Journal of Function Spaces
Volume 2015 (2015), Article ID 426576, 7 pages
http://dx.doi.org/10.1155/2015/426576

Research Article

Some New Results on Fixed Points of Meromorphic Functions Defined in Annuli

Zhaojun Wu,¹ Zuxing Xuan,² and Yuxian Chen³

¹School of Mathematics and Statistics, Hubei University of Science and Technology, Xianning 437100, China
²Beijing Key Laboratory of Information Service Engineering, Department of General Education, Beijing Union University, No. 97 Bei Si Huan Dong Road, Chaoyang District, Beijing 100101, China
³School of Mathematics and Computer Science, Xinyu University, Xinyu 338004, China

Received 16 July 2015; Accepted 9 November 2015

Academic Editor: Adrian Petrusel

Copyright © 2015 Zhaojun Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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. Consider

where . This leads to

Applying the first fundamental theorem, we get

Substituting (21) into (20), noting that , , we have

Lemma 6. Suppose that is an admissible meromorphic function in , . Then

where 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 have

Now we estimate . Firstly, we have

In [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 have

From (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 have

Denoting , by (31) we derive

If 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. Consider

This leads to

Applying the first fundamental theorem, we get

Substituting (36) into (35), we have

Denoting , by (37) we derive

Since , by (38), we can get

Since , then there is a positive number such that

If 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 have

Since

therefore,

It follows from (43)–(45) and Theorem D that

In particular, we have

Applying (46) to the function we obtain

Further application of (46) gives

On the other hand, in view of Theorems C and D, one has

If is a pole of order , it is a pole of or order . So and by the given condition . Therefore, we get from (50) that

Combining (47), (48), and (49) with (51), we have

By Theorem C and (10), we see that

If 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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Zuxing Xuan is the corresponding author and he is supported in part by NNSFC (no. 91420202) and the Project of Construction of Innovative Teams and Teacher Career Development for Universities and Colleges under Beijing Municipality (CIT and TCD201504041, IDHT20140508). This research was partly supported by the National Natural Science Foundation of China (Grant no. 11201395) and by the Science Foundation of Educational Commission of Hubei Province (Grant no. Q20132801). Yuxian Chen is supported in part by University Science and Technology Project of Jiangxi Province (Grant no. GJJ14774).

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