The Shared Set and Uniqueness of Meromorphic Functions on Annuli

Abstract

The purpose of this paper is to deal with the shared set and uniqueness of meromorphic functions on annulus. The set of this paper is different from the set of the paper by Cao and Deng, and our theorems are improvement of the results given by Cao and Deng.

1. Introduction

In 1929, Nevanlinna [1] first investigated the uniqueness of meromorphic functions in the whole complex plane and obtained the well-known result—5 IM theorem of two meromorphic functions sharing five distinct values.

After his theorem, there are vast references on the uniqueness of meromorphic functions sharing values and sets in the whole complex plane, the unit disc; and angular domain (see [2–8]).

The notations of the Nevanlinna theory such as 𝑇(𝑟,𝑓), 𝑚(𝑟,𝑓), and 𝑁(𝑟,𝑓) were usually used in those papers (see [5, 9, 10]). We use ℂ to denote the open complex plane, ℂ∶=ℂ∪{∞} to denote the extended complex plane, and 𝕏 to denote the subset of ℂ. Let 𝑆 be a set of distinct elements in ℂ and 𝕏⊆ℂ. Define 𝐸^𝕏𝑆,𝑓=_𝑎∈𝑆𝑧∈𝕏∣𝑓_𝑎(𝑧)=0,countingmultiplicities,𝐸^𝕏𝑆,𝑓=_𝑎∈𝑆𝑧∈𝕏∣𝑓_𝑎(𝑧)=0,ignoringmultiplicities, (1) where 𝑓_𝑎(𝑧)=𝑓(𝑧)−𝑎 if 𝑎∈ℂ and 𝑓_∞(𝑧)=1/𝑓(𝑧). We also define 𝐸^𝕏₁𝑆,𝑓=_𝑎∈𝑆𝑧∈𝕏∶allthesimplezerosof𝑓_𝑎(𝑧). (2)

For 𝑎∈ℂ, we say that two meromorphic functions 𝑓 and 𝑔 share the value 𝑎𝐶𝑀(𝐼𝑀) in 𝕏 (or ℂ), if 𝑓(𝑧)−𝑎 and 𝑔(𝑧)−𝑎 have the same zeros with the same multiplicities (ignoring multiplicities) in 𝕏 (or ℂ).

The whole complex plane ℂ, the unit disc, and angular domain all can be regarded as simply connected regions those results of the uniqueness of shared values and sets in the above cases can also be regarded as the uniqueness of meromorphic functions in simply connected regions.

Thus, it raises naturally an interesting subject on the uniqueness of the meromorphic functions in the multiply connected region.

The main purpose of this paper is to study the uniqueness of meromorphic functions in doubly connected domains of complex plane ℂ. From the doubly connected mapping theorem [11], we can get that each doubly connected domain is conformally equivalent to the annulus {𝑧∶𝑟<|𝑧|<𝑅}, 0≤𝑟<𝑅≤+∞. There are two cases: (1) 𝑟=0 and 𝑅=+∞ and (2) 0<𝑟<𝑅<+∞; for case (2) the homothety 𝑧↦𝑧/√𝑟𝑅 reduces the given domain to the annulus {𝑧∶1/𝑅₀<|𝑧|<𝑅₀}, where 𝑅₀=𝑅/𝑟. Thus, every annulus is invariant with respect to the inversion 𝑧↦1/𝑧 in two cases. The basic notions of the Nevanlinna theory on annuli will be showed in the next section.

Recently, there are some results on the Nevanlinna Theory of meromorphic functions on the annulus (see [12–19]). In 2005, Khrystiyanyn and Kondratyuk [13, 14] proposed the Nevanlinna theory for meromorphic functions on annuli (see also [20]). Lund and Ye [16] in 2009 studied meromorphic functions on the annuli with the form {𝑧∶𝑅₁<|𝑧|<𝑅₂}, where 𝑅₁≥0 and 𝑅₂≤∞. However, there are few results about the uniqueness of meromorphic functions on the annulus. In 2009 and 2011, Cao et al. [21, 22] investigated the uniqueness of meromorphic functions on annuli sharing some values and some sets and obtained an analog of Nevanlinna's famous five-value theorem as follows.

Theorem 1 ([22, Theorem 3.2] or [21, Corollary 3.3]). Let 𝑓₁ and 𝑓₂ be two transcendental or admissible meromorphic functions on the annulus 𝔸={𝑧∶1/𝑅₀<|𝑧|<𝑅₀}, where 1<𝑅₀≤+∞. Let 𝑎_𝑗 (𝑗=1,2,3,4,5) be five distinct complex numbers in ℂ. If 𝑓₁, 𝑓₂ share 𝑎_𝑗𝐼𝑀 for 𝑗=1,2,3,4,5, then 𝑓₁(𝑧)≡𝑓₂(𝑧).

Remark 2. For the case 𝑅₀=+∞, the assertion was proved by Kondratyuk and Laine [20].

In 2012, Cao and Deng [23] investigated the uniqueness of two meromorphic functions in 𝔸 sharing three or two finite sets; we obtain the following theorems which are an analog of results on ℂ according to Lin and Yi [24].

Theorem 3. Let 𝑓 and 𝑔 be two admissible meromorphic functions in the annulus 𝔸. Put 𝑆₁={0}, 𝑆₂={∞}, and 𝑆₃={𝑤∶𝑃(𝑤)=0}, where 𝑃(𝑤)=𝑎𝑤^𝑛+𝑛(𝑛−1)𝑤²()−()𝑏+2𝑛𝑛−2𝑏𝑤𝑛−1)(𝑛−2², (3) where 𝑛≥5 is an integer and 𝑎 and 𝑏 are two nonzero complex numbers satisfying 𝑎𝑏^𝑛−2≠1,2. If 𝐸(𝑆₂,𝑓)=𝐸(𝑆₂,𝑔) and 𝐸(𝑆_𝑗,𝑓)=𝐸(𝑆_𝑗,𝑔) (𝑗=1,3), then 𝑓≡𝑔.

Theorem 4. Let 𝑓 and 𝑔 be two admissible meromorphic functions in the annulus 𝔸. Put 𝑆₁={∞} and 𝑆₂={𝑤∶𝑃(𝑤)=0}, where 𝑃(𝑤) is stated as in Theorem 3, and 𝑎 and 𝑏 are two nonzero complex numbers satisfying 𝑎𝑏^𝑛−2≠2, 𝑛≥8 is an integer. If 𝐸(𝑆₁,𝑓)=𝐸(𝑆₁,𝑔) and 𝐸(𝑆₂,𝑓)=𝐸(𝑆₂,𝑔), then 𝑓≡𝑔.

In this paper, we will focus on the uniqueness problem of shared set of meromorphic functions on the annuli. In fact, we will study the uniqueness of meromorphic functions on the annuli sharing one set 𝑆={𝑤∈𝔸∶𝑃₁(𝑤)=0}, where 𝑃₁(𝑤)=(𝑛−1)(𝑛−2)2𝑤^𝑛−𝑛(𝑛−2)𝑤^𝑛−1+𝑛(𝑛−1)2𝑤^𝑛−2−𝑐, (4) and 𝑐 is a complex number satisfying 𝑐≠0,1 and we obtain the following results.

Theorem 5. Let 𝑓 and 𝑔 be two admissible meromorphic functions in the annulus 𝔸. If 𝐸^𝔸(𝑆,𝑓)=𝐸^𝔸(𝑆,𝑔) and 𝑛 is an integer ≥11, then 𝑓≡𝑔.

A set 𝑆 is called a unique range set for meromorphic functions on annulus 𝔸, if, for any two nonconstant meromorphic functions 𝑓 and 𝑔, the condition 𝐸^𝔸(𝑆,𝑓)=𝐸^𝔸(𝑆,𝑔) implies 𝑓≡𝑔. We denote by ♯𝑆 the cardinality of a set 𝑆. Thus, from Theorem 5, we can get the following corollary.

Corollary 6. There exists one finite set 𝑆 with ♯𝑆=7, such that any two admissible meromorphic functions 𝑓 and 𝑔 on 𝔸 must be identical if 𝐸^𝔸(𝑆,𝑓)=𝐸^𝔸(𝑆,𝑔).

Theorem 7. Let 𝑓 and 𝑔 be two admissible meromorphic functions in the annulus 𝔸. If 𝑛 is an integer ≥7, 𝐸^𝔸(𝑆,𝑓)=𝐸^𝔸(𝑆,𝑔), and Θ₀(∞,𝑓)>3/4, Θ₀(∞,𝑔)>3/4, then 𝑓≡𝑔.

Corollary 8. There exists one finite set 𝑆 with ♯𝑆=7, such that any two admissible analytic functions 𝑓 and 𝑔 on 𝔸 must be identical if 𝐸^𝔸₁(𝑆,𝑓)=𝐸^𝔸₁(𝑆,𝑔).

Theorem 9. Let 𝑓 and 𝑔 be two admissible meromorphic functions in the annulus 𝔸. If 𝐸^𝔸₁(𝑆,𝑓)=𝐸^𝔸₁(𝑆,𝑔) and 𝑛 is an integer ≥15, then 𝑓≡𝑔.

A set 𝑆 is called a unique range set, with weight 1 for meromorphic functions on annulus 𝔸, if for any two nonconstant meromorphic functions 𝑓 and 𝑔, the condition 𝐸^𝔸₁(𝑆,𝑓)=𝐸^𝔸₁(𝑆,𝑔) implies 𝑓≡𝑔. Thus, from Theorem 9, we can get the following corollary.

Corollary 10. There exists one finite set 𝑆 with ♯𝑆=15, such that any two admissible meromorphic functions 𝑓 and 𝑔 on 𝔸 must be identical if 𝐸^𝔸₁(𝑆,𝑓)=𝐸^𝔸₁(𝑆,𝑔).

Theorem 11. Let 𝑓 and 𝑔 be two admissible meromorphic functions in the annulus 𝔸. Let 𝑛 be an integer ≥9 and 𝑆={𝑤∈𝔸∶𝑃₁(𝑤)=0}, where 𝑃₁(𝑤) and 𝑐 are stated as in Theorem 5. If 𝐸^𝔸₁(𝑆,𝑓)=𝐸^𝔸₁(𝑆,𝑔) and Θ₀(∞,𝑓)>5/6, Θ₀(∞,𝑔)>5/6, then 𝑓≡𝑔.

From Theorem 11, we can get the corollary as follows.

Corollary 12. There exists one finite set 𝑆 with ♯𝑆=9, such that any two admissible analytic functions 𝑓 and 𝑔 on 𝔸 must be identical if 𝐸^𝔸₁(𝑆,𝑓)=𝐸^𝔸₁(𝑆,𝑔).

2. Preliminaries and Some Lemmas

Letting 𝑓 be a meromorphic function on whole plane ℂ, the classical notations of the Nevanlinna theory are denoted as follows:𝑁𝑅,𝑓=^𝑅₀𝑛𝑡,𝑓−𝑛0,𝑓𝑡𝑚=1𝑑𝑡+𝑛0,𝑓log𝑅,𝑅,𝑓2𝜋₀^2𝜋log⁺𝑓𝑅𝑒^𝑖𝜃𝑇,𝑑𝜃,𝑅,𝑓=𝑁𝑅,𝑓+𝑚𝑅,𝑓 (5) where log⁺𝑥=max{log𝑥,0} and 𝑛(𝑡,𝑓) is the counting function of poles of the function 𝑓 in {𝑧∶|𝑧|≤𝑡}.

Letting 𝑓 be a meromorphic function on the annulus 𝔸={𝑧∶1/𝑅₀<|𝑧|<𝑅₀}, where 1<𝑅<𝑅₀≤+∞, the notations of the Nevanlinna theory on annuli had been introduced in [13, 20], such as 𝑁₀(𝑅,𝑓),𝑚₀(𝑅,𝑓),𝑇₀(𝑅,𝑓),…. In addition, we define Θ₀∞,𝑓=1−limsup_𝑅→∞𝑁₀𝑅,𝑓𝑇₀.𝑅,𝑓 (6) We also use 𝑛₁^𝑘)(𝑡,1/(𝑓−𝑎)) (or 𝑛₁^(𝑘(𝑡,1/(𝑓−𝑎))) to denote the counting function of poles of the function 1/(𝑓−𝑎) with multiplicities ≤𝑘 (or >𝑘) in {𝑧∶𝑡<|𝑧|≤1}, with each point being counted only once. Similarly, we have the notations 𝑁₁^𝑘)(𝑡,𝑓), 𝑁₁^(𝑘(𝑡,𝑓), 𝑁₂^𝑘)(𝑡,𝑓), 𝑁₂^(𝑘(𝑡,𝑓), 𝑁₀^𝑘)(𝑡,𝑓), 𝑁₀^(𝑘(𝑡,𝑓).

For a nonconstant meromorphic function 𝑓 on the annulus 𝔸={𝑧∶1/𝑅₀<|𝑧|<𝑅₀}, where 1<𝑅<𝑅₀≤+∞, the following properties will be used in this paper (see [13]): (i)𝑇₀𝑅,𝑓=𝑇₀1𝑅,𝑓, (7)(ii)max𝑇₀𝑅,𝑓₁⋅𝑓₂,𝑇₀𝑓𝑅,¹𝑓₂,𝑇₀𝑅,𝑓₁+𝑓₂≤𝑇₀𝑅,𝑓₁+𝑇₀𝑅,𝑓₂+𝑂(1), (8)(iii)𝑇₀1𝑅,𝑓−𝑎=𝑇₀𝑅,𝑓+𝑂(1),foreveryfixed𝑎∈ℂ. (9)

Khrystiyanyn and Kondratyuk [14] also obtained the lemma on the logarithmic derivative on the annulus 𝔸.

Lemma 13 (see [14], lemma on the logarithmic derivative). Let 𝑓 be a nonconstant meromorphic function on the annulus 𝔸={𝑧∶1/𝑅₀<|𝑧|<𝑅₀}, where 𝑅₀≤+∞, and let 𝜆>0. Then,(i) in the case 𝑅₀=+∞, 𝑚₀𝑓𝑅,^′𝑓=𝑂log𝑅𝑇₀𝑅,𝑓, (10) for 𝑅∈(1,+∞) except for the set Δ_𝑅 such that ∫_Δ𝑅𝑅^𝜆−1𝑑𝑅<+∞;(ii) if 𝑅₀<+∞, then 𝑚₀𝑓𝑅,^′𝑓𝑇=𝑂log⁰𝑅,𝑓𝑅₀,−𝑅 (11) for 𝑅∈(1,𝑅₀) except for the set Δ^′_𝑅 such that ∫_Δ′𝑅(𝑑𝑅/(𝑅₀−𝑅)^𝜆−1)<+∞.

Remark 14. From [14, 20], the conclusions still hold if 𝑚₀(𝑅,𝑓^′/𝑓) is replaced by 𝑚₀(𝑅,𝑓^(𝑘)/𝑓), 𝑘∈𝑁₊.

In 2005, the second fundamental theorem on the annulus 𝔸 was first obtained by Khrystiyanyn and Kondratyuk [14]. Later, Cao et al. [22] introduced other forms of the second fundamental theorem on annuli as follows.

Lemma 15 ([22, Theorem 2.3], the second fundamental theorem). Let 𝑓 be a nonconstant meromorphic function on the annulus 𝔸={𝑧∶1/𝑅₀<|𝑧|<𝑅₀}, where 1<𝑅₀≤+∞. Let 𝑎₁,𝑎₂,…,𝑎_𝑞 be 𝑞 distinct complex numbers in the extended complex plane ℂ. Then, 𝑞−2𝑇₀𝑅,𝑓<^𝑞_𝑗=1𝑁₀1𝑅,𝑓−𝑎_𝑗+𝑆𝑅,𝑓, (12) where (i) in the case 𝑅₀=+∞, 𝑆𝑅,𝑓=𝑂log𝑅𝑇₀𝑅,𝑓, (13) for 𝑅∈(1,+∞) except for the set Δ_𝑅 such that ∫_Δ𝑅𝑅^𝜆−1𝑑𝑅<+∞.
(ii) If 𝑅₀<+∞, then 𝑆𝑇𝑅,𝑓=𝑂log⁰𝑅,𝑓𝑅₀−𝑅, (14) for 𝑅∈(1,𝑅₀) except for the set Δ^′_𝑅 such that ∫_Δ′𝑅(𝑑𝑅/(𝑅₀−𝑅)^𝜆−1)<+∞.

Remark 16. In fact, from the proof of Theorem 2.3 in [22], under the assumptions of Lemma 15, we can get the following conclusion: 𝑞−2𝑇₀<𝑅,𝑓_𝑞𝑗=1𝑁₀1𝑅,𝑓−𝑎_𝑗−𝑁⁰₀1𝑅,𝑓^′+𝑆𝑅,𝑓, (15) where 𝑆(𝑅,𝑓) is stated as in Lemma 15 and 𝑁⁰₀(𝑅,1/𝑓^′) is the counting function for the zeros of 𝑓^′ in 𝔸, where 𝑓 does not take one of the values 𝑎_𝑗 (𝑗=1,2,…,𝑞).

Definition 17. Let 𝑓(𝑧) be a nonconstant meromorphic function on the annulus 𝔸={𝑧∶1/𝑅₀<|𝑧|<𝑅₀}, where 1<𝑅₀≤+∞. The function 𝑓 is called an admissible meromorphic function on the annulus 𝔸 provided that limsup_𝑅→∞𝑇₀𝑅,𝑓log𝑅=∞,1<𝑅<𝑅₀=+∞, (16) or limsup_𝑅→𝑅0𝑇₀𝑅,𝑓−log𝑅₀−𝑅=∞,1<𝑅<𝑅₀<+∞, (17) respectively.

Thus, for an admissible meromorphic function on the annulus 𝔸, 𝑆(𝑅,𝑓)=𝑜(𝑇₀(𝑅,𝑓)) holds for all 1<𝑅<𝑅₀ except for the set Δ_𝑅 or the set Δ^′_𝑅 mentioned in Lemma 13, respectively.

The following result can be derived from the proof of Frank-Reinders' theorem in [25].

Lemma 18. Let 𝑛≥6 and 𝐻(𝑤)=(𝑛−1)(𝑛−2)2𝑤^𝑛−𝑛(𝑛−2)𝑤^𝑛−1+𝑛(𝑛−1)2𝑤^𝑛−2. (18) Then, 𝐻(𝑤) is a unique polynomial for admissible meromorphic functions; that is, for any two admissible meromorphic functions 𝑓 and 𝑔 in 𝔸, 𝐻(𝑓)≡𝐻(𝑔) implies 𝑓≡𝑔.

By a similar discussion to the one in [26], one can obtain a stand and Valiron-Mohoko type result in 𝔸 as follows.

Lemma 19 (see [23]). Let 𝑓 be a nonconstant meromorphic function in 𝔸, 𝑄₁(𝑓) and let 𝑄₂(𝑓) be two mutually prime polynomials in 𝑓 with degree 𝑚 and 𝑛, respectively. Then, 𝑇₀𝑄𝑅,¹𝑓𝑄₂𝑓=max{𝑚,𝑛}𝑇₀𝑅,𝑓+𝑆𝑅,𝑓. (19)

Lemma 20. Suppose 𝑓 is a nonconstant meromorphic function in 𝔸. Then, 𝑁₀1𝑅,𝑓^′≤𝑁₀1𝑅,𝑓+𝑁₀𝑅,𝑓+𝑆𝑅,𝑓+𝑂(1), (20) where 𝑆(𝑅,𝑓) is stated as in Lemma 15.

Proof. Since 𝑚₀1𝑅,𝑓≤𝑚₀1𝑅,𝑓^′+𝑚₀𝑓𝑅,^′𝑓=𝑚₀1𝑅,𝑓^′+𝑆𝑅,𝑓. (21) Then, from properties of 𝑇₀(𝑅,𝑓), we have 𝑇₀𝑅,𝑓−𝑁₀1𝑅,𝑓≤𝑇₀𝑅,𝑓^′−𝑁₀1𝑅,𝑓^′+𝑆𝑟,𝑓+𝑂(1); (22) that is, 𝑁₀1𝑅,𝑓^′≤𝑇₀𝑅,𝑓^′−𝑇₀𝑅,𝑓+𝑁₀1𝑅,𝑓+𝑆𝑟,𝑓+𝑂(1). (23) Since 𝑇₀𝑅,𝑓^′=𝑚₀𝑅,𝑓^′+𝑁₀𝑅,𝑓^′−2𝑚1,𝑓^′≤𝑚₀𝑅,𝑓+𝑚₀𝑓𝑅,^′𝑓+𝑁₀+𝑅,𝑓𝑁₀𝑅,𝑓−2𝑚1,𝑓^′≤𝑇₀𝑅,𝑓+𝑁₀𝑅,𝑓+𝑆𝑅,𝑓+𝑂(1). (24) Then, from (23) and (24), we can get the conclusion of this lemma.

Next, we will give the two main lemmas of this paper as follows.

Lemma 21. Let 𝐹 and 𝐺 be admissible meromorphic functions in 𝐴 satisfying 𝐸^𝔸(𝐹,0)=𝐸^𝔸(𝐺,0) and let 𝑐₁,𝑐₂,…,𝑐_𝑞 be 𝑞 (≥2) distinct nonzero complex numbers. If limsup_{𝑅→∞,𝑅∈𝐼}3𝑁₀(𝑅,𝐹)+^𝑞_𝑗=1𝑁₀⁽²⁾1𝑅,𝐹−𝑐_𝑗+𝑁₀1𝑅,𝐹^′×𝑇₀(𝑅,𝐹)⁻¹<𝑞,limsup_{𝑅→∞,𝑅∈𝐼}3𝑁₀(𝑅,𝐺)+^𝑞_𝑗=1𝑁₀⁽²⁾1𝑅,𝐺−𝑐_𝑗+𝑁₀1𝑅,𝐺^′×𝑇₀(𝑅,𝐺)⁻¹<𝑞, (25) where 𝑁₀⁽²⁾(𝑅,⋅)=𝑁₀(𝑅,⋅)+𝑁₀⁽²(𝑅,⋅), 𝑁₀⁽²(𝑟,⋅)=𝑁₀(𝑟,⋅)−𝑁₀¹⁾(𝑟,⋅), and 𝐼 is some set of 𝑅 of infinite linear measure, then 𝐹=𝑎𝐺+𝑏,𝑐𝐺+𝑑 (26) where 𝑎,𝑏,𝑐,𝑑∈ℂ are constants with 𝑎𝑑−𝑏𝑐≠0.

Proof. Set 𝐹𝐻≡^′′𝐹^′𝐹−2^′𝐹−𝐺^′′𝐺^′𝐺−2^′𝐺. (27)
Supposing that 𝐻≢0, from Lemma 13 and Remark 14, we have 𝑚₀(𝑅,𝐻)=𝑆(𝑅), (28) where 𝑆(𝑅)∶=𝑜{𝑇₀(𝑅)}, 𝑇₀(𝑅)=max{𝑇₀(𝑅,𝐹),𝑇₀(𝑅,𝐺)}. Since 𝐸^𝔸(𝐹,0)=𝐸^𝔸(𝐺,0), and by an elementary calculation, we can conclude that if 𝑧₀ is a common simple zero of 𝐹 and 𝐺 in 𝔸, then 𝐻(𝑧₀)=0. Thus, we have 𝑁₀¹⁾(𝑅)≤𝑁₀1𝑅,𝐻≤𝑇₀(𝑅,𝐻)+𝑂(1)≤𝑁₀(𝑅,𝐻)+𝑆(𝑅), (29) where 𝑁₀¹⁾(𝑅)=𝑁₀¹⁾(𝑅,1/𝐹)=𝑁₀¹⁾(𝑅,1/𝐺). The poles of 𝐻 in 𝔸 can only occur at zeros of 𝐹^′ and 𝐺^′ in 𝔸 or poles of 𝐹 and 𝐺 in 𝔸. Moreover, 𝐻 only has simple zeros in 𝔸. Hence, from (29), we have 𝑁₀¹⁾(𝑅)≤𝑁₀(𝑅,𝐹)+𝑁₀(𝑅,𝐺)+𝑁⁰₀1𝑅,𝐹^′+𝑁⁰₀1𝑅,𝐺^′+^𝑞_𝑗=1𝑁₀⁽²1𝑅,𝐹−𝑐_𝑗+^𝑞_𝑗=1𝑁₀⁽²1𝑅,𝐺−𝑐_𝑗+𝑆(𝑅), (30) where 𝑁⁰₀(𝑅,1/𝐹^′) is the reduced counting function for the zeros of 𝐹^′ in 𝐴, where 𝐹 does not take one of the values 0,𝑐₁,𝑐₂,…,𝑐_𝑞.
Since 𝑁₀1𝑅,𝐹+𝑁₀1𝑅,𝐺=2𝑁₀¹⁾(𝑅)+𝑁₀⁽²1𝑅,𝐹+𝑁₀⁽²1𝑅,𝐺. (31) Then, from (30) and (31), we have 𝑁₀1𝑅,𝐹+𝑁₀1𝑅,𝐺≤2𝑁₀(𝑅,𝐹)+2𝑁₀(𝑅,𝐺)+2𝑁⁰₀1𝑅,𝐹^′+2𝑁⁰₀1𝑅,𝐺^′+𝑁₀⁽²1𝑅,𝐹+𝑁₀⁽²1𝑅,𝐺+2^𝑞_𝑗=1𝑁₀⁽²1𝑅,𝐹−𝑐_𝑗+2^𝑞_𝑗=1𝑁₀⁽²1𝑅,𝐺−𝑐_𝑗+𝑆(𝑅). (32) From Remark 16, we have 𝑞𝑇₀(𝑅,𝐹)≤𝑁₀(𝑅,𝐹)+𝑁₀1𝑅,𝐹+^𝑞_𝑗=1𝑁₀1𝑅,𝐹−𝑐_𝑗−𝑁⁰₀1𝑅,𝐹^′+𝑆(𝑅),𝑟∉𝐸,𝑞𝑇₀(𝑅,𝐺)≤𝑁₀(𝑅,𝐺)+𝑁₀1𝑅,𝐺+^𝑞_𝑗=1𝑁₀1𝑅,𝐺−𝑐_𝑗−𝑁⁰₀1𝑅,𝐺^′+𝑆(𝑅),𝑟∉𝐸, (33) where 𝐸 is a set of 𝑟 of finite linear measure and it needs not to be the same at each occurrence. From (32)-(33), it follows that, for 𝑟∉𝐸, 𝑞𝑇₀(𝑅,𝐹)+𝑇₀(𝑅,𝐺)≤3𝑁₀(𝑅,𝐹)+3𝑁₀(𝑅,𝐺)+^𝑞_𝑗=1𝑁₀1𝑅,𝐹−𝑐_𝑗+^𝑞_𝑗=1𝑁₀1𝑅,𝐺−𝑐_𝑗+2^𝑞_𝑗=1𝑁₀⁽²1𝑅,𝐹−𝑐_𝑗+2^𝑞_𝑗=1𝑁₀⁽²1𝑅,𝐺−𝑐_𝑗+𝑁⁰₀1𝑅,𝐹^′+𝑁⁰₀1𝑅,𝐺^′+𝑁₀⁽²1𝑅,𝐹+𝑁₀⁽²1𝑅,𝐺+𝑆(𝑅). (34) Since ^𝑞_𝑗=1𝑁₀⁽²1𝑅,𝐹−𝑐_𝑗+𝑁₀⁽²1𝑅,𝐹+𝑁⁰₀1𝑅,𝐹^′=𝑁₀1𝑅,𝐹^′_𝑞𝑗=1,𝑁₀⁽²1𝑅,𝐺−𝑐_𝑗+𝑁₀⁽²1𝑅,𝐺+𝑁⁰₀1𝑅,𝐺^′=𝑁₀1𝑅,𝐺^′. (35) From (34)-(35), we can get that, for 𝑅∉𝐸, 𝑞𝑇₀(𝑅,𝐹)+𝑇₀(𝑅,𝐺)≤3𝑁₀(𝑅,𝐹)+3𝑁₀(𝑅,𝐺)+^𝑞_𝑗=1𝑁₀⁽²⁾1𝑅,𝐹−𝑐_𝑗+^𝑞_𝑗=1𝑁₀⁽²⁾1𝑅,𝐺−𝑐_𝑗+𝑁₀1𝑅,𝐹^′+𝑁₀1𝑅,𝐺^′+𝑆(𝑅). (36) From (25) and (36), since 𝑓, 𝑔 are admissible functions in 𝔸, we can get that 𝑇₀(𝑅,𝐹)+𝑇₀(𝑅,𝐺)≤𝑜𝑇₀(𝑅,𝐹)+𝑇₀(𝑅,𝐺),𝑅∉𝐸,𝑅∈𝐼. (37) Thus, we can get a contradiction. Therefore, 𝐻(𝑧)≡0; that is, 𝐹^′′𝐹^′𝐹−2^′𝐹≡𝐺^′′𝐺^′𝐺−2^′𝐺. (38) For the above equality, by integration, we can get 𝐹≡𝑎𝐺+𝑏,𝑐𝐺+𝑑 (39) where 𝑎,𝑏,𝑐,𝑑∈ℂ and 𝑎𝑑−𝑏𝑐≠0.

Lemma 22. Let 𝐹 and 𝐺 be admissble meromorphic functions in 𝔸 satisfying 𝐸^𝔸₁(𝐹,0)=𝐸^𝔸₁(𝐺,0) and let 𝑐₁,𝑐₂,…,𝑐_𝑞 be 𝑞 (≥2) distinct nonzero complex numbers. If limsup_{𝑅→∞,𝑅∈𝐼}3𝑁₀(𝑅,𝐹)+^𝑞_𝑗=1𝑁₀⁽²⁾1𝑅,𝐹−𝑐_𝑗+𝑁₀1𝑅,𝐹^′+2𝑁₀⁽²1𝑅,𝐹×𝑇₀(𝑅,𝐹)⁻¹<𝑞,limsup_{𝑅→∞,𝑅∈𝐼}3𝑁₀(𝑅,𝐺)+^𝑞_𝑗=1𝑁₀⁽²⁾1𝑅,𝐺−𝑐_𝑗+𝑁₀1𝑅,𝐺^′+2𝑁₀⁽²1𝑅,𝐺×𝑇₀(𝑅,𝐺)⁻¹<𝑞, (40) where 𝑁₀⁽²⁾(𝑅,⋅), 𝑁₀⁽²(𝑟,⋅), and 𝐼 are stated as in Lemma 21; then 𝐹=𝑎𝐺+𝑏,𝑐𝐺+𝑑 (41) where 𝑎,𝑏,𝑐,𝑑∈ℂ are constants with 𝑎𝑑−𝑏𝑐≠0.

Proof. Let 𝐻 be stated as in the proof of Lemma 21, since 𝐸^𝔸₁(𝐹,0)=𝐸^𝔸₁(𝐺,0), we can get that 𝑁₀¹⁾(𝑅)≤𝑁₀(𝑅,𝐹)+𝑁₀(𝑅,𝐺)+𝑁⁰₀1𝑅,𝐹^′+𝑁⁰₀1𝑅,𝐺^′+𝑁₀⁽²1𝑅,𝐹+𝑁₀⁽²1𝑅,𝐺+^𝑞_𝑗=1𝑁₀⁽²1𝑅,𝐹−𝑐_𝑗+^𝑞_𝑗=1𝑁₀⁽²1𝑅,𝐺−𝑐_𝑗. (42) Similar to the argument in Lemma 21, we can get that, for 𝑅∉𝐸𝑞𝑇₀(𝑅,𝐹)+𝑇₀(𝑅,𝐺)≤3𝑁₀(𝑅,𝐹)+3𝑁₀(𝑅,𝐺)+^𝑞_𝑗=1𝑁₀⁽²⁾1𝑅,𝐹−𝑐_𝑗+^𝑞_𝑗=1𝑁₀⁽²⁾1𝑅,𝐺−𝑐_𝑗+2𝑁₀⁽²1𝑅,𝐹+2𝑁₀⁽²1𝑅,𝐺+𝑁₀1𝑅,𝐹^′+𝑁₀1𝑅,𝐺^′+𝑆(𝑅). (43) From (40) and (43), since 𝑓, 𝑔 are admissible functions in 𝔸, we can get that 𝑇₀(𝑅,𝐹)+𝑇₀(𝑅,𝐺)≤𝑜𝑇₀(𝑅,𝐹)+𝑇₀(𝑅,𝐺),𝑅∉𝐸,𝑅∈𝐼. (44) Thus, we can get a contradiction. Therefore, 𝐻(𝑧)≡0; that is, 𝐹^′′𝐹^′𝐹−2^′𝐹≡𝐺^′′𝐺^′𝐺−2^′𝐺. (45) From the above equality, by integration, we can get 𝐹≡𝑎𝐺+𝑏,𝑐𝐺+𝑑 (46) where 𝑎,𝑏,𝑐,𝑑∈ℂ and 𝑎𝑑−𝑏𝑐≠0.

3. Proofs of Theorems 5 and 7

3.1. The Proof of Theorem 5

From the definition of 𝑃₁(𝑤), we can get that 𝑃₁(1)=1−𝑐∶=𝑐₁≠0, 𝑃₁(0)=−𝑐∶=𝑐₂≠0, and 𝑃^′₁(𝑤)=𝑛(𝑛−1)(𝑛−2)2(𝑤−1)²𝑤^𝑛−3,𝑃₁(𝑤)−𝑐₁=(𝑤−1)³𝑄₁(𝑤),𝑄₁(1)𝑃≠0,₁(𝑤)−𝑐₂=𝑤^𝑛−2𝑄₂(𝑤),𝑄₂(0)≠0, (47)(48)(49) where 𝑄₁, 𝑄₂ are polynomials of degrees 𝑛−3 and 2, respectively. We also see that 𝑄_𝑖 (𝑖=1,2) and 𝑃₁ have only simple zeros.

Let 𝐹 and 𝐺 be defined as 𝐹=𝑃₁(𝑓) and 𝐺=𝑃₁(𝑔). Since 𝐸^𝔸(𝑓,𝑆)=𝐸^𝔸(𝑔,𝑆), we have 𝐸^𝔸(𝐹,0)=𝐸^𝔸(𝐺,0). From (48) and (49), we have 𝑁₀⁽²⁾1𝑅,𝐹−𝑐₁=𝑁₀1𝑅,𝐹−𝑐₁+𝑁₀⁽²1𝑅,𝐹−𝑐₁≤2𝑁₀1𝑅,+𝑓−1^𝑛−3_𝑖=1𝑁₀1𝑅,𝑓−𝑎_𝑖≤(𝑛−1)𝑇₀𝑅,𝑓+𝑆(𝑅),𝑁₀⁽²⁾1𝑅,𝐹−𝑐₂=𝑁₀1𝑅,𝐹−𝑐₂+𝑁₀⁽²1𝑅,𝐹−𝑐₂≤2𝑁₀1𝑅,𝑓+²_𝑗=1𝑁₀1𝑅,𝑓−𝑏_𝑗≤4𝑇₀𝑅,𝑓+𝑆(𝑅), (50) where 𝑎_𝑖 (𝑖=1,…,𝑛−3) and 𝑏_𝑗 (𝑗=1,2) are the zeros of 𝑄₁(𝑤) and 𝑄₂(𝑤) in 𝔸, respectively.

From (47), we have 𝑁₀1𝑅,𝐹^′≤𝑁₀1𝑅,𝑓+𝑁₀1𝑅,+𝑓−1𝑁1𝑅,𝑓^′. (51) From Lemma 19, we have 𝑇₀(𝑅,𝐹)=𝑛𝑇₀(𝑅,𝑓)+𝑆(𝑅). Thus, combining (50) and (51), by Lemmas 21 and 20 and 𝑛≥11, we have limsup_{𝑅→∞,𝑅∉𝐸}3𝑁₀(𝑅,𝐹)+²_𝑗=1𝑁₀⁽²⁾1𝑅,𝐹−𝑐_𝑗+𝑁₀1𝑅,𝐹^′×𝑇₀(𝑅,𝐹)⁻¹≤limsup_{𝑅→∞,𝑅∉𝐸}4𝑁₀𝑅,𝑓+(𝑛+6)𝑇₀𝑅,𝑓𝑛𝑇₀𝑅,𝑓<2. (52) Similarly, we can obtain limsup_{𝑅→∞,𝑅∉𝐸}3𝑁₀(𝑅,𝐺)+²_𝑗=1𝑁₀⁽²⁾1𝑅,𝐺−𝑐_𝑗+𝑁₀1𝑅,𝐺^′×𝑇₀(𝑅,𝐺)⁻¹≤limsup_{𝑅→∞,𝑅∉𝐸}4𝑁₀𝑅,𝑔+(𝑛+6)𝑇₀𝑅,𝑔𝑛𝑇₀𝑅,𝑔<2. (53) Thus, by Lemma 21, we have 𝐹^′′𝐹^′𝐹−2^′𝐹≡𝐺^′′𝐺^′𝐺−2^′𝐺. (54) From the previous equality, by integration, we can get 𝐹≡𝑎𝐺+𝑏,𝑐𝐺+𝑑 (55) where 𝑎,𝑏,𝑐,𝑑∈ℂ and 𝑎𝑑−𝑏𝑐≠0. Since 𝐸^𝔸(𝑓,𝑆) is nonempty and 𝐸^𝔸(𝑓,𝑆)=𝐸^𝔸(𝑔,𝑆), we have 𝑏=0, 𝑎≠0. Hence, 𝐹≡𝑎𝐺≡𝐺𝑐𝐺+𝑑,𝐴𝐺+𝐵 (56) where 𝐴=𝑐/𝑎, 𝐵=𝑑/𝑎≠0.

Two cases will be considered as follows.

Case 1 (𝐴≠0). From the definition of 𝑃₁(𝑤) and (56), we can see that every zero of 𝑃₁(𝑔)+𝐵/𝐴 in 𝔸 has a multiplicity of at least 𝑛. Here, three following subcases will be discussed.

Subcase 1 (𝐵/𝐴=−𝑐₁). From (48), we have 𝑃₁𝑔+𝐵𝐴=𝑔−1³𝑔−𝑎₁𝑔−𝑎₂⋯𝑔−𝑎_𝑛−3, (57) where 𝑎_𝑖≠0,1 are distinct values. It follows that Θ₀𝑎_𝑖,𝑓=1−limsup_𝑅→∞𝑁₀(𝑅,𝑎)𝑇₀𝑅,𝑓≥1−limsup_𝑅→∞𝑁₀(𝑅,𝑎)𝑁₀≥1𝑅,𝑓2. (58) We can see that 𝑃₁(𝑔)+𝐵/𝐴 has 𝑛−2 values satisfying the above inequality. Thus, from Lemma 15 and 𝑛≥11, we can get a contradiction.

Subcase 2 (𝐵/𝐴=−𝑐₂). From (48), we have 𝑃₁𝑔+𝐵𝐴=𝑔^𝑛−2𝑔−𝑏₁𝑔−𝑏₂, (59) where 𝑏₁≠𝑏₂, 𝑏_𝑖≠0,1 (𝑖=1,2). It follows that every zero of 𝑔 in 𝔸 has a multiplicity of at least 2 and every zero of 𝑔−𝑏_𝑖 (𝑖=1,2) in 𝔸 has a multiplicity of at least 𝑛. Then, by Lemma 15, we have 𝑇₀𝑅,𝑔≤𝑁₀1𝑅,𝑔+𝑁₀1𝑅,𝑔−𝑏₁+𝑁₀1𝑅,𝑔−𝑏₂(𝑅)≤1+𝑆2𝑁₀1𝑅,𝑔+1𝑛𝑁₀1𝑅,𝑔−𝑏₁+1𝑛𝑁₀1𝑅,𝑔−𝑏₁(𝑅)≤1+𝑆2+2𝑛𝑇₀𝑅,𝑔+𝑆(𝑅). (60) Since 𝑔 is an admissible function in 𝔸 and 𝑛≥11, we can get a contradiction.

Subcase 3 (𝐵/𝐴≠−𝑐₁,−𝑐₂). By using the same argument as in Subcases 1 or 2, we can get a contradiction.

Case 2 (𝐴=0). If 𝐵≠1, from (56); we have 𝐹=𝐺/𝐵; that is, 𝑃₁𝑓=1𝐵𝑃₁𝑔. (61) From (49) and (61), we have 𝑃₁𝑓−𝑐₂𝐵=1𝐵𝑃₁𝑔−𝑐₂=1𝐵𝑔^𝑛−2𝑔−𝑏₁𝑔−𝑏₂. (62) Since 𝑐₂/𝐵≠𝑐₂, from (47), it follows that 𝑃₁(𝑓)−𝑐₂/𝐵 has at least 𝑛−2 distinct zeros 𝑒₁,𝑒₂,…,𝑒_𝑛−2. Then, by applying Lemma 15, we have (𝑛−4)𝑇₀≤𝑅,𝑓_{𝑛−2𝑖=1}𝑁₀1𝑅,𝑓−𝑒_𝑖(𝑅)≤+𝑆𝑁₀1𝑅,𝑔+𝑁₀1𝑅,𝑔−𝑏₁+𝑁₀1𝑅,𝑔−𝑏₂+𝑆(𝑅)≤3𝑇₀𝑅,𝑔+𝑆(𝑅). (63)
By applying Lemma 21 to (61), and from (63), since 𝑛≥11 and 𝑓 is an admissible function in 𝔸, we can get a contradiction.
Thus, we have 𝐴=0 and 𝐵=1; that is, 𝑃₁(𝑓)≡𝑃₁(𝑔). Noting the form of 𝑃₁(𝑤); we can get that 𝑃₁(𝑓)≡𝑃₁(𝑔), that is, (𝑛−1)(𝑛−2)2𝑓^𝑛−𝑛(𝑛−2)𝑓^𝑛−1+𝑛(𝑛−1)2𝑓^𝑛−2≡(𝑛−1)(𝑛−2)2𝑔^𝑛−𝑛(𝑛−2)𝑔^𝑛−1+𝑛(𝑛−1)2𝑔^𝑛−2. (64) Since 𝑓, 𝑔 are admissible functions in 𝔸, then it follows by Lemma 18 that 𝑓≡𝑔.

Therefore, the proof of Theorem 5 is completed.

3.2. The Proof of Theorem 7

Since Θ₀(∞,𝑓)>3/4 and Θ₀(∞,𝑔)>3/4, it follows that limsup_𝑅→∞𝑁₀𝑅,𝑓𝑇₀<1𝑅,𝑓4,limsup_𝑅→∞𝑁₀𝑅,𝑔𝑇₀<1𝑅,𝑔4. (65) By applying (65), from (52) and (53), and since 𝑛≥7, we can get limsup_{𝑅→∞,𝑅∉𝐸}3𝑁₀(𝑅,𝐹)+²_𝑗=1𝑁₀⁽²⁾1𝑅,𝐹−𝑐_𝑗+𝑁₀1𝑅,𝐹^′×𝑇₀(𝑅,𝐹)⁻¹≤limsup_{𝑅→∞,𝑅∉𝐸}4𝑁₀𝑅,𝑓+(𝑛+6)𝑇₀𝑅,𝑓𝑛𝑇₀𝑅,𝑓<2,limsup_{𝑅→∞,𝑅∉𝐸}3𝑁₀(𝑅,𝐺)+²_𝑗=1𝑁₀⁽²⁾1𝑅,𝐺−𝑐_𝑗+𝑁₀1𝑅,𝐺^′×𝑇₀(𝑅,𝐺)⁻¹≤limsup_{𝑅→∞,𝑅∉𝐸}4𝑁₀𝑅,𝑔+(𝑛+6)𝑇₀𝑅,𝑔𝑛𝑇₀𝑅,𝑔<2. (66) Then, from Lemma 21, we have 𝐹≡(𝑎𝐺+𝑏)/(𝑐𝐺+𝑑), where 𝑎,𝑏,𝑐,𝑑∈ℂ and 𝑎𝑑−𝑏𝑐≠0. Thus, by using the same argument as that in Theorem 5, we can prove the conclusion of Theorem 7.

4. Proofs of Theorems 9 and 11

4.1. The Proof of Theorem 9

Since 𝐸^𝔸₁(𝑓,𝑆)=𝐸^𝔸₁(𝑔,𝑆), we have 𝐸^𝔸₁(𝐹,0)=𝐸^𝔸₁(𝐺,0). From (47)–(49), we can get 𝑁₀⁽²1𝑅,𝐹=^𝑛_𝑖=1𝑁₀1𝑅,𝑓−𝑑_𝑖≤𝑁1𝑅,𝑓^′, (67) where 𝑑_𝑖 (𝑖=1,…,𝑛) are the distinct zeros of 𝑃₁(𝑤). And from (51) and (67), by Lemma 20, we have 𝑁₀1𝑅,𝐹^′+2𝑁₀⁽²1𝑅,𝐹≤𝑁₀1𝑅,𝑓+𝑁₀1𝑅,𝑓−1+3𝑁₀1𝑅,𝑓+3𝑁₀𝑅,𝑓≤5𝑇₀𝑅,𝑓+3𝑁₀𝑅,𝑓+𝑆(𝑅). (68) Then, from (50) and (68), since 𝑇₀(𝑅,𝐹)=𝑛𝑇₀(𝑅,𝑓)+𝑆(𝑅) and 𝑛≥15, we have limsup_{𝑅→∞,𝑅∉𝐸}3𝑁₀(𝑅,𝐹)+²_𝑗=1𝑁₀⁽²⁾1𝑅,𝐹−𝑐_𝑗+𝑁₀1𝑅,𝐹^′+2𝑁₀⁽²1𝑅,𝐹×𝑇₀(𝑅,𝐹)⁻¹≤limsup_{𝑅→∞,𝑅∉𝐸}6𝑁₀𝑅,𝑓+(𝑛+8)𝑇₀𝑅,𝑓𝑛𝑇₀𝑅,𝑓<2. (69) Similarly, we can get limsup_{𝑅→∞,𝑅∉𝐸}3𝑁₀(𝑅,𝐺)+²_𝑗=1𝑁₀⁽²⁾1𝑅,𝐺−𝑐_𝑗+𝑁₀1𝑅,𝐺^′+2𝑁₀⁽²1𝑅,𝐺×𝑇₀(𝑅,𝐺)⁻¹≤limsup_{𝑅→∞,𝑅∉𝐸}6𝑁₀𝑅,𝑔+(𝑛+8)𝑇₀𝑅,𝑔𝑛𝑇₀𝑅,𝑔<2. (70) Thus, by Lemma 22, we have 𝐹≡𝑎𝐺+𝑏,𝑐𝐺+𝑑 (71) where 𝑎,𝑏,𝑐,𝑑∈ℂ and 𝑎𝑑−𝑏𝑐≠0. By using arguments similar to those in the proof of Theorem 5, we can get that 𝑓≡𝑔.

Therefore, this completes the proof of Theorem 9.

4.2. The Proof of Theorem 11

Since Θ₀(∞,𝑓)>5/6 and Θ₀(∞,𝑔)>5/6, it follows that limsup_𝑅→∞𝑁₀𝑅,𝑓𝑇₀<1𝑅,𝑓6,limsup_𝑅→∞𝑁₀𝑅,𝑔𝑇₀<1𝑅,𝑔6. (72) By applying (72), from (69) and (70), since 𝑛≥7, we can get limsup_{𝑅→∞,𝑅∉𝐸}3𝑁₀(𝑅,𝐹)+²_𝑗=1𝑁₀⁽²⁾1𝑅,𝐹−𝑐_𝑗+𝑁₀1𝑅,𝐹^′+2𝑁₀⁽²1𝑅,𝐹×𝑇₀(𝑅,𝐹)⁻¹≤limsup_{𝑅→∞,𝑅∉𝐸}6𝑁₀𝑅,𝑓+(𝑛+8)𝑇₀𝑅,𝑓𝑛𝑇₀𝑅,𝑓<2,limsup_{𝑅→∞,𝑅∉𝐸}3𝑁₀(𝑅,𝐺)+²_𝑗=1𝑁₀⁽²⁾1𝑅,𝐺−𝑐_𝑗+𝑁₀1𝑅,𝐺^′+2𝑁₀⁽²1𝑅,𝐺×𝑇₀(𝑅,𝐺)⁻¹≤limsup_{𝑅→∞,𝑅∉𝐸}6𝑁₀𝑅,𝑔+(𝑛+8)𝑇₀𝑅,𝑔𝑛𝑇₀𝑅,𝑔<2. (73) Then, from Lemma 22, we have 𝐹≡(𝑎𝐺+𝑏)/(𝑐𝐺+𝑑), where 𝑎,𝑏,𝑐,𝑑∈ℂ and 𝑎𝑑−𝑏𝑐≠0. Thus, by using the same argument as in Theorem 5, we can prove the conclusion of Theorem 11.

Acknowledgments

This work was supported by the NNSF of China (11301233, 11201395, and 61202313), the NSF of Jiangxi of China (Grant no. 2010GQS0119 and no. 20132BAB211001), and the Science Foundation of Educational Commission of Hubei Province (Grant nos. Q20132801 and D20132804).