Borel Directions and Uniqueness of Meromorphic Functions

Abstract

We investigate the relationship between Borel directions and uniqueness of meromorphic functions and obtain some results of meromorphic functions sharing four distinct values IM and one set in an angular domain containing a Borel line. Our result is an improvement of a recent theorem given by Long and Wu (2012).

1. Introduction and Main Results

We use ℂ to denote the open complex plane, ℂ(=ℂ⋃{∞}) to denote the extended complex plane, and Ω (⊂ℂ) to denote an angular domain. The fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used in [1, 2]. In addition, the order of meromorphic function 𝑓 is defined by 𝜌=𝜌𝑓=limsup_𝑟→∞log𝑇𝑟,𝑓.log𝑟 (1)

Let 𝑆 be a set of distinct elements in ℂ and Ω∶={𝑧∶𝛼≤arg𝑧≤𝛽}⊆ℂ. Define 𝐸=𝑆,Ω,𝑓_𝑎∈𝑆𝑧∈Ω∣𝑓_𝑎(𝑧)=0,countingmultiplicities,𝐸=𝑆,Ω,𝑓_𝑎∈𝑆𝑧∈Ω∣𝑓_𝑎(𝑧)=0,ignoringmultiplicities, (2) where 𝑓_𝑎(𝑧)=𝑓(𝑧)−𝑎 if 𝑎∈ℂ and 𝑓_∞(𝑧)=1/𝑓(𝑧).

Let 𝑓 and 𝑔 be two nonconstant meromorphic functions in ℂ. If 𝐸(𝑆,Ω,𝑓)=𝐸(𝑆,Ω,𝑔), then we say 𝑓 and 𝑔 share the set 𝑆 CM (counting multiplicities) in Ω. If 𝐸(𝑆,Ω,𝑓)=𝐸(𝑆,Ω,𝑔), then we say 𝑓 and 𝑔 share the set 𝑆 IM (ignoring multiplicities) in Ω. In particular, when 𝑆={𝑎}, 𝑎∈ℂ, we say 𝑓 and 𝑔 share the value 𝑎 CM in Ω if 𝐸(𝑆,Ω,𝑓)=𝐸(𝑆,Ω,𝑔), and we say 𝑓 and 𝑔 share the value 𝑎 IM in Ω if 𝐸(𝑆,Ω,𝑓)=𝐸(𝑆,Ω,𝑔). When Ω=ℂ, we give the simple notation as before, 𝐸(𝑆,𝑓),𝐸(𝑆,𝑓), and so on (see [3]).

Nevanlinna (see [4]) proved the following well-known theorems.

Theorem 1 (see [4]). If 𝑓 and 𝑔 are two nonconstant meromorphic functions that share five distinct values 𝑎₁,𝑎₂,𝑎₃,𝑎₄,𝑎₅ IM in ℂ, then 𝑓(𝑧)≡𝑔(𝑧).

Theorem 2 (see [4]). If 𝑓 and 𝑔 are two distinct nonconstant meromorphic functions that share four distinct values 𝑎₁,𝑎₂,𝑎₃,𝑎₄ CM in ℂ, then 𝑓 is a Möbius transformation of 𝑔, two of the shared values, say 𝑎₁ and 𝑎₂, are Picard values, and the cross ratio (𝑎₁,𝑎₂,𝑎₃,𝑎₄)=−1.

After their very work, many investigations studied the uniqueness of meromorphic functions with shared values in the whole complex plane (see [5]). Zheng studied the uniqueness problem under the condition that five values and four values are shared in some angular domain in ℂ around 2003 (see [6, 7]). It is an interesting topic to investigate the uniqueness with shared values in the angular domain; see [3, 6–12]. The basic notations and definitions of meromorphic functions in an angular domain will be introduced as follows (see [1, 6, 7]).

Let 𝑓 be a meromorphic function on the angular domain Ω(𝛼,𝛽)={𝑧∶𝛼≤arg𝑧≤𝛽} and 0<𝛽−𝛼≤2𝜋. Define 𝐴_𝛼,𝛽=𝜔𝑟,𝑓𝜋^𝑟₁1𝑡^𝜔−𝑡^𝜔𝑟^2𝜔×log⁺𝑓𝑡𝑒^𝑖𝛼+log⁺𝑓𝑡𝑒^𝑖𝛽𝑑𝑡𝑡,𝐵_𝛼,𝛽𝑟,𝑓=2𝜔𝜋𝑟^𝜔𝛽_𝛼log⁺𝑓𝑟𝑒^𝑖𝜃()𝐶sin𝜔𝜃−𝛼𝑑𝜃,_𝛼,𝛽𝑟,𝑓=2_{𝑏1<𝜇<𝑟}1|𝑏_𝜇|^𝜔−|𝑏_𝜇|^𝜔𝑟^2𝜔sin𝜔𝜃_𝜇,𝑆−𝛼_𝛼,𝛽𝑟,𝑓=𝐴_𝛼,𝛽𝑟,𝑓+𝐵_𝛼,𝛽𝑟,𝑓+𝐶_𝛼,𝛽𝑟,𝑓, (3) where 𝜔=𝜋/(𝛽−𝛼) and 𝑏_𝜇=|𝑏_𝜇|𝑒^𝑖𝜃𝜇(𝜇=1,2,…) are the poles of 𝑓 on Ω(𝛼,𝛽) counted according to their multiplicities. 𝑆_𝛼,𝛽(𝑟,𝑓) is called the Nevanlinna’s angular characteristic, and 𝐶_𝛼,𝛽(𝑟,𝑓) is called the angular counting function of the poles of 𝑓 on Ω(𝛼,𝛽), and 𝐶_𝛼,𝛽(𝑟,𝑓) is the reduced function of 𝐶_𝛼,𝛽(𝑟,𝑓). Similarly, when 𝑎≠∞, we will use the notations 𝐴_𝛼,𝛽(𝑟,1/(𝑓−𝑎)), 𝐵_𝛼,𝛽(𝑟,1/(𝑓−𝑎)), 𝐶_𝛼,𝛽(𝑟,1/(𝑓−𝑎)), 𝑆_𝛼,𝛽(𝑟,1/(𝑓−𝑎)), and so on.

It is well known that angular distribution is an interesting topic of value distribution of meromorphic function in complex analysis, and Borel directions played an important role in the topic of angular distribution (see [13–24]). Valiron [16] proved that every meromorphic function of finite order 𝜌>0 has at least one Borel direction of order 𝜌. Chuang [25] investigated the existence of Borel direction of meromorphic function of infinite order. To state the Chuang’s results, we will introduce the definition as follows.

Definition 3 (see [25]). Let 𝑓 be a meromorphic function of infinite order, 𝜌(𝑟) is a real function satisfying the following conditions: (i)𝜌(𝑟) is continuous, nondecreasing for 𝑟≥𝑟₀ and 𝜌(𝑟)→∞ as 𝑟→∞; (ii)lim_𝑟→∞log𝑈(𝑅)(𝑟)𝑟log𝑈=1,𝑅=𝑟+(𝑟),log𝑈 (4) where 𝑈(𝑟)=𝑟^𝜌(𝑟)(𝑟≥𝑟₀); (iii) limsup_𝑟→∞log𝑇𝑟,𝑓log𝑈(𝑟)=1. (5)Then 𝜌(𝑟) is called infinite order of meromorphic function 𝑓. This definition is given by Chuang [25].

Let 𝜌(𝑟) be infinite order of meromorphic function 𝑓; we will denote by 𝑀(𝜌(𝑟)) the set of meromorphic function 𝑔 satisfying limsup_𝑟→∞(log𝑇(𝑟,𝑔)/𝜌(𝑟)log𝑟)≤1; that is, 𝑀𝜌(𝑟)∶=𝑔∶limsup_𝑟→∞log𝑇𝑟,𝑔𝜌(𝑟)log𝑟≤1. (6)

Let 𝛼<𝛽, 𝛽−𝛼<2𝜋, 𝑟>0, and Ω(𝛼,𝛽,𝑟)∶={𝑧∶𝛼≤arg𝑧≤𝛽,0<|𝑧|≤𝑟}. The definition of Borel direction of meromorphic functions 𝑓 of infinite order 𝜌(𝑟) is given as follows.

Definition 4 (see [25]). Let 𝑓 be meromorphic functions of infinite order 𝜌(𝑟); if for any 𝜀(0<𝜀<𝜋), the equality limsup_𝑟→∞log𝑛Ω(𝜃−𝜀,𝜃+𝜀,𝑟),𝑓=𝑎𝜌(𝑟)log𝑟=1 (7) holds for any complex number 𝑎∈ℂ, at most except two exceptions, where 𝑛(Ω(𝜃−𝜀,𝜃+𝜀,𝑟),𝑓=𝑎) is the counting function of zeros of the function 𝑓−𝑎 in the angular domain Ω(𝜃−𝜀,𝜃+𝜀), counting multiplicities, then the ray arg𝑧=𝜃 is called a Borel direction of 𝜌(𝑟) order of meromorphic function 𝑓.

Remark 5. Chuang [25] proved that every meromorphic function 𝑓 of infinite order 𝜌(𝑟) has at least one Borel direction of infinite order 𝜌(𝑟).

In 2012, Long and Wu [26] investigated the problem concerning Borel direction and shared value of meromorphic functions and obtained the following theorems.

Theorem 6 (see [26, Theorem 1.1]). Let 𝑓 be a meromorphic function of infinite order 𝜌(𝑟), and let 𝑔∈𝑀(𝜌(𝑟)), arg𝑧=𝜃(0≤𝜃<2𝜋) be one Borel direction of 𝜌(𝑟) order of meromorphic function 𝑓; let 𝑎_𝑖∈ℂ(𝑖=1,2,3,4,5) be five distinct complex numbers. If 𝑓 and 𝑔 share 𝑎_𝑖(𝑖=1,2,3,4,5) IM in the angular domain Ω(𝜃−𝜀,𝜃+𝜀) for any 𝜀(0<𝜀<𝜋), then 𝑓≡𝑔.

Theorem 7 (see [26, Theorem 1.2]). Let 𝑓 be a meromorphic function of infinite order 𝜌(𝑟), and let 𝑔∈𝑀(𝜌(𝑟)), arg𝑧=𝜃(0≤𝜃<2𝜋) be one Borel direction of 𝜌(𝑟) order of meromorphic function 𝑓; let 𝑎_𝑖∈ℂ(𝑖=1,2,3,4) be four distinct complex numbers. If 𝑓 and 𝑔 share 𝑎_𝑖(𝑖=1,2,3,4) CM in the angular domain Ω(𝜃−𝜀,𝜃+𝜀) for any 𝜀(0<𝜀<𝜋), then 𝑓 is a Möbius transformation of 𝑔.
Thus, a question arises naturally: could the nature of sharing the values 𝛼_𝑖 be further relaxed in Theorems 6 and 7?

In this paper, we will deal with the above question and obtain the following result which is an improvement of Theorem 6.

Theorem 8. Let 𝑓 be a meromorphic function of infinite order 𝜌(𝑟), and let 𝑔∈𝑀(𝜌(𝑟)), arg𝑧=𝜃(0≤𝜃<2𝜋) be one Borel direction of 𝜌(𝑟) order of meromorphic function 𝑓; we assume that 𝑓 and 𝑔 share four distinct values 𝑎_𝑗(𝑗=1,2,3,4) IM in Ω(𝜃−𝜀,𝜃+𝜀), and 𝐸(𝑆,Ω(𝜃−𝜀,𝜃+𝜀),𝑓)⊂𝐸(𝑆,Ω(𝜃−𝜀,𝜃+𝜀),𝑔), for any 𝜀(0<𝜀<𝜋), where 𝑆={𝑏₁,…,𝑏_𝑚}, 𝑚≥1 and 𝑏₁,…,𝑏_𝑚∈ℂ⧵{𝑎₁,𝑎₂,𝑎₃,𝑎₄}. Then 𝑓 and 𝑔 share all values CM; thus, it follows that either 𝑓≡𝑔 or 𝑓 is a Möbius transformation of 𝑔. Furthermore, if the number of the values in 𝑆 is odd, then 𝑓≡𝑔.

Remark 9. The special case 𝑚=1 of this theorem immediately yields Theorem 6. In fact, when 𝑚=1, set 𝑆={𝑎₅}. If 𝑓, 𝑔 share 𝑎₅ IM, which implies 𝐸(𝑆,Ω(𝜃−𝜀,𝜃+𝜀),𝑓)⊂𝐸(𝑆,Ω(𝜃−𝜀,𝜃+𝜀),𝑔), then by Theorem 8, we can get 𝑓≡𝑔.

2. Some Lemmas

To prove our result, we require the following lemmas.

Lemma 10 (see [27]). Let 𝑓 be a nonconstant meromorphic function on Ω(𝛼,𝛽). Then for arbitrary complex number 𝑎, we have 𝑆_𝛼,𝛽1𝑟,𝑓−𝑎=𝑆_𝛼,𝛽𝑟,𝑓+𝜀(𝑟,𝑎), (8) where 𝜀(𝑟,𝑎)=𝑂(1) as 𝑟→∞.

Lemma 11 (see [7, 28]). Suppose that 𝑓 is a nonconstant meromorphic function in one angular domain Ω(𝛼,𝛽) with 0<𝛽−𝛼≤2𝜋; then, for arbitrary 𝑞 distinct 𝑎_𝑗∈ℂ(1≤𝑗≤𝑞), we have 𝑞−2𝑆_𝛼,𝛽𝑟,𝑓≤^𝑞_𝑗=1𝐶_𝛼,𝛽1𝑟,𝑓−𝑎_𝑗+𝑅_𝛼,𝛽𝑟,𝑓, (9) where the term 𝐶_𝛼,𝛽(𝑟,1/(𝑓−𝑎_𝑗)) will be replaced by 𝐶_𝛼,𝛽(𝑟,𝑓) when some 𝑎_𝑗=∞ and 𝑅_𝛼,𝛽𝑟,𝑓=𝐴_𝛼,𝛽𝑓𝑟,^′𝑓+𝐵_𝛼,𝛽𝑓𝑟,^′𝑓+^𝑞_𝑗=1𝐴_𝛼,𝛽𝑓𝑟,^′𝑓−𝑎_𝑗+𝐵_𝛼,𝛽𝑓𝑟,^′𝑓−𝑎_𝑗+𝑂(1). (10)

Lemma 12 (see [27, Page 138]). Let 𝑓 be a nonconstant meromorphic function in the whole complex plane ℂ. Given one angular domain on Ω(𝛼,𝛽). Then, for any 1≤𝑟<𝑅, we have 𝐴_𝛼,𝛽𝑓𝑟,^′𝑓𝑅≤𝐾𝑟^𝜔𝑅₁log⁺𝑇𝑟,𝑓𝑡^1+𝜔𝑑𝑡+log⁺𝑟𝑅𝑅−𝑟+log𝑟,𝐵+1_𝛼,𝛽𝑓𝑟,^′𝑓≤4𝜔𝑟^𝜔𝑚𝑓𝑟,^′𝑓, (11)(12) where 𝜔=𝜋/(𝛽−𝛼) and 𝐾 is a positive constant not depending on 𝑟 and 𝑅.

Remark 13. Nevanlinna conjectured that 𝐴_𝛼,𝛽𝑓𝑟,^′𝑓+𝐵_𝛼,𝛽𝑓𝑟,^′𝑓=𝑜𝑆_𝛼,𝛽𝑟,𝑓 (13) when 𝑟 tends to +∞ outside an exceptional set of finite linear measure, and he proved that 𝐴_𝛼,𝛽(𝑟,𝑓^′/𝑓)+𝐵_𝛼,𝛽(𝑟,𝑓^′/𝑓)=𝑂(1) when the function 𝑓 is meromorphic in ℂ and has finite order. In 1975, Goldberg [28] constructed a counter example to show that (13) is not valid.

Remark 14. From Lemmas 11 and 12, we can get the following conclusion: 𝑅_𝛼,𝛽=𝑂(1)𝑂(𝑟)𝑟,𝑓,𝑓isoffiniteorder,log𝑈,𝑟∉𝐸,𝑓isofinfiniteorder, (14) where 𝑅_𝛼,𝛽(𝑟,𝑓) is stated as in (10), 𝑈(𝑟)=𝑟^𝜌(𝑟),𝜌(𝑟) is infinite order of meromorphic function 𝑓, and 𝐸 is a set of finite linear measure.

Remark 15. From the definition of 𝐴_𝛼,𝛽(𝑟,𝑓), 𝐵_𝛼,𝛽(𝑟,𝑓), 𝐶_𝛼,𝛽(𝑟,𝑓), 𝑆_𝛼,𝛽(𝑟,𝑓), and Lemmas 10–12, we can see that the properties of 𝐶_𝛼,𝛽(𝑟,𝑓),(𝐴+𝐵)_𝛼,𝛽(𝑟,𝑓), and 𝑆_𝛼,𝛽(𝑟,𝑓) are the same as for the more familiar quantities 𝑁(𝑟,𝑓), 𝑚(𝑟,𝑓), and 𝑇(𝑟,𝑓), respectively.

Lemma 16 (see [29]). Let 𝑓 be meromorphic function of infinite order 𝜌(𝑟). Then the ray arg𝑧=𝜃 is one Borel direction of 𝜌(𝑟) order of meromorphic function 𝑓 if and only if 𝑓 satisfies the equality limsup_𝑟→∞log𝑆_{𝜃−𝜖,𝜃+𝜖}𝑟,𝑓𝜌(𝑟)log𝑟=1, (15) for any 𝜖(0<𝜖<𝜋/2).

By using the same argument as in [8, Lemma 1] and [5], we can get the lemma below easily.

Lemma 17. Suppose that 𝑓 is a nonconstant meromorphic function with infinite order 𝜌(𝑟), and the ray arg𝑧=𝜃 is one Borel direction of 𝜌(𝑟) order of meromorphic function 𝑓. Let 𝑃(𝑓)=𝑎₀𝑓^𝑝+𝑎₁𝑓^𝑝−1+⋯+𝑎_𝑝(𝑎₀≠0) be a polynomial of 𝑓 with degree 𝑝, where the coefficients 𝑎_𝑗(𝑗=0,1,…,𝑝) are constants, and let 𝑏_𝑗(𝑗=1,2,…,𝑞) be 𝑞(𝑞≥𝑝+1) distinct finite complex numbers. Then for any 𝜀(0<𝜀<𝜋/2), (𝐴+𝐵)_{𝜃−𝜀,𝜃+𝜀}𝑃𝑓𝑟,⋅𝑓^′𝑓−𝑏₁𝑓−𝑏₂⋯𝑓−𝑏_𝑞=𝑅_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓. (16)

Lemma 18. Suppose that 𝑓 is a nonconstant meromorphic function with infinite order 𝜌(𝑟), and the ray arg𝑧=𝜃 is one Borel direction of 𝜌(𝑟) order of meromorphic function 𝑓. We assume that for any for any 𝜀(0<𝜀<𝜋/2), 𝑓 and 𝑔 share four distinct values 𝑎_𝑗(𝑗=1,2,3,4)  IM in angular domain Ω(𝜃−𝜀,𝜃+𝜀) and 𝑓≢𝑔. Let Ψ=𝑓^′𝑔^′𝑓−𝑔²×𝑓−𝑎₁𝑓−𝑎₂𝑓−𝑎₃𝑓−𝑎₄×𝑔−𝑎₁𝑔−𝑎₂×𝑔−𝑎₃𝑔−𝑎₄⁻¹. (17) Then, 𝑆_{𝜃−𝜀,𝜃+𝜀}(𝑟,Ψ)=𝑅_{𝜃−𝜀,𝜃+𝜀}(𝑟,𝑓)+𝑅_{𝜃−𝜀,𝜃+𝜀}(𝑟,𝑔).

Proof. Since the ray arg𝑧=𝜃 is one Borel direction of meromorphic function 𝑓 of 𝜌(𝑟) order, thus, for any 𝜀(0<𝜀<𝜋/2), we can assume that 𝑧₀∈Ω(𝜃−𝜀,𝜃+𝜀) and 𝑓(𝑧₀)=𝑎₁ (or 𝑎₂,𝑎₃,𝑎₄) with multiplicity 𝑝 and 𝑔(𝑧₀)=𝑎₁ (or 𝑎₂,𝑎₃,𝑎₄) with multiplicity 𝑞. From (17), we can get Ψ(𝑧)=𝑂(𝑧−𝑧₀)^{2min(𝑝,𝑞)−2}. (18) Hence, Ψ is analytic in Ω(𝜃−𝜀,𝜃+𝜀). By Lemmas 12 and 17, we have 𝑆_{𝜃−𝜀,𝜃+𝜀}(𝑟,Ψ)=(𝐴+𝐵)_{𝜃−𝜀,𝜃+𝜀}()≤()𝑟,Ψ𝐴+𝐵_{𝜃−𝜀,𝜃+𝜀}×𝑓𝑟,^′𝑓−𝑎₂𝑓−𝑎₃𝑓−𝑎₄+(𝐴+𝐵)_{𝜃−𝜀,𝜃+𝜀}×𝑓𝑟,^′𝑓−𝑎₁𝑓−𝑎₂𝑓−𝑎₃+(𝐴+𝐵)_{𝜃−𝜀,𝜃+𝜀}×𝑓𝑟,^′𝑓−𝑎₁𝑓−𝑎₂𝑓−𝑎₃𝑓−𝑎₄+(𝐴+𝐵)_{𝜃−𝜀,𝜃+𝜀}×𝑓𝑟,^′𝑃₁𝑓𝑓−𝑎₁𝑓−𝑎₂𝑓−𝑎₃𝑓−𝑎₄+(𝐴+𝐵)_{𝜃−𝜀,𝜃+𝜀}×𝑔𝑟,^′𝑔−𝑎₂𝑔−𝑎₃𝑔−𝑎₄+(𝐴+𝐵)_{𝜃−𝜀,𝜃+𝜀}×𝑔𝑟,^′𝑔−𝑎₁𝑔−𝑎₂𝑔−𝑎₃+(𝐴+𝐵)_{𝜃−𝜀,𝜃+𝜀}×𝑔𝑟,^′𝑔−𝑎₁𝑔−𝑎₂𝑔−𝑎₃𝑔−𝑎₄+(𝐴+𝐵)_{𝜃−𝜀,𝜃+𝜀}×𝑔𝑟,^′𝑃₂𝑔𝑔−𝑎₁𝑔−𝑎₂𝑔−𝑎₃𝑔−𝑎₄+𝑂(1)=𝑅_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓+𝑅_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑔, (19) where 𝑃₁(𝑓) is a polynomial of degree no more than 2 in 𝑓 and 𝑃₂(𝑔) is a polynomial of degree no more than 2 in 𝑔.
Thus, we complete the proof of this lemma.

3. Proof of Theorem 8

Proof. Since 𝑓 is a meromorphic function of infinite order 𝜌(𝑟) and arg𝑧=𝜃(0≤𝜃<2𝜋) is one Borel direction of 𝜌(𝑟) order of the meromorphic function 𝑓, by Lemma 16, we can get for any 𝜀(0<𝜀<𝜋)limsup_𝑟→∞log𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓𝜌(𝑟)log𝑟=1. (20) And since 𝑔∈𝑀(𝜌(𝑟)), we have limsup_𝑟→∞log𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑔𝜌(𝑟)log𝑟≤1. (21) Set 𝑅(𝑟)=𝑂(𝜌(𝑟)log𝑟) as 𝑟→∞,𝑟∉𝐸 where 𝐸 is a set of finite linear measure; then 𝑅(𝑟)=𝑜(𝑆_{𝜃−𝜀,𝜃+𝜀}(𝑟,𝑓)) as 𝑟→∞, 𝑟∉𝐸.
Since 𝑓 and 𝑔 share four distinct values 𝑎_𝑗(𝑗=1,2,3,4) IM in Ω(𝜃−𝜀,𝜃+𝜀), suppose that 𝑓≢𝑔 and none of the 𝑎_𝑗(𝑗=1,2,3,4) is ∞. By Lemma 11, we have 2𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓≤⁴_𝑗=1𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑎_𝑗+𝑅_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓≤𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑔+𝑅_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓≤𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓+𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑔+𝑅_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓, (22) and by interchanging 𝑓 and 𝑔, we can get that 𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓=𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑔+𝑅_{𝜃−𝜀,𝜃+𝜀},𝑆𝑟,𝑓_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑔=𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓+𝑅_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑔. (23) Thus, it follows 𝑅_{𝜃−𝜀,𝜃+𝜀}(𝑟,𝑔)=𝑅_{𝜃−𝜀,𝜃+𝜀}(𝑟,𝑓)=𝑅(𝑟) and ⁴_𝑗=1𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑎_𝑗=2𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓+𝑅(𝑟). (24)
Let Ψ be the function expressed in Lemma 18. Then, Ψ≢0. By Lemma 11 and (24), for any 𝑏_𝑗∈𝑆(𝑗=1,2,…,𝑚), we have 3𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓≤⁴_𝑗=1𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑎_𝑗+𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑏_𝑗+𝑅_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓≤2𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓+𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑏_𝑗+𝑅(𝑟); (25) that is, 𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑏_𝑗=𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓+𝑅(𝑟). (26) Similarly, we have 𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑔−𝑏_𝑗=𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑔+𝑅(𝑟). (27) From (26) and (27), it follows (𝐴+𝐵)_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑏_𝑗=𝑅(𝑟),(𝐴+𝐵)_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑔−𝑏_𝑗=𝑅(𝑟), (28) for any 𝑏_𝑗∈𝑆(𝑗=1,2,…,𝑚).
Set Ψ₁∶=𝑔−𝑏₁⋯𝑔−𝑏_𝑚𝑓−𝑏₁⋯𝑓−𝑏_𝑚⋅𝑔^′𝑓−𝑔𝑔−𝑎₁⋯𝑔−𝑎₄^𝑚,Ψ₂∶=𝑓−𝑏₁⋯𝑓−𝑏_𝑚𝑔−𝑏₁⋯𝑔−𝑏_𝑚⋅𝑓^′(𝑓−𝑔)(𝑓−𝑎₁)⋯(𝑓−𝑎₄)^𝑚. (29) By Lemma 17 and (28), we can get that (𝐴+𝐵)_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑏_𝑗⋅𝑔^′𝑓−𝑔𝑔−𝑏_𝑗𝑔−𝑎₁⋯𝑔−𝑎₄=𝑅(𝑟),(𝐴+𝐵)_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑔−𝑏_𝑗⋅𝑓^′𝑓−𝑔𝑓−𝑏_𝑗𝑓−𝑎₁⋯𝑓−𝑎₄=𝑅(𝑟). (30) Thus, it follows that (𝐴+𝐵)_{𝜃−𝜀,𝜃+𝜀}(𝑟,Ψ_𝑗)=𝑅(𝑟), 𝑗=1,2. From (28), we see that “almost all” of poles and 𝑏_𝑗-points of 𝑓 and 𝑔 in the angular domain Ω(𝜃−𝜀,𝜃+𝜀) are simple. Since 𝑓, 𝑔 share the four distinct values 𝑎_𝑗, 𝑗=1,2,3,4 in the angular domain Ω(𝜃−𝜀,𝜃+𝜀) and 𝐸(𝑆,Ω(𝜃−𝜀,𝜃+𝜀),𝑓)⊂𝐸(𝑆,Ω(𝜃−𝜀,𝜃+𝜀),𝑔), we can easily get that 𝐶_{𝜃−𝜀,𝜃+𝜀}(𝑟,Ψ₁)=𝑅(𝑟). Therefore, we have 𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,Ψ₁=𝑅(𝑟). (31)
Since Ψ₁Ψ₂≡Ψ^𝑚, we can have 𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,Ψ₂=𝑅(𝑟). (32)
Let Ξ_Ω^𝑝𝑞(𝑎_𝑗) be the set of those 𝑎_𝑗-points of 𝑓 and 𝑔 in the angular domain Ω(𝜃−𝜀,𝜃+𝜀) in which the multiplicities of 𝑓 and 𝑔 at these points are 𝑝 and 𝑞, respectively. For any 𝑧₀∈Ξ_Ω^𝑝𝑞(𝑎₁), by simple computation, we have Ψ₁𝑧₀=𝑓𝑞⋅^′𝑧₀−𝑔^′𝑧₀𝑎₁−𝑎₂𝑎₁−𝑎₃𝑎₁−𝑎₄^𝑚,Ψ₂𝑧₀=𝑓𝑝⋅^′(𝑧₀)−𝑔^′(𝑧₀)(𝑎₁−𝑎₂)(𝑎₁−𝑎₃)(𝑎₁−𝑎₄)^𝑚. (33) Hence, 1𝑞^𝑚Ψ₁𝑧₀−1𝑝^𝑚Ψ₂𝑧₀=0. (34) Similarly, we can see that (34) holds for any 𝑧₀∈Ξ_Ω^𝑝𝑞(𝑎_𝑗),𝑗=2,3,4.
Now, two cases will be considered below.
Case  1. Suppose that Ψ^𝑝𝑞∶=(1/𝑞^𝑚)Ψ₁−(1/𝑝^𝑚)Ψ₂≢0, for all positive integers 𝑝, 𝑞.
First, we use 𝐶^𝑝𝑞_{𝜃−𝜀,𝜃+𝜀}(𝑟,1/(𝑓−𝑎_𝑗)) to denote the counting function of 𝑓 in Ω(𝜃−𝜀,𝜃+𝜀) with respect to the set Ξ_Ω^𝑝𝑞(𝑎_𝑗), and we also use 𝐶^𝑝𝑞_{𝜃−𝜀,𝜃+𝜀}(𝑟,1/(𝑓−𝑎_𝑗)) to denote the corresponding reduced counting function. Thus, we have 𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑎_𝑗=^∞_𝑝,𝑞=1𝐶^𝑝𝑞_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑎_𝑗,𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑎_𝑗=^∞_𝑝,𝑞=1𝐶^𝑝𝑞_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑎_𝑗. (35)
From the above two equations and (28)–(32), we can see that 𝑆_{𝜃−𝜀,𝜃+𝜀}(𝑟,Ψ^𝑝𝑞)=𝑅(𝑟). And by (34) each zero of 𝑓−𝑎_𝑗 is a zero of Ψ^𝑝𝑞, so with the help of Lemma 10 and Ψ^𝑝𝑞≢0, we can get 𝐶^𝑝𝑞_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑎_𝑗≤𝐶^𝑝𝑞_{𝜃−𝜀,𝜃+𝜀}1𝑟,Ψ^𝑝𝑞≤𝑆_{𝜃−𝜀,𝜃+𝜀}1𝑟,Ψ^𝑝𝑞≤𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,Ψ^𝑝𝑞+𝑂(1)=𝑅(𝑟), (36) for some 𝑝,𝑞. Since 𝑆_{𝜃−𝜀,𝜃+𝜀}(𝑟,𝑓)=𝑆_{𝜃−𝜀,𝜃+𝜀}(𝑟,𝑔)+𝑅(𝑟), it follows 𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑎_𝑗=_{max𝑝,𝑞≥5}𝐶^𝑝𝑞_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑎_𝑗≤1+𝑅𝑟,𝑓5_{max𝑝,𝑞≥5}𝐶^𝑝𝑞_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑎_𝑗+15_{max𝑝,𝑞≥5}𝐶^𝑝𝑞_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑔−𝑎_𝑗(𝑟)≤1+𝑅5𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑓−𝑎_𝑗+𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑔−𝑎_𝑗(𝑟)≤2+𝑅5𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓+𝑅(𝑟),𝑗=1,2,3,4. (37) From the above inequality and (24), we can get 2𝑆_{𝜃−𝜀,𝜃+𝜀}≤8𝑟,𝑓5𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑓+𝑅(𝑟). (38) Since 𝑓 is of infinite order 𝜌(𝑟) and 𝑅(𝑟)=𝑂(𝜌(𝑟)log𝑟), we can get a contradiction to (38).
Case  2. Suppose that Ψ^𝑝𝑞∶=(1/𝑞^𝑚)Ψ₁−(1/𝑝^𝑚)Ψ₂≡0, for some positive integers 𝑝, 𝑞. From the definitions of Ψ₁ and Ψ₂, we have 𝑝𝑞^𝑚⋅𝑔−𝑏₁²⋯𝑔−𝑏_𝑚²𝑓−𝑏₁²⋯𝑓−𝑏_𝑚²≡𝑓^′𝑔−𝑎₁⋯𝑔−𝑎₄𝑔^′𝑓−𝑎₁⋯𝑓−𝑎₄^𝑚. (39)
Next, we take the following two subcases into consideration.
Subcase  2.1. Suppose that 𝑝≠𝑞. Without loss of generality, we may assume that 𝑝<𝑞. For some two positive integers 𝑝₁ and 𝑞₁, if 𝑧₁∈Ξ^𝑝1𝑞1_Ω(𝑎_𝑗) for some 𝑗∈{1,2,3,4}, then (39) implies that 𝑝/𝑞=𝑝₁/𝑞₁. Hence, 𝑞₁>𝑝₁≥1, and 𝑞₁≥2 which means that any 𝑎_𝑗-points (𝑗=1,2,3,4) of 𝑔 in Ω(𝜃−𝜀,𝜃+𝜀) are multiple. By Lemma 11 and (24), we can get 2𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑔=⁴_𝑗=1𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑔−𝑎_𝑗(𝑟)≤1+𝑅2⁴_𝑗=1𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑔−𝑎_𝑗+𝑅(𝑟)≤2𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑔+𝑅(𝑟); (40) it follows that 𝑆_{𝜃−𝜀,𝜃+𝜀}𝑟,𝑔=𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑔−𝑎_𝑗(𝑟),𝐶+𝑅_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑔−𝑎_𝑗=2𝐶_{𝜃−𝜀,𝜃+𝜀}1𝑟,𝑔−𝑎_𝑗+𝑅(𝑟). (41)(42)
From (41) and (42), we can see that “almost all” of 𝑎_𝑗-points of 𝑔 have multiplicity 2, and “almost all” of 𝑎_𝑗-points of 𝑓 are simple in Ω(𝜃−𝜀,𝜃+𝜀). Without loss of generality, we may assume that 𝑓 and 𝑔 attain the values 𝑎₃ and 𝑎₄ in Ω(𝜃−𝜀,𝜃+𝜀). Set Φ₁∶=2𝑓^′𝑓−𝑎₄𝑓−𝑎₁𝑓−𝑎₂𝑓−𝑎₃−𝑔^′𝑔−𝑎₄𝑔−𝑎₁𝑔−𝑎₂𝑔−𝑎₃,Φ₂∶=2𝑓^′𝑓−𝑎₃𝑓−𝑎₁𝑓−𝑎₂𝑓−𝑎₄−𝑔^′𝑔−𝑎₃𝑔−𝑎₁𝑔−𝑎₂𝑔−𝑎₄. (43)
Since Φ_𝑖(𝑖=1,2) is analytic at the poles of 𝑓 and of 𝑔 and also at those common 𝑎_𝑗-points of 𝑓 and 𝑔 which have multiplicity 1 with respect to 𝑓 and multiplicity 2 with respect to 𝑔, by Lemma 17, we have 𝑆_{𝜃−𝜀,𝜃+𝜀}(𝑟,Φ_𝑖)=𝑅(𝑟),𝑖=1,2. If Φ_𝑖≢0, then 𝐶_{𝜃−𝜀,𝜃+𝜀}(𝑟,1/(𝑔−𝑎₄))≤𝐶_{𝜃−𝜀,𝜃+𝜀}(𝑟,1/𝜙₁)=𝑅(𝑟), which contradicts to (41). Then, Φ₁≡0. Similarly, we have Φ₂≡0. Therefore, from the definitions of Φ₁ and Φ₂, we have 𝑓−𝑎₄𝑓−𝑎₃²≡𝑔−𝑎₄𝑔−𝑎₃². (44)
Since 𝑓≢𝑔, from (44), we have 𝑓−𝑎₄𝑓−𝑎₃≡−𝑔−𝑎₄𝑔−𝑎₃, (45) which implies that 𝑓 and 𝑔 share 𝑎₃,𝑎₄ CM in Ω(𝜃−𝜀,𝜃+𝜀). Since 𝑓 and 𝑔 assume the value 𝑎₃, there exist positive integers 𝑝₁,𝑞₁ such that Ξ^𝑝1𝑞1_Ω(𝑎₃)≠∅. From the considerations above we get 𝑞₁>𝑝₁, contradicting the fact that 𝑓 and 𝑔 share 𝑎₃ CM.
Subcase  2.2. Suppose that 𝑝=𝑞.
In this subcase, (39) becomes (𝑔−𝑏₁)²⋯(𝑔−𝑏_𝑚)²(𝑓−𝑏₁)²⋯(𝑓−𝑏_𝑚)²≡𝑓^′(𝑔−𝑎₁)⋯(𝑔−𝑎₄)𝑔^′𝑓−𝑎₁⋯(𝑓−𝑎₄)^𝑚. (46) which implies that 𝑓 and 𝑔 share the four values 𝑎_𝑗(𝑗=1,2,3,4) CM in Ω(𝜃−𝜀,𝜃+𝜀). Then by applying Theorem 7, 𝑔 is a Möbius transformation of 𝑓. Furthermore, two of the four values, say 𝑎₁, 𝑎₂, are Picard exceptional values of 𝑓 and 𝑔 in Ω(𝜃−𝜀,𝜃+𝜀). Set Λ₁𝑓∶=^′𝑓−𝑎₄𝑓−𝑎₁𝑓−𝑎₂𝑓−𝑎₃−𝑔^′𝑔−𝑎₄𝑔−𝑎₁𝑔−𝑎₂𝑔−𝑎₃,Λ₂𝑓∶=^′𝑓−𝑎₃𝑓−𝑎₁𝑓−𝑎₂𝑓−𝑎₄−𝑔^′𝑔−𝑎₃𝑔−𝑎₁𝑔−𝑎₂𝑔−𝑎₄. (47) Similar to the discussion in Subcase 2.1 for Λ₁, Λ₂, we can get 𝑓−𝑎₃𝑓−𝑎₄≡−𝑔−𝑎₃𝑔−𝑎₄. (48) We define the Möbius transformations 𝑇, 𝐻, and 𝐽 by 𝑇(𝜔)∶=𝑤−𝑎₃𝑤−𝑎₄,𝐻(𝑤)∶=−𝑤,𝐽∶=𝑇⁻¹∘𝐻∘𝑇. (49) Then, we have 𝑇∘𝑓=−𝑇∘𝑔;hence,𝑔=𝐽∘𝑓. (50) Obviously, 𝑎₃ and 𝑎₄ are the fixed points of 𝐽. Therefore, there exist no fixed points of 𝐽 in the set 𝑆. Let some 𝑏∈𝑆 be given. Then, in vies of 𝑏≠𝑎₁,𝑎₂, there exists a 𝑧₀∈ℂ such that 𝑏=𝑓(𝑧₀), and from 𝐸(𝑆,Ω(𝜃−𝜀,𝜃+𝜀),𝑓)⊆𝐸(𝑆,Ω(𝜃−𝜀,𝜃+𝜀),𝑔) we obtain 𝐽(𝑏)=𝐽𝑓𝑧₀=𝑔𝑧₀∈𝑆. (51) So 𝑆 is invariant under 𝐽. Furthermore, we have 𝐽∘𝐽=𝐼 where 𝐼 denotes the identical transformation. Hence, 𝑆 must contain an even number of values. Thus, the proof of Theorem 8 is completed.

Acknowledgments

This project is supported by the National Natural Science Foundation of China (project nos. 11171184, 61202313) and the Natural Science Foundation of Jiangxi Province in China (nos. 2010GQS0119 and 20132BAB211001).