#### Abstract

The purpose of this paper is to deal with the shared values and uniqueness of analytic functions on annulus. Two theorems about analytic functions on annulus sharing four distinct values are obtained, and these theorems are improvement of the results given by Cao and Yi.

#### 1. Introduction

In this paper, we will study the uniqueness problem of analytic functions in the field of complex analysis and adopt the standard notations of the Nevanlinna theory of meromorphic functions as explained (see [1–3]).

We use to denote the open complex plane, to denote the extended complex plane, and to denote the subset of . For , we say that and have the same zeros with the same multiplicities (ignoring multiplicities) in (or ) if two meromorphic functions and share the value in (or ). In addition, we also use in (or ) to express that and share the value in (or ), in (or ) to express that and share the value in (or ), and in (or ) to express that implies in (or ).

In 1929, Nevanlinna (see [4]) proved the following well-known theorem.

Theorem 1.1 (see [4]). *If and are two nonconstant meromorphic functions that share five distinct values , and in , then . *

After his theorem, the uniqueness theory of meromorphic functions sharing values in the whole complex plane attracted many investigations (see [2]). In 2003, Zheng [5] studied the uniqueness problem under the condition that five values are shared in some angular domain in . There were many results in the field of the uniqueness with shared values in the complex plane and angular domain, see ([5–12]). The whole complex plane and angular domain all can be regarded as simply connected region. *Thus, it is interesting to consider the uniqueness theory of meromorphic functions in the multiply connected region.* Here, we will mainly study the uniqueness of meromorphic functions in doubly connected domains of complex plane . By the doubly connected mapping theorem [13] each doubly connected domain is conformally equivalent to the annulus , . We consider only two cases: , simultaneously and . In the latter case the homothety reduces the given domain to the annulus , where . Thus, every annulus is invariant with respect to the inversion in two cases.

In 2005, Khrystiyanyn and Kondratyuk [14, 15] proposed the Nevanlinna theory for meromorphic functions on annuli (see also [16]). We will show the basic notions of the Nevanlinna theory on annuli in the next section. In 2009, Cao et al. [17, 18] 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 1.2 (see [18, Theorem 3.2]). *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 1.3. *For the case , the assertion was proved by Kondratyuk and Laine [16].

From Theorem 1.2, we can get the following results easily.

Theorem 1.4. *Under the assumptions of Theorem 1.2, if are two transcendental or admissible analytic functions on annulus and share for , then .*

In fact, we will prove some general theorems on the uniqueness of analytic functions on the annuli sharing four values in this paper (see Section 3), and these theorems improve Theorem 1.4.

#### 2. Basic Notions in the Nevanlinna Theory on Annuli

Let be a meromorphic function on the annulus , where . We recall the classical notations of the Nevanlinna theory as follows: where and is the counting function of poles of the function in . We here show the notations of the Nevanlinna theory on annuli. Let where and are the counting functions of poles of the function in and , respectively. The Nevanlinna characteristic of on the annulus is defined by and has the following properties.

Proposition 2.1 (see [14]). *Let be a nonconstant meromorphic function on the annulus , where . Then,*(i),
(ii).

By Proposition 2.1, the first fundamental theorem on the annulus is immediately obtained.

Theorem 2.2 (see [14] (the first fundamental theorem)). *Let f be a nonconstant meromorphic function on the annulus , where . Then
**
for every fixed .*

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

Theorem 2.3 (see [15] (lemma on the logarithmic derivative)). *Let f 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 .*

We denote the deficiency of with respect to a meromorphic function on the annulus by and denote the reduced deficiency by where in which each zero of the function is counted only once. In addition, we use (or ) to denote the counting function of poles of the function with multiplicities (or ) in , each point counted only once. Similarly, we can give the notations , , , , , and .

Khrystiyanyn and Kondratyuk [15] first obtained the second fundamental theorem on the the annulus . Later, Cao et al. [18] introduced other forms of the second fundamental theorem on annuli as follows.

Theorem 2.4 (see [18, 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 . Let . Then, *(i)*,
*(ii)*,
** where
**
and (i) in the case ,
**
for except for the set such that ; (ii) if , then
**
for except for the set such that .*

*Definition 2.5. *Let be a nonconstant meromorphic function on the annulus , where . The function is called a transcendental or admissible meromorphic function on the annulus provided that
or
respectively.

Thus, for a transcendental or admissible meromorphic function on the annulus , holds for all except for the set or the set mentioned in Theorem 2.3, respectively.

#### 3. The Main Theorems and Some Lemmas

Now we show our main results, which improve Theorem 1.4.

Theorem 3.1. *Let be two analytic functions on the annulus , where , and let be four distinct values. If and share the two distinct values in and in and in , and is transcendental or admissible on , then . *

Theorem 3.2. *Under the assumptions of Theorem 3.1, with replaced by , we have either or
**
and , , and are exceptional values of and in , respectively. *

*Remark 3.3. *It is easily seen that Theorems 3.1 and 3.2 are improvement of Theorem 1.4.

To prove the above theorems, we need some lemmas as follows.

Lemma 3.4. *Let be two distinct analytic functions on the annulus , where , and let be four distinct complex numbers. If in for and if is transcendental or admissible on , then is also transcendental or admissible. *

* Proof. *By the assumption of Lemma 3.4 and applying Theorem 2.4(ii), we can get

Therefore
holds for all except for the set or the set mentioned in Theorem 2.3, respectively. Then, from Definition 2.5, we get that is transcendental or admissible on .

Lemma 3.5. *Suppose that is a transcendental or admissible meromorphic function on the annulus , where . Let be a polynomial of with degree , where the coefficients are constants, and let be distinct finite complex numbers. Then,
*

*Proof. *From Theorem 2.3 and the definition of , transcendental and admissible function, we can get this lemma by using the same argument as in Lemma 4.3 in [2].

Lemma 3.6. *Let be two distinct analytic functions on the annulus , where . Suppose that and share in , and in and in , and are four distinct finite complex numbers. If is a transcendental or admissible function on , then is also transcendental or admissible, and*(i)*,
*(ii)*;
*(iii)*,
*(iv)*,
*(v)*,
*(vi)*, **where . *

* Proof. *By the assumption of this lemma and by Theorem 2.4(ii), we have and . Thus, we can get .

Let

From the conditions of this lemma, we can get that is analytic on and unless . By Lemma 3.5, we have . Thus, we can get .

Since are two nonconstant analytic functions on annulus and share in and and in , again by Theorem 2.4, we have

From (3.8) and (3.11), we can get (i), and from (3.7), (3.8), and (i), we can get (ii), and from (3.6), (3.8), (3.10), (3.11), and (i), we can get (iii). Thus, we can deduce that (iv) and (v) hold easily from (3.6)–(3.11) and (i)–(iii). Now, we will prove that (vi) holds as follows.

First, we can rewrite (3.5) as

From (3.12) and Lemma 3.5, we can get . Since is analytic on , we have
From the fact that is transcendental or admissible, we have
On the other hand, since , from Theorem 2.3, we have . Thus, we can get

From (3.14), (3.15) and the fact that is transcendental or admissible, we can get . Similarly, we can get .

Thus, we complete the proof of this lemma.

#### 4. The Proof of Theorem 3.1

Suppose . By the assumption of Theorem 3.1, we can get the conclusions (i)–(vi) of Lemma 3.6 and that is transcendental or admissible on . Set

By Lemma 3.4, we can get Moreover, we can prove . In fact, the poles of in only can occur at the zeros of and in . Since share in , we can see that if is a zero of with multiplicity , then is a zero of with multiplicity . Suppose that where are analytic functions in and ; by a simple calculation, we have where is a constant. Therefore, we can get that are analytic in . Thus, from (4.2), we can get .

If , then we have From (4.5) and Lemma 3.6(iv), we have . Thus, since are transcendental or admissible functions on , that is, and are of unbounded characteristic, and from the definition of , we can get a contradiction.

Assume that one of and is identically zero, say ; then we have

From (3.5), we can see that implies that for such satisfying . Since , we have

From (4.6) and (4.7), we can get

Similarly, when , we can get

From (4.8), (4.9), and Lemma 3.6(i), (v), we can get or Since are transcendental or admissible functions on the annulus , we can get a contradiction again.

Thus, we complete the proof of Theorem 3.1.

#### 5. The Proof of Theorem 3.2

Suppose that . By Theorem 2.4(ii) and the fact that is transcendental or admissible on , we have Therefore, we have Similarly, we have

From (5.2) and (5.3), we can see that , and

Thus, from (5.2), (5.3), and the definition of , we can get that is also transcendental or admissible on when is transcendental or admissible on .

From (5.1)–(5.4), we can also get

From (5.5), we can see that “almost all” of zeros of in are simple. Similarly, “almost all” of zeros of in are simple, too. Let By Lemma 3.5, we can get that . Since share in and from (5.2), we have . Therefore, we can get .

If , then we have . Thus, from (5.5), we can get a contradiction easily. Similarly, when , we can get a contradiction, too. Hence, are identically equal to 0. Then, we have , that is, which implies that where is a nonzero constant. Rewrite (5.8) as where . The discriminant of (5.9) is where is a polynomial of degree 4 in . If is a zero of in , obviously . Then, from (5.9), implies that Set

By Lemma 3.5, we can get . And by a simple calculation, we can get . Then we have , thus we have .

Assume that is not a Möbius transformation of ; then is a nonconstant function. Since From and (5.8), we can get where . Since is transcendental or admissible analytic in , by Theorem 2.4(ii) and (5.4), we can get

Since are transcendental or admissible functions on , from the above inequality, we can get a contradiction. Therefore, we can get that is a Möbius transformation of on . Since are transcendental or admissible functions on , by a simple calculation, we can get easily that and Furthermore, are Picard exceptional values of and in , respectively.

Thus, we complete the proof of Theorem 3.2.

#### Acknowledgments

Z.-X. Xuan is supported in part by Beijing Municipal Research Foundation for The Excellent Scholars Ministry (2011D005022000009); Science and Technology Research Program of Beijng Municipal Commission of Education (KM201211417011); Funding Project for Academic Human Resources Development in Beijing Union University. This work was supported by the NSF of Jiangxi of China (Grant no. 2010GQS0119).