued evB for 𝑓 for distinct zeros if limsup_𝑟→+∞log⁺𝑉𝑎,𝑓+𝑁(𝑟,𝑎)log𝑈(𝑟)<1; (3.22)(iii)𝐸-valued evB for 𝑓 (for the whole aggregate of zeros) if limsup_𝑟→+∞log⁺𝑉𝑎,𝑓+𝑁(𝑟,𝑎)log𝑈(𝑟)<1. (3.23)

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 𝑘_𝑗(𝑗=1,2,…,𝑞) any system such that 𝑘_𝑗 is a positive integer or +∞. If 𝑎^[𝑗] is an 𝐸-valued evB for 𝑓 for distinct zeros of order ≤𝑘_𝑗(𝑗=1,2…,𝑞), then ^𝑞_𝑗=111−𝑘_𝑗+1≤2. (3.24)

Proof. By Corollary 3.5, we have 𝑞−^𝑞_𝑗=11𝑘_𝑗𝑇≤+1−2𝑟,𝑓^𝑞_𝑗=1𝑘_𝑗𝑘_𝑗𝑉+1𝑟,𝑎^[𝑗]+𝑁_𝑘𝑗𝑟,𝑎^[𝑗]+𝑆𝑟,𝑓. (3.25) By hypothesis, there exists a positive number 𝜂<1 such that 𝑉𝑟,𝑎^[𝑗]+𝑁_𝑘𝑗𝑟,𝑎^[𝑗]<𝑈^𝜂(𝑟),𝑗=1,2,…,𝑞. (3.26) Using (3.25) to (3.26), we have ^𝑞_𝑗=111−𝑘_𝑗𝑇≤+1−2𝑟,𝑓^𝑞_𝑗=1𝑘_𝑗𝑘_𝑗𝑈+1^𝜂(𝑟)+𝑆𝑟,𝑓. (3.27) If ∑^𝑞_𝑗=1(1−(1/(𝑘_𝑗+1)))>2, then by Theorem 1.4 and (3.27), we can get a contradiction. So ^𝑞_𝑗=111−𝑘_𝑗+1≤2. (3.28)

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).