Abstract

The purpose of this paper is to investigate the fixed points of meromorphic functions in annuli. Some well-known facts of fixed points for meromorphic functions in the plane will be considered in annuli.

1. Introduction and Main Results

Let be a meromorphic function. A point is called a fixed point if . There are a considerable number of results on the fixed points for meromorphic functions in the plane; we refer the reader to Chuang and Yang [1]. In 1988, Zhu [2] has proved the following.

Theorem A. Let be a transcendental meromorphic function in the plane. Then either or has infinitely many fixed points.
In the 1960s, Baker [3] proved that if is a transcendental entire function in the plane such that there exists with , then has fixed points of order one. In 1993, Lahiri [4] gave an extension of Baker’s results and proved the following theorem.

Theorem B. Let be a transcendental meromorphic function in the plane. Suppose that there exists with and . Then has infinitely many fixed points.
In recent ten years some well-known facts of the value distribution theory for meromorphic function in the plane were extended for the meromorphic function in the annuli: In 2005, Khrystiyanyn and Kondratyuk [5, 6] gave an extension of the Nevanlinna value distribution theory for meromorphic functions in annuli. In their extension the main characteristics of meromorphic functions are one-parameter and possess the same properties as in the classical case of a simply connected domain. In [5, 6], we can get the analogues of Jensen’s formula, the first fundamental theorem, the lemma on logarithmic derivative, and the second fundamental theorem of the Nevanlinna theory for meromorphic functions in annuli. Lund and Ye [7] investigated logarithmic derivatives in the annuli . After [5, 6], Cao et al. [8–11] study the uniqueness of the meromorphic functions in annuli, Chen and Wu [12] study the exceptional values for meromorphic function and its derivatives in annuli, and Fernández [13] studies the value distribution of meromorphic functions in the punctured plane . The main purpose of this paper is to study the fixed points of meromorphic functions in annuli. We will prove the following theorems by using the similar method as in [2, 4].

Theorem 1. Let be an admissible meromorphic function in , . Then either or has infinitely many fixed points.

Theorem 2. Let be an admissible meromorphic function in , . Suppose that there exists with and . Then both and have infinitely many fixed points.

2. Nevanlinna Theory in Annuli

In the following, we introduce the definitions, notations, and results of [5, 6, 8, 14] which will be used in this paper. Let be a meromorphic function in . Denote where and . Let Put where , is the counting function of poles of the function in , and is the counting function of poles of the function in . Denote also where , is the counting function of poles of the function in , and is the counting function of poles of the function in . Let Finally, we define the Nevanlinna characteristic of in by where . Suppose that , are two meromorphic functions in , where . Then

Definition 3. Let be a nonconstant meromorphic function in . One calls admissible if or

Definition 4. Let be a nonconstant meromorphic function in , , and . Then the value is called the deficiency of the function for the value . For , one sets

Throughout, we denote by quantities that satisfy the following:(1)for the case ,  for except for the set such that ;(2)for the case ,  for except for the set such that .Thus, for an admissible meromorphic function in , , holds for all except for the set or the set mentioned above, respectively. Under the above notations, we give the following theorems which will be used in the proof of Theorems 1 and 2.

Theorem C (the first fundamental theorem). Let be a nonconstant meromorphic function in , where . Then for every fixed

Theorem D (lemma on the logarithmic derivative). Let be an admissible meromorphic function in , . Then holds for every positive integer .

3. Proof of Theorem 1

Lemma 5. Let be an admissible meromorphic function in , . Then

Proof. Considerwhere . This leads toApplying the first fundamental theorem, we getSubstituting (21) into (20), noting that , , we have

Lemma 6. Suppose that is an admissible meromorphic function in , . Thenwhere is the function of which denote the number of simple poles of .

Proof. We define where . Let be a simple pole of . Then as in [2] we have . Therefore, when , by the first fundamental theorem and (10), noting that , , we haveNow we estimate . Firstly, we haveIn [2], Zhu indicated that poles of can occur only at zeros of or . But every zero with multiplicity of gives a pole of once at most, for it must be the zero of with multiplicity being at least . So we haveFrom (25), (26), and (27) we derive By a similar discussion as in [2], one can prove Lemma 6 in the case of .

We are now in the position to prove Theorem 1.

Proof. Note , where is the function of which denote the number of multiple poles of , ignoring multiplicity. By (18) we have Thus,By (18), (23), and (30) we haveDenoting , by (31) we deriveIf both and have only a finite number of fixed points, then from (32) we would have This leads to a contradiction and Theorem 1 is proved.

4. Proof of Theorem 2

Proof. ConsiderThis leads toApplying the first fundamental theorem, we getSubstituting (36) into (35), we haveDenoting , by (37) we deriveSince , by (38), we can getSince , then there is a positive number such thatIf have only a finite number of fixed points, then from (39) and (40) we would have This leads to a contradiction and has infinitely many fixed points. In the following, we will prove that also has infinitely many fixed points under the condition of Theorem 2.
If are distinct finite complex numbers, put Then as in [15] we haveSince therefore,It follows from (43)–(45) and Theorem D thatIn particular, we haveApplying (46) to the function we obtainFurther application of (46) givesOn the other hand, in view of Theorems C and D, one hasIf is a pole of order , it is a pole of or order . So and by the given condition . Therefore, we get from (50) thatCombining (47), (48), and (49) with (51), we haveBy Theorem C and (10), we see thatIf has only a finite number of fixed points, then from (40) and (53) we would have This leads to a contradiction and has infinitely many fixed points.