#### 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 , , and were usually used in those papers (see [5, 9, 10]). We use to denote the open complex plane, to denote the extended complex plane, and to denote the subset of . Let be a set of distinct elements in and . Define where if and . We also define

For , we say that two meromorphic functions and share the value in (or ), if and have the same zeros with the same multiplicities (ignoring multiplicities) in (or ).

The whole complex plane , 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 . From the doubly connected mapping theorem [11], we can get that each doubly connected domain is conformally equivalent to the annulus , . There are two cases: (1) and and (2) ; for case (2) the homothety reduces the given domain to the annulus , where . Thus, every annulus is invariant with respect to the inversion 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 , where and . 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 and be two transcendental or admissible meromorphic functions on the annulus , where . Let () be five distinct complex numbers in . If , share for , then .*

*Remark 2. *For the case , the assertion was proved by Kondratyuk and Laine [20].

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

Theorem 3. *Let and be two admissible meromorphic functions in the annulus . Put , , and , where
**
where is an integer and and are two nonzero complex numbers satisfying . If and (), then .*

Theorem 4. *Let and be two admissible meromorphic functions in the annulus . Put and , where is stated as in Theorem 3, and and are two nonzero complex numbers satisfying , is an integer. If and , then . *

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 , where and is a complex number satisfying and we obtain the following results.

Theorem 5. *Let and be two admissible meromorphic functions in the annulus . If and is an integer ≥11, then . *

A set is called a unique range set for meromorphic functions on annulus , if, for any two nonconstant meromorphic functions and , the condition implies . We denote by the cardinality of a set . Thus, from Theorem 5, we can get the following corollary.

Corollary 6. * There exists one finite set with , such that any two admissible meromorphic functions and on must be identical if . *

Theorem 7. *Let and be two admissible meromorphic functions in the annulus . If is an integer ≥7, , and , , then . *

Corollary 8. *There exists one finite set with , such that any two admissible analytic functions and on must be identical if . *

Theorem 9. *Let and be two admissible meromorphic functions in the annulus . If and is an integer ≥15, then . *

A set is called a unique range set, with weight 1 for meromorphic functions on annulus , if for any two nonconstant meromorphic functions and , the condition implies . Thus, from Theorem 9, we can get the following corollary.

Corollary 10. *There exists one finite set with , such that any two admissible meromorphic functions and on must be identical if . *

Theorem 11. *Let and be two admissible meromorphic functions in the annulus . Let be an integer ≥9 and , where and are stated as in Theorem 5. If and , , then . *

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

Corollary 12. *There exists one finite set with , such that any two admissible analytic functions and on must be identical if . *

#### 2. Preliminaries and Some Lemmas

Letting be a meromorphic function on whole plane , the classical notations of the Nevanlinna theory are denoted as follows: where and is the counting function of poles of the function in .

Letting be a meromorphic function on the annulus , where , the notations of the Nevanlinna theory on annuli had been introduced in [13, 20], such as . In addition, we define We also use (or ) to denote the counting function of poles of the function with multiplicities (or ) in , with each point being counted only once. Similarly, we have the notations , , , , , .

For a nonconstant meromorphic function on the annulus , where , the following properties will be used in this paper (see [13]): (i)(ii)(iii)

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

Lemma 13 (see [14], lemma on the logarithmic derivative). *Let be a nonconstant meromorphic function on the annulus , where , and let . Then,*(i)* in the case ,
* *for except for the set such that ;*(ii)* if , then
* *for except for the set such that .*

*Remark 14. *From [14, 20], the conclusions still hold if is replaced by , .

In 2005, the second fundamental theorem on the annulus 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 be a nonconstant meromorphic function on the annulus , where . Let be distinct complex numbers in the extended complex plane . Then,
**
where (i) in the case ,
**
for except for the set such that .**(ii) If , then
**
for except for the set such that .*

*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:
where is stated as in Lemma 15 and is the counting function for the zeros of in , where does not take one of the values ().

*Definition 17. *Let be a nonconstant meromorphic function on the annulus , where . The function is called an admissible meromorphic function on the annulus provided that
or
respectively.

Thus, for an admissible meromorphic function on the annulus , holds for all except for the set or the set mentioned in Lemma 13, respectively.

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

Lemma 18. *Let and
**
Then, is a unique polynomial for admissible meromorphic functions; that is, for any two admissible meromorphic functions and in , implies . *

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

Lemma 19 (see [23]). * Let be a nonconstant meromorphic function in , and let be two mutually prime polynomials in with degree and , respectively. Then,
*

Lemma 20. *Suppose is a nonconstant meromorphic function in . Then,
**
where is stated as in Lemma 15. *

* Proof. *Since
Then, from properties of , we have
that is,
Since
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 and be admissible meromorphic functions in satisfying and let be (≥2) distinct nonzero complex numbers. If
**
where , , and is some set of of infinite linear measure, then
**
where are constants with . *

*Proof. *Set

Supposing that , from Lemma 13 and Remark 14, we have
where , . Since , and by an elementary calculation, we can conclude that if is a common simple zero of and in , then . Thus, we have
where . The poles of in can only occur at zeros of and in or poles of and in . Moreover, only has simple zeros in . Hence, from (29), we have
where is the reduced counting function for the zeros of in , where does not take one of the values .

Since
Then, from (30) and (31), we have
From Remark 16, we have
where is a set of of finite linear measure and it needs not to be the same at each occurrence. From (32)-(33), it follows that, for ,
Since
From (34)-(35), we can get that, for ,
From (25) and (36), since , are admissible functions in , we can get that
Thus, we can get a contradiction. Therefore, ; that is,
For the above equality, by integration, we can get
where and .

Lemma 22. *Let and be admissble meromorphic functions in satisfying and let be (≥2) distinct nonzero complex numbers. If
**
where , , and are stated as in Lemma 21; then
**
where are constants with . *

*Proof. *Let be stated as in the proof of Lemma 21, since , we can get that
Similar to the argument in Lemma 21, we can get that, for
From (40) and (43), since , are admissible functions in , we can get that
Thus, we can get a contradiction. Therefore, ; that is,
From the above equality, by integration, we can get
where and .

#### 3. Proofs of Theorems 5 and 7

##### 3.1. The Proof of Theorem 5

From the definition of , we can get that , , and where , are polynomials of degrees and 2, respectively. We also see that () and have only simple zeros.

Let and be defined as and . Since , we have . From (48) and (49), we have where () and () are the zeros of and in , respectively.

From (47), we have From Lemma 19, we have . Thus, combining (50) and (51), by Lemmas 21 and 20 and , we have Similarly, we can obtain Thus, by Lemma 21, we have From the previous equality, by integration, we can get where and . Since is nonempty and , we have , . Hence, where , .

Two cases will be considered as follows.

*Case 1 (). *From the definition of and (56), we can see that every zero of in has a multiplicity of at least . Here, three following subcases will be discussed.

*Subcase 1 (). *From (48), we have
where are distinct values. It follows that
We can see that has values satisfying the above inequality. Thus, from Lemma 15 and , we can get a contradiction.

*Subcase 2 (). *From (48), we have
where , (). It follows that every zero of in has a multiplicity of at least 2 and every zero of () in has a multiplicity of at least . Then, by Lemma 15, we have
Since is an admissible function in and , we can get a contradiction.

*Subcase 3 (). *By using the same argument as in Subcases 1 or 2, we can get a contradiction.

*Case 2 (). *If , from (56); we have ; that is,
From (49) and (61), we have
Since , from (47), it follows that has at least distinct zeros . Then, by applying Lemma 15, we have

By applying Lemma 21 to (61), and from (63), since and is an admissible function in , we can get a contradiction.

Thus, we have and ; that is, . Noting the form of ; we can get that , that is,
Since , are admissible functions in , then it follows by Lemma 18 that .

Therefore, the proof of Theorem 5 is completed.

##### 3.2. The Proof of Theorem 7

Since and , it follows that By applying (65), from (52) and (53), and since , we can get Then, from Lemma 21, we have , where and . 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 , we have . From (47)–(49), we can get where () are the distinct zeros of . And from (51) and (67), by Lemma 20, we have Then, from (50) and (68), since and , we have Similarly, we can get Thus, by Lemma 22, we have where and . By using arguments similar to those in the proof of Theorem 5, we can get that .

Therefore, this completes the proof of Theorem 9.

##### 4.2. The Proof of Theorem 11

Since and , it follows that By applying (72), from (69) and (70), since , we can get Then, from Lemma 22, we have , where and . 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).