Exceptional Values of Meromorphic Function on Annulus

Abstract

The main purpose of this paper is to study the exceptional values of meromorphic function and its derivative on annulus. We also give some theorems and corollaries about exceptional values of meromorphic function on the annulus, which are the improvement of the previous results given by Chen and Wu.

1. Introduction

It is a very interesting problem on the exceptional values of meromorphic functions in the value distribution theory and argument distributed theory such as the Picard exceptional value and the Borel exceptional value. It is well known that every class of exceptional value is always responding to a singular direction, such as, the Picard exceptional value relating with Julia’s direction and the Borel exceptional value relating with Borel’s direction. In the 1970s, Gopalakrishna and Bhoosnurmath [1, 2] and Singh and Gopalakrishna [3] investigated Borel exceptional values of meromorphic function and its derivative on the whole complex plane. In 2011, Peng and Sun [4] gave some examples on T exceptional value which is an exceptional value relating with T direction.

In fact, Peng, Sun, Singh, Gopalakrishna, and so forth only studied exceptional values of meromorphic functions in the whole complex plane—single-connected region. For meromorphic function on the double-connected domain and several-connected region, there were few papers about its exceptional values. In 2005, Khrystiyanyn and Kondratyuk [5, 6] proposed the Nevanlinna theory for meromorphic functions on annuli (see also [7]). In 2010, Fernández [8] further investigated the value distribution of meromorphic functions on annuli. In 2012, Xu and Xuan [9] studied the uniqueness of meromorphic functions sharing some values on annuli. At the same year, Chen and Wu [10] firstly studied the Borel exceptional values of meromorphic function and its derivative on annulus. In this paper, we will further investigate the exceptional value of meromorphic function and its derivative on annulus and obtain a series of results which are improvement of previous theorems given by Chen and Wu [10].

The structure of this paper is as follows. In Section 2, we introduce the basic notations and fundamental theorems of meromorphic functions on annulus. Sections 3 and 4 are devoted to study the Borel exceptional values of meromorphic function on annulus and give some consequences of our theorems. Section 5 are devoted to study the Borel exceptional values of meromorphic function and its derivative on annulus.

2. Basic Notions in the Nevanlinna Theory on Annuli

Let 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 will be introduced as follows. Let 𝑁₁𝑅,𝑓=¹_1/𝑅𝑛₁𝑡,𝑓𝑡𝑑𝑡,𝑁₂𝑅,𝑓=^𝑅₁𝑛₂𝑡,𝑓𝑡𝑚𝑑𝑡,₀1𝑅,𝑓=𝑚𝑅,𝑓+𝑚𝑅,𝑁,𝑓₀𝑅,𝑓=𝑁₁𝑅,𝑓+𝑁₂𝑅,𝑓, (1) where 𝑛₁(𝑡,𝑓) and 𝑛₂(𝑡,𝑓) are the counting functions of poles of the function 𝑓 in {𝑧∶𝑡<|𝑧|≤1} and {𝑧∶1<|𝑧|≤𝑡}, respectively. The Nevanlinna characteristic of 𝑓 on the annulus 𝔸 is defined by 𝑇₀𝑅,𝑓=𝑚₀𝑅,𝑓−2𝑚1,𝑓+𝑁₀𝑅,𝑓. (2) Similarly, for 𝑎∈ℂ, we have 𝑁₀1𝑟,=𝑓−𝑎𝑁₁1𝑅,+𝑓−𝑎𝑁₂1𝑅,=𝑓−𝑎_11/𝑅𝑛₁𝑡,1/𝑓−𝑎𝑡+𝑑𝑡_𝑅1𝑛₂𝑡,1/𝑓−𝑎𝑡𝑑𝑡 (3) in which each zero of the function 𝑓−𝑎 is counted only once. In addition, we 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}, each point counted only once. Similarly, we have the notations 𝑁₁^𝑘)(𝑡,𝑓), 𝑁₁^(𝑘(𝑡,𝑓), 𝑁₂^𝑘)(𝑡,𝑓), 𝑁₂^(𝑘(𝑡,𝑓), and 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 [5]): (i)𝑇₀(𝑅,𝑓)=𝑇₀(𝑅,1/𝑓), (ii)max{𝑇₀(𝑅,𝑓₁⋅𝑓₂),𝑇₀(𝑅,𝑓₁/𝑓₂),𝑇₀(𝑅,𝑓₁+𝑓₂)}≤𝑇₀(𝑅,𝑓₁)+𝑇₀(𝑅,𝑓₂)+𝑂(1), (iii)𝑇₀(𝑅,1/(𝑓−𝑎))=𝑇₀(𝑅,𝑓)+𝑂(1), for every fixed 𝑎∈ℂ.

Khrystiyanyn and Kondratyuk [6] also obtained the lemma on the logarithmic derivative on the annulus 𝔸.

Theorem 1 (see [6] 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 (4)for 𝑅∈(1,+∞) except for the set Δ_𝑅 such that ∫_Δ𝑅𝑅^𝜆−1𝑑𝑅<+∞,(ii)if 𝑅₀<+∞, then 𝑚₀𝑓𝑅,^′𝑓𝑇=𝑂log⁰𝑅,𝑓𝑅₀−𝑅 (5)for 𝑅∈(1,𝑅₀) except for the set Δ^′_𝑅 such that ∫_Δ′𝑅(𝑑𝑅/(𝑅₀−𝑅)^𝜆−1)<+∞.

In 2005, The second fundamental theorem on the annulus 𝔸 was first obtained by Khrystiyanyn and Kondratyuk [6]. Cao et al. [11] introduce other forms of the second fundamental theorem on annuli as follows.

Theorem 2 ([11, 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. Let 𝑘₁,𝑘₂,…,𝑘_𝑞 be 𝑞 positive integers, and let 𝜆≥0. Then 𝑞−2𝑇₀𝑅,𝑓<^𝑞_𝑗=1𝑁₀1𝑅,𝑓−𝑎_𝑗+𝑆𝑅,𝑓, (6) where (i) in the case 𝑅₀=+∞, 𝑆𝑅,𝑓=𝑂log𝑅𝑇₀𝑅,𝑓 (7) for 𝑅∈(1,+∞) except for the set Δ_𝑅 such that ∫_Δ𝑅𝑅^𝜆−1𝑑𝑅<+∞,(ii) if 𝑅₀<+∞, then 𝑆𝑇𝑅,𝑓=𝑂log⁰𝑅,𝑓𝑅₀−𝑅 (8)for 𝑅∈(1,𝑅₀) except for the set Δ^′_𝑅 such that ∫_Δ′𝑅(𝑑𝑅/(𝑅₀−𝑅)^𝜆−1)<+∞.

Definition 3. 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 a transcendental or admissible meromorphic function on the annulus 𝔸 which provided that limsup_𝑅→∞𝑇₀𝑅,𝑓log𝑅=∞,1<𝑅<𝑅₀=+∞ (9) or limsup_𝑅→𝑅0𝑇₀𝑅,𝑓−log𝑅₀−𝑅=∞,1<𝑅<𝑅₀<+∞, (10) respectively.

Thus for a transcendental or 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 Theorem 1, respectively.

Definition 4 (see [10], Definition 2.1). Let 𝑓(𝑧) be a nonconstant meromorphic function on the annulus 𝔸={𝑧∶1/𝑅₀<|𝑧|<𝑅₀}, where 𝑅₀=+∞. Then the order of 𝑓(𝑧) is defined by 𝜌𝑓=limsup_𝑅→+∞𝑇₀𝑅,𝑓.log𝑅 (11)

Definition 5 (see [10], Definition 3.1). Let 𝑓(𝑧) be a nonconstant meromorphic function of order 𝜌(0<𝜌<∞) on the annulus 𝔸={𝑧∶1/𝑅₀<|𝑧|<𝑅₀}, where 1<𝑅₀≤+∞, and let 𝑎∈ℂ. Then we say that 𝑎 is(i) an evB (exceptional value in the sense of Borel) for 𝑓 on 𝔸 for distinct zeros of order ≤𝑘 if 𝜌_𝑘(𝑎,𝑓)<𝜌;(ii)an evB (exceptional value in the sense of Borel) for 𝑓 on 𝔸 for distinct zeros if 𝜌(𝑎,𝑓)<𝜌;(iii)an evB (the Borel exceptional value) for 𝑓 on 𝔸 if 𝜌(𝑎,𝑓)<𝜌, where 𝜌_𝑘𝑎,𝑓=limsup_𝑅→+∞𝑁₀^𝑘)𝑅,1/𝑓−𝑎,log𝑅𝜌𝑎,𝑓=limsup_𝑅→+∞𝑁₀𝑅,1/𝑓−𝑎,𝜌log𝑅𝑎,𝑓=limsup_𝑅→+∞𝑁₀𝑅,1/𝑓−𝑎.log𝑅 (12)

In particular, we say that 𝑎 is an evB for 𝑓 on 𝔸 for simple zeros if 𝑘=1 and 𝑎 is an evB for 𝑓 on 𝔸 for simple and double zeros if 𝑘=2.

3. The Main Results

Now, the main theorems of this paper are listed as follows.

Theorem 6. Let 𝑓 be transcendental meromorphic function of order 𝜌(0<𝜌<∞) on the annulus 𝔸∶={𝑧∶1/𝑅₀<|𝑧|<𝑅₀}, where 𝑅₀=+∞. If there exist 𝑎¹₁,…,𝑎¹_𝑝1,𝑎²₁,…,𝑎²_𝑝2,…,𝑎^𝜈₁,…,𝑎^𝜈_𝑝𝜈∈ℂ such that 𝑎^𝑖₁,𝑎^𝑖₂,…,𝑎^𝑖_𝑝𝑖 are evBs for 𝑓 on 𝔸 for distinct zeros of order ≤𝑘_𝑖, 𝑖=1,2,…,𝜈, where 𝑝₁,…,𝑝_𝜈,𝜈∈ℕ₊, and 𝑘₁,𝑘₂,…,𝑘_𝜈 are positive integers or infinity. Then Ξ∶=^𝜈_𝑖=1𝑝_𝑖𝑘_𝑖1+𝑘_𝑖+^𝜈_𝑖=111+𝑘_𝑖^𝑝𝑖_𝑗=1𝛿^𝑘𝑖₀𝑎^𝑖_𝑗,𝑓≤2. (13)

Remark 7. Under the assumptions of Theorem 6, if 𝜈=3, 𝑝₁=𝑟,𝑝₂=𝑡,𝑝₃=𝑠 and 𝑘₁=𝑘, 𝑘₂=𝑙, 𝑘₃=𝑚, since 𝛿^𝑘𝑖₀(𝑎^𝑖_𝑗,𝑓)≥0, then we can get 𝑘𝑟+𝑘+1𝑡𝑙+𝑙+1𝑠𝑚≤𝑚+1^𝜈_𝑖=1𝑝_𝑖𝑘_𝑖1+𝑘_𝑖+^𝜈_𝑖=111+𝑘_𝑖^𝑝𝑖_𝑗=1𝛿^𝑘𝑖₀𝑎^𝑖_𝑗,𝑓≤2. (14) This shows that Theorem 6 is an improvement of results given by Chen and Wu [10].

Definition 8. For positive integers 𝑘,𝑚, we define that 𝛿^𝑘₀𝑎,𝑓=1−limsup_𝑅→+∞𝑁^𝑘₀𝑅,1/𝑓−𝑎𝑇₀,Θ𝑅,𝑓₀𝑎,𝑓=1−limsup_𝑅→+∞𝑁₀𝑅,1/𝑓−𝑎𝑇₀,𝑅,𝑓 (15) where 𝑁^𝑘₀(𝑅,1/(𝑓−𝑎)) is the counting function of 𝑎-points of 𝑓 on 𝔸 where an 𝑎-point of multiplicity 𝑚 is counted 𝑚 times if 𝑚≤𝑘 and 1+𝑘 times if 𝑚>𝑘. In particular, if 𝑘=∞, then 𝛿^∞₀𝑎,𝑓=𝛿₀𝑎,𝑓=liminf_𝑅→+∞𝑚₀𝑅,1/𝑓−𝑎𝑇₀𝑅,𝑓=1−limsup_𝑅→+∞𝑁₀𝑅,1/𝑓−𝑎𝑇₀.𝑅,𝑓 (16)

Theorem 9. Let 𝑓 be transcendental meromorphic function of order 𝜌(0<𝜌<∞) on the annulus 𝔸∶={𝑧∶1/𝑅₀<|𝑧|<𝑅₀}, where 𝑅₀=+∞. If there exist 𝑎∈ℂ and two positive integers 𝑘 and 𝑝 such that (1+𝑘)Θ₀𝑎,𝑓+_𝑏≠𝑎𝛿₀𝑏,𝑓>2−𝑘𝑝−1, (17) then there exist at most 𝑝 elements ℂ⧵{𝑎} which are evBs for 𝑓 on 𝔸 for distinct zeros of order not exceeding 𝑘.

3.1. Proof of Theorem 6

Proof. For any positive integer 𝑘 or ∞ and 𝑎∈ℂ, we have 𝑁₀1𝑅,≤𝑘𝑓−𝑎1+𝑘𝑁₀^𝑘)1𝑅,+1𝑓−𝑎𝑁1+𝑘^𝑘01𝑅,,𝑓−𝑎 (18) where 𝑘/(𝑘+1)=1 and 1/(𝑘+1)=0 if 𝑘=∞. Then, from (18) and Theorem 2, we have ^𝜈_𝑖=1𝑝_𝑖−2𝑇₀≤𝑅,𝑓_𝜈𝑖=1𝑘_𝑖1+𝑘_𝑖^𝑝𝑖_𝑗=1𝑁^𝑘𝑖)₀1𝑅,𝑓−𝑎^𝑖_𝑗+^𝜈_𝑖=111+𝑘_𝑖^𝑝𝑖_𝑗=1𝑁^𝑘𝑖₀1𝑅,𝑓−𝑎^𝑖_𝑗+𝑆𝑅,𝑓. (19) From Definition 5 and the assumptions of Theorem 6, there exists a constant 𝜂(0<𝜂<𝜌) such that for sufficiently large 𝑅, 𝑁^𝑘𝑖)₀1𝑅,𝑓−𝑎^𝑖_𝑗<𝑅^𝜂,𝑗=1,2,…,𝑝_𝑖,𝑖=1,2,…,𝜈. (20) Hence, from (19) and (20) and for sufficiently large 𝑅, we have ^𝜈_𝑖=1𝑝_𝑖−2𝑇₀𝑅,𝑓≤^𝜈_𝑖=111+𝑘_𝑖^𝑝𝑖_𝑗=1𝑁^𝑘𝑖₀1𝑅,𝑓−𝑎^𝑖_𝑗+𝑂𝑅^𝜂+𝑆𝑅,𝑓. (21) Thus, for sufficiently large 𝑅 and taking arbitrary 𝜀(>0), we can get from (21) and the definition of 𝛿_𝑘(𝑎,𝑓) that ^𝜈_𝑖=1𝑝_𝑖−2𝑇₀≤𝑅,𝑓_𝜈𝑖=111+𝑘_𝑖^𝑝𝑖_𝑗=11−𝛿^𝜃_𝑘𝑖𝑎^𝑖_𝑗,𝑓+𝜀𝑇₀𝑅𝑅,𝑓+𝑂^𝜂+𝑆𝑅,𝑓. (22) Since 𝑅=+∞, it follows that (Ξ−2−𝜀)𝑇₀𝑅,𝑓≤𝑂𝑅^𝜂+𝑂log𝑅𝑇₀𝑅,𝑓, (23) for 𝑅∈(1,+∞) except for the set Δ_𝑅 such that ∫_Δ𝑅𝑅^𝜆−1𝑑𝑅<+∞. If 𝐼=[𝑅,𝑅^′)∈Δ_𝑅, we have 𝑅^′𝜆−𝑅^𝜆=𝜆^𝑅′_𝑅𝑥^𝜆−1𝑑𝑥≤𝜆_Δ𝑅𝑅^𝜆−1𝑑𝑅<𝑀, (24) where 𝑀 is a constant. Thus, (Ξ−2−𝜀)𝑇₀𝑅,𝑓≤𝑂𝑅^𝜂+𝑂log𝑅𝑇₀𝑅,𝑓 (25) holds for all 𝑅. If Ξ>2, we can choose an arbitrary 𝜀(0<𝜀<Ξ−2) satisfying 𝜂+𝜀<𝜌. Thus, from (25) and for sufficiently large 𝑅, we can get a contradiction to (13) easily.
Therefore, we can get the conclusion of Theorem 6.

3.2. The Proof of Theorem 9

Lemma 10. Let 𝑓 be transcendental meromorphic function of order 𝜌(0<𝜌<∞) on the annulus 𝔸∶={𝑧∶1/𝑅₀<|𝑧|<𝑅₀}, where 1<𝑅₀≤+∞. Then limsup_𝑅→+∞𝑚₀𝑅,1/𝑓^′𝑇₀𝑅,𝑓≤2−Θ₀∞,𝑓−_𝑏∈ℂ𝛿₀𝑏,𝑓. (26)

Proof. Suppose that 𝑏₁,𝑏₂,…,𝑏_𝑡∈ℂ are 𝑡 distinct complex constants. From the definitions of 𝑚₀(𝑅,𝑓),𝑇₀(𝑅,𝑓), we have ^𝑡_𝑖=1𝑚₀1𝑟,𝑓−𝑏_𝑖≤𝑚₀1𝑟,𝑓^′+𝑆𝑅,𝑓≤𝑇₀𝑅,𝑓^′−𝑁₀1𝑅,𝑓^′+𝑆𝑅,𝑓. (27) And since 𝑇₀(𝑅,𝑓^′)≤𝑇₀(𝑅,𝑓)+𝑁₀(𝑅,𝑓)+𝑆(𝑅,𝑓), then we have ^𝑡_𝑖=1𝑚₀1𝑅,𝑓−𝑏_𝑖+𝑁₀1𝑅,𝑓^′≤𝑇₀+𝑅,𝑓𝑁₀𝑅,𝑓+𝑆𝑅,𝑓. (28) It follows from the definitions of 𝛿₀(𝑎,𝑓),Θ₀(𝑎,𝑓) that ^𝑡_𝑖=1𝛿₀𝑏_𝑖,𝑓+limsup_𝑅→+∞𝑁₀𝑅,1/𝑓^′𝑇₀𝑅,𝑓≤2−Θ₀∞,𝑓. (29) From (29) and 𝑡 is arbitrary, the proof of Lemma 10 is completed easily.

Proof of Theorem 9. Without loss of generality, we assume that 𝑎=∞. Next, we will employ reductio ad absurdum to prove the conclusion of Theorem 9. Suppose that there exist 𝑝+1 elements 𝑎₁,𝑎₂,…,𝑎_𝑝+1∈ℂ which are evBs for 𝑓 on 𝔸 for distinct zeros of multiplicity ≤𝑘. We know that if 𝑧₀ is a zero of 𝑓−𝑏 on 𝔸 of multiplicity 𝑠(>1) for 𝑏∈ℂ, then 𝑧₀ is a zero of 𝑓^′ on 𝔸 of multiplicity 𝑠−1. It follows that ^𝑝+1_𝑖=1𝑁₀1𝑅,𝑓−𝑎_𝑖≤^𝑝+1_𝑖=1𝑁₀^𝑘)1𝑅,𝑓−𝑎_𝑖+1𝑘𝑁₀1𝑅,𝑓^′. (30) Therefore, by Theorem 2 we have 𝑝𝑇₀𝑅,𝑓≤^𝑝+1_𝑖=1𝑁₀1𝑅,𝑓−𝑎_𝑖+𝑁₀≤𝑅,𝑓+𝑆𝑅,𝑓_𝑝+1𝑖=1𝑁₀^𝑘)1𝑅,𝑓−𝑎_𝑖+1𝑘𝑁₀1𝑅,𝑓^′+𝑁₀𝑅,𝑓+𝑆𝑅,𝑓. (31) From the definitions of order and evB of 𝑓 on 𝔸, there exists a constant 𝜂(0<𝜂<𝜌) such that for sufficiently large 𝑅, we have 𝑁₀^𝑘)1𝑅,𝑓−𝑎_𝑖<𝑅^𝜂,𝑖=1,2,…,𝑝+1. (32) From (31), (32), and Lemma 10, for sufficiently large 𝑅, it follows that 1𝑝−1−𝑘2−Θ₀∞,𝑓−_𝑏≠∞𝛿₀𝑏,𝑓+Θ₀∞,𝑓×𝑇₀𝑅,𝑓≤𝑅^𝜂+𝑆𝑅,𝑓, (33) for 𝑅∈(1,+∞) except for the set Δ_𝑅 such that ∫_Δ𝑅𝑅^𝜆−1𝑑𝑅<+∞. Since 𝑅₀=+∞, similar to the argument as in Theorem 6, from (33) we find that ()Θ𝑝−1−2−𝑘+1₀∞,𝑓−∑_𝑏≠∞𝛿₀𝑏,𝑓𝑘𝑇₀𝑅,𝑓≤𝑅^𝜂+𝑂log𝑅𝑇₀𝑅,𝑓 (34) holds for all 𝑅. Since 𝜂<𝜌, from (34) and for sufficiently large 𝑅, we can get (1+𝑘)Θ₀∞,𝑓+_𝑏≠∞𝛿₀𝑏,𝑓>2−𝑘𝑝−1, (35) which is contradiction with the assumption of Theorem 9.
Thus, this completes the proof of Theorem 9.

4. Some Consequences of Theorems 6 and 9

In this section, we will give some consequences of Theorems 6 and 9. Before we give these results, some definitions will be introduced below.

Definition 11. Let 𝑓 be meromorphic function on the annulus 𝔸∶={𝑧∶1/𝑅₀<|𝑧|<𝑅₀}, where 𝑅₀=+∞. For 𝑎∈ℂ, then we say that(i)𝑎 is called an exceptional value in the sense of Nevanlinna (evN for short) for 𝑓 on 𝔸, if 𝛿₀(𝑎,𝑓)>0;(ii)𝑎 is called a normal value in the sense of Nevanlinna (nvN for short) for 𝑓 on 𝔸, if 𝛿₀(𝑎,𝑓)=0.
In addition, similar to the Picard exceptional value in the whole complex plane, we define that 𝑎 is called an exceptional value in the sense of Picard (evP for short) for 𝑓 on 𝔸, if 𝑓 has at most a finite number of 𝑎-points on 𝔸.

Consequence 1. Under the assumptions of Theorem 6, if 𝑘₁=1, from Theorem 6, we get 𝑝₁+2^𝜈_𝑖=2𝑝_𝑖𝑘_𝑖1+𝑘_𝑖+^𝑝1_𝑗=1𝛿¹₀𝑎¹_𝑗,𝑓+^𝜈_𝑖=221+𝑘_𝑖^𝑝𝑖_𝑗=1𝛿¹₀𝑎^𝑖_𝑗,𝑓≤4. (36) Since 𝑝_𝑖≥0 and 𝛿¹₀(𝑎^𝑖_𝑗,𝑓)≥0 for 𝑖=1,…,𝜈, it follows that(i)if 𝑓 has an evB for simple zeros which is also an evN for 𝑓 on 𝔸, then 𝑓 has at most three evBs for simple zeros on 𝔸;(ii)if 𝑎₁ and 𝑎₂ are two evPs for 𝑓 on 𝔸 then no other element is an evB for 𝑓 for simple zeros on 𝔸;(iii) there exist at most four elements which are evBs for 𝑓 simple zeros on 𝔸 since 𝛿₁(𝑎¹_𝑗,𝑓)≥0. Moreover, all these four values are nvNs for 𝑓 on 𝔸.

Consequence 2. Under the assumptions of Theorem 6, let 𝑘₁=1,𝑝₁=1, then(a) if 𝑘₂=2, we have 2𝑝₂3+^𝜈_𝑖=3𝑝_𝑖𝑘_𝑖1+𝑘_𝑖+12𝛿¹₀𝑎¹₁+1,𝑓3^𝑝2_𝑗=1𝛿²₀𝑎²_𝑗+,𝑓_𝜈𝑖=311+𝑘_𝑖^𝑝𝑖_𝑗=1𝛿^𝑘𝑖₀𝑎^𝑖_𝑗≤3,𝑓2. (37)Since 𝑝_𝑖≥0 and 𝛿^𝑘𝑖₀(𝑎^𝑖_𝑗,𝑓)≥0, it follows from (37) that 𝑝₂<3. Thus, if 𝑎¹₁ is an evB for 𝑓 on 𝔸 for simple zeros, that is, 𝛿¹₀(𝑎¹₁,𝑓)>0, then there exist at most two other elements which are evBs for 𝑓 on 𝔸 for distinct simple zeros and double zeros. Furthermore,(i)if 𝛿¹₀(𝑎¹₁,𝑓)=1/3, then two other elements evBs are also evNs for 𝑓 on 𝔸;(ii)if any of the two other elements 𝑎²₁ and 𝑎²₂, say 𝑎²₁, satisfies 𝛿¹₀(𝑎²₁,𝑓)=1/2, then 𝑎¹₁, 𝑎²₂ are also evNs for 𝑓 on 𝔸;(b)if 𝑘₂=3, we have 3𝑝₂4+^𝜈_𝑖=3𝑝_𝑖𝑘_𝑖1+𝑘_𝑖+12𝛿¹₀𝑎₁+1,𝑓4^𝑝2_𝑗=1𝛿³₀𝑎²_𝑗+,𝑓_𝜈𝑖=311+𝑘_𝑖^𝑝𝑖_𝑗=1𝛿^𝑘𝑖₀𝑎^𝑖_𝑗≤3,𝑓2. (38)From the above inequality and 𝛿₀(𝑎,𝑓)≥0 for 𝑎∈ℂ, we get that 𝑝₁=1, 𝑝₂≤2 and 𝑝₁=1, 𝑝₂=2, 𝑝_𝑖=0(𝑖=3,…,𝜈). Thus, if 𝑓 has an evB for simple zeros on 𝔸, then there exist at most two other elements which are evBs for 𝑓 distinct zeros of multiplicity ≤3 on 𝔸. Moreover, all these exceptional values are nvNs for 𝑓 on 𝔸.

Consequence 3. Under the assumptions of Theorem 6, if 𝑘₁=2, then we have 2𝑝₁3+^𝜈_𝑖=2𝑝_𝑖𝑘_𝑖1+𝑘_𝑖+13^𝑝1_𝑗=1𝛿²₀𝑎¹_𝑗+,𝑓_𝜈𝑖=211+𝑘_𝑖^𝑝𝑖_𝑗=1𝛿^𝑘𝑖₀𝑎^𝑖_𝑗,𝑓≤2. (39) Since ∑^𝜈_𝑖=2(𝑝_𝑖𝑘_𝑖/(1+𝑘_𝑖))+∑^𝜈_𝑖=2(1/(1+𝑘_𝑖))∑^𝑝𝑖_𝑗=1𝛿^𝑘𝑖₀(𝑎^𝑖_𝑗,𝑓)≥0, then from (39) we have 2𝑝₁3+13^𝑝1_𝑗=1𝛿²₀𝑎¹_𝑗,𝑓≤2. (40) Thus, it follows that 𝑝₁≤3 and 𝑝₁=3,𝑝_𝑖=0, 𝑖=2,…,𝜈. Hence, we have that 𝑓 has at most three evBs for distinct simple and double zeros on 𝔸. Moreover, all three evBs for distinct simple and double zeros are nvNs for 𝑓 on 𝔸.

Consequence 4. Under the assumptions of Theorem 6, let 𝑘₁=2,𝑝₁=1.(a) If 𝑘₂=1, then we have 𝑝₂+2^𝜈_𝑖=3𝑝_𝑖𝑘_𝑖1+𝑘_𝑖+23𝛿²₀𝑎¹₁,𝑓+^𝑝2_𝑗=1𝛿¹₀𝑎²_𝑗+,𝑓_𝜈𝑖=321+𝑘_𝑖^𝑝𝑖_𝑗=1𝛿^𝑘𝑖₀𝑎^𝑖_𝑗≤8,𝑓3. (41)Thus, it follows that 𝑝₂≤2. So, if there exists an evB for 𝑓 on 𝔸 for distinct and double zeros, say 𝑎¹₁, then there exist at most two other evBs for 𝑓 on 𝔸 for simple zeros, say 𝑎²₁, 𝑎²₂. Furthermore, if 𝑝₁=1, 𝑝₂=2, it follows that 2𝛿²₀𝑎¹₁,𝑓+3𝛿¹₀𝑎²₁,𝑓+𝛿¹₀𝑎²₂,𝑓≤2. (42) Thus, we can see that any one of 𝑎²₁, 𝑎²₂ may not be an evP for 𝑓 on 𝔸. Furthermore, if 𝑎¹₁ is an evP for 𝑓 on 𝔸. Then 𝑎²₁, 𝑎²₂ are nvNs for 𝑓 on 𝔸;(b) if 𝑝₂=1, 𝑘₂=3,𝑘₃=1, then we have 𝑝₃+^𝜈_𝑖=42𝑝_𝑖𝑘_𝑖1+𝑘_𝑖+23𝛿²₀𝑎¹₁+1,𝑓2𝛿³₀𝑎²₁+,𝑓_𝑝3𝑗=1𝛿¹₀𝑎³_𝑗,𝑓+^𝜈_𝑖=421+𝑘_𝑖^𝑝𝑖_𝑗=1𝛿^𝑘𝑖₀𝑎^𝑖_𝑗≤7,𝑓6. (43)Thus, we can get that 𝑝₃≤1. That is, if 𝑓 have two evBs on 𝔸, say 𝑎¹₁ and 𝑎²₁ and 𝑎¹₁ is for distinct simple and double zeros, 𝑎²₁ for distinct zeros of order ≤3, then 𝑓 has at most one evB for simple zeros on 𝔸. Furthermore, if 𝑝₃=1, then 23𝛿²₀𝑎¹₁+1,𝑓2𝛿³₀𝑎²₁,𝑓+𝛿¹₀𝑎³₁≤1,𝑓6. (44)

It follows that 𝛿²₀(𝑎¹₁,𝑓)≤1/4,𝛿³₀(𝑎²₁,𝑓)≤1/3 and 𝛿¹₀(𝑎³₁,𝑓)≤1/6. Thus, if any equality of these three inequalities holds, then the other two are nvNs for 𝑓 on 𝔸.

Now, we will give some consequences of Theorem 9 as follows.

Consequence 5. Under the assumptions of Theorem 9, if 𝑘=1 and 2Θ₀𝑎,𝑓+_𝑏≠𝑎𝛿₀𝑏,𝑓>3−𝑝, (45) we have the following:(i) if 𝑝=1 and 2Θ₀(𝑎,𝑓)+∑_𝑏≠𝑎𝛿₀(𝑏,𝑓)>2, then there exists at most one element 𝑏≠𝑎 which is an evB for 𝑓 on 𝔸 for simple zeros; in particular, this holds if there exists an 𝑎∈ℂ satisfying Θ₀(𝑎,𝑓)=1;(ii)if 𝑝=2 and 2Θ₀(𝑎,𝑓)+∑_𝑏≠𝑎𝛿₀(𝑏,𝑓)>1, then there exists at most two elements 𝑏₁,𝑏₂≠𝑎 which are evBs for 𝑓 on 𝔸 for simple zeros; in particular, this holds if there exists an 𝑎∈ℂ satisfying Θ₀(𝑎,𝑓)>1/2;(iii)if 𝑝=3 and 2Θ₀(𝑎,𝑓)+∑_𝑏≠𝑎𝛿₀(𝑏,𝑓)>0, then there exists at most three elements 𝑏₁,𝑏₂,𝑏₃≠𝑎 which are evBs for 𝑓 on 𝔸 for simple zeros; in particular, this holds if there exists an 𝑎∈ℂ satisfying Θ₀(𝑎,𝑓)>0.

Remark 12. Under the assumptions of Theorem 9, from Consequence 5, we have that if there exist four distinct elements 𝑏₁,𝑏₂,𝑏₃,𝑏₄∈ℂ which are evBs for 𝑓 on 𝔸 for simple zeros, then Θ₀(𝑏_𝑖,𝑓)≤1/2 and Θ₀(𝑎,𝑓)=0 for 𝑎≠𝑏_𝑖 and 𝑖=1,2,3,4.

Consequence 6. Under the assumptions of Theorem 9, if 𝑘=2 and 3Θ₀𝑎,𝑓+_𝑏≠𝑎𝛿₀𝑏,𝑓>4−2𝑝, (46) we have the following:(i) if 𝑝=1 and 3Θ₀(𝑎,𝑓)+∑_𝑏≠𝑎𝛿₀(𝑏,𝑓)>2, then there exists at most one element 𝑏≠𝑎 which is an evB for 𝑓 on 𝔸 for distinct simple and double zeros; in particular, this holds if there exists an 𝑎∈ℂ satisfying Θ₀(𝑎,𝑓)>2/3;(ii)if 𝑝=2 and 3Θ₀(𝑎,𝑓)+∑_𝑏≠𝑎𝛿₀(𝑏,𝑓)>0, then there exists at most two elements 𝑏₁,𝑏₂≠𝑎 which are evBs for 𝑓 on 𝔸 for distinct simple and double zeros; in particular, this holds if there exists an 𝑎∈ℂ satisfying Θ₀(𝑎,𝑓)>0.

Remark 13. Under the assumptions of Theorem 9, from Consequence 6, we have that if there exist four distinct elements 𝑏₁,𝑏₂,𝑏₃∈ℂ which are evBs for 𝑓 on 𝔸 for distinct simple and double zeros, then Θ₀(𝑏_𝑖,𝑓)≤2/3 and Θ₀(𝑎,𝑓)=0 for 𝑎≠𝑏_𝑖 and 𝑖=1,2,3.

5. Exceptional Value of Meromorphic Function and Its Derivative

In this section, we will study the exceptional value of meromorphic function 𝑓 on 𝔸 and its differential polynomial of the form 𝑀𝑓∶=𝑀𝑓,𝑓^′,…,𝑓^(𝑘)=𝑎𝑓(𝑧)^𝑖0𝑓^′𝑖1⋯𝑓^{(𝑘)𝑖𝑘}, (47) where 𝑖₀,𝑖₁,…,𝑖_𝑘∈ℕ, 𝑘∈ℕ₊, and 𝑎(≠0)∈ℂ is a complex constant.

Theorem 14. Let 𝑓 be transcendental meromorphic function of order 𝜌(0<𝜌<∞) on the annulus 𝔸∶={𝑧∶1/𝑅₀<|𝑧|<𝑅₀}, where 𝑅₀=+∞. If 0 and ∞ are evBs for 𝑓 on 𝔸 for distinct zeros, then 𝑀[𝑓] has no evB for simple zeros in ℂ⧵{0,∞}.

Proof. From (21), set 𝜎=𝑖₀+𝑖₁+⋯+𝑖_𝑘. Since 𝑓 is a transcendental meromorphic function on 𝔸, similar to the method of [12, Theorem 1.21], we can get that 𝑓 and 𝑀[𝑓] have the same order 𝜌 on 𝔸. By applying Theorem 1 for 𝑀[𝑓], for 𝑏(≠0)∈ℂ we have 𝑇₀𝑅,𝑀𝑓≤𝑁₀1𝑅,𝑀𝑓+𝑁₀𝑓+𝑅,𝑀𝑁₀1𝑅,𝑀𝑓𝑓≤−𝑏+𝑆𝑅,𝑀𝑁₀1𝑅,𝑀𝑓+𝑁₀+1𝑅,𝑓2𝑁₀¹⁾1𝑅,𝑀𝑓+1−𝑏2𝑁¹₀1𝑅,𝑀𝑓≤−𝑏+𝑆𝑅,𝑓𝑁₀1𝑅,𝑀𝑓+𝑁₀+1𝑅,𝑓2𝑁₀¹⁾1𝑅,𝑀𝑓+1−𝑏2𝑇₀𝑅,𝑓+𝑆𝑅,𝑓. (48) Since 𝑁₀1𝑅,𝑀𝑓≤𝑁₀𝑓𝑅,^𝜎𝑀𝑓+𝑁₀1𝑅,𝑓^𝜎≤𝑇₀𝑀𝑓𝑅,𝑓^𝜎+𝑁₀1𝑅,𝑓+𝑆𝑅,𝑓≤𝑁₀𝑀𝑓𝑅,𝑓^𝜎+𝑁₀1𝑅,𝑓+𝑆𝑅,𝑓≤𝑂𝑁₀1𝑅,𝑓+𝑁₀𝑅,𝑓+𝑆𝑅,𝑓. (49) and 0 and ∞ are evBs for 𝑓 on 𝔸 for distinct zeros, then there exists a constant 𝜂(0<𝜂<𝜌) such that for sufficiently large 𝑅, we have 𝑁₀1𝑅,𝑓<𝑅^𝜂,𝑁₀𝑅,𝑓<𝑅^𝜂. (50)
Suppose that 𝑏 is an evB for 𝑀[𝑓] on 𝔸 for simple zeros, since 𝑀[𝑓] is of order 𝜌. Then for sufficiently large 𝑅, there exists a constant 𝛽<𝜌 such that 𝑁₀¹⁾1𝑅,𝑀𝑓−𝑏<𝑅^𝛽. (51) Thus, from (48)–(50), it follows that 12𝑇₀𝑅,𝑀𝑓≤𝑂𝑟^𝛽′+𝑂log𝑅𝑇₀𝑅,𝑓, (52) where 𝛽^′=max{𝜂,𝛽} and for 𝑅∈(1,+∞) except for the set Δ_𝑅 such that ∫_Δ𝑅𝑅^𝜆−1𝑑𝑅<+∞. Since 𝑀[𝑓] is of order 𝜌, then there exists a sequence {𝑅_𝑛} tending to ∞ such that 𝑇₀𝑅_𝑛,𝑀𝑓>𝑅^𝛼_𝑛, (53) where 𝛼(𝛽^′<𝛼<𝜌) is a constant. Thus, we can get a contradiction by using the same argument as in Theorem 6.
Therefore, 𝑀[𝑓] has no evB for simple zeros in ℂ⧵{0,∞}.

Acknowledgments

The author was supported by the NNSF of China (no. 61202313) and the Natural Science Foundation of Jiang-Xi Province in China (2010GQS0119 and 20132BAB211001).