Fixed Points of Difference Operator of Meromorphic Functions

Abstract

Let f be a transcendental meromorphic function of order less than one. The authors prove that the exact difference has infinitely many fixed points, if and are Borel exceptional values (or Nevanlinna deficiency values) of f. These results extend the related results obtained by Chen and Shon.

1. Introduction and Main Results

In this paper, we assume that the reader is familiar with the notations of frequency use in Nevanlinna theory (see [1–3]). Let be a meromorphic function in the complex plane and . We use the notations to denote the order of , , and , respectively, to denote the exponent of convergence of zeros of and poles of . Especially, if , we denote . A point is called as a fixed point of if . There is a considerable number of results on the fixed points for meromorphic functions in the plane; we refer the reader to Chuang and Yang [4]. It follows Chen and Shon [5]; we use the notation to denote the exponent of convergence of fixed points of that is defined as

Let be a transcendental meromorphic function in the complex plane . The exact differences are defined by .

Recently, there are a number of papers (including [6–16]) focusing on the differences analogues of Nevanlinna's theory and its application on the complex difference equations. For the fixed points of the difference operator , Chen and Shon have proved the following.

Theorem A (see [17]). Let be a transcendental entire function of order of growth and have infinitely many zeros with the exponent of convergence of zeros . Then has infinitely many zeros and infinitely many fixed points.

When the order of is less than 1, Chen and Shon have proved the following.

Theorem B (see [5]). Let be a transcendental meromorphic function of order of growth . Suppose that satisfies or has infinitely many zeros (with ) and finitely many poles. Then has infinitely many fixed points and satisfies the exponent of convergence of fixed points .

A natural question is, letting be a transcendental meromorphic function of order of growth , is there a similar result as that in Theorem B if or has infinitely many zeros (with ) and infinitely many poles?

In this paper, we will prove the following theorem to answer the question.

Theorem 1 (main). Let be a transcendental meromorphic function of order of growth and . Suppose that satisfies and . Then has infinitely many fixed points and satisfies the exponent of convergence of fixed points .

From Theorem 1, we can get the following corollary.

Corollary 2. Let be a transcendental meromorphic function of order of growth . Suppose that satisfies . Then has infinitely many fixed points and satisfies the exponent of convergence of fixed points .

In Theorem 1, we suppose that satisfies and . That is to say and are Borel exceptional values of . If we suppose that and are Nevanlinna deficiency values of , is there a similar result as that in Theorem B? In the following, we give Theorem 3 to answer this question.

Let be a meromorphic function in the complex plane and . Nevanlinna’s deficiency of with respect to is defined by

If , then one should replace in the above formula by . If , then is called a Nevanlinna deficiency value of .

Theorem 3 (main). Let be a transcendental meromorphic function of order of growth and . Suppose that satisfies and is a Nevanlinna deficiency value of . Then has infinitely many fixed points.

Corollary 4. Let be a transcendental entire function of order of growth and . Suppose that . Then has infinitely many fixed points.

2. Some Lemmas

Lemma 1 (lemma on the logarithmic derivative). Let be a meromorphic function. If the function has finite order, then holds for any positive integer .

Lemma 2 (see [18]). Let be a meromorphic function with the exponent of convergence of poles and let be a nonzero complex number. Then for each , we have

Lemma 3. Let be a transcendental meromorphic function of order of growth and let be a nonzero complex number. Then

Proof. Since the order , then . Therefore, for any , it follows from Lemma 2 that That is,

Lemma 4 (see [6]). Let be a function transcendental and meromorphic in the plane which satisfies Then is transcendental.

Lemma 5. Let be a transcendental meromorphic function of order of growth . Then is transcendental.

Proof. Since the order , then, for any positive , there exists such that for any we have
Therefore, Lemma 5 follows Lemma 4.

Lemma 6 (see [7]). Let be a meromorphic function of finite order, then .

Lemma 7 (see [7]). Let be a transcendental meromorphic function of order of growth . Then for any and any positive integer , there exists a set that depends on and has finite logarithmic measure, such that for all satisfying we have

It is easy to derive the following lemma from Lemma 1 and Lemma 7.

Lemma 8. Let be a transcendental meromorphic function of order of growth . Then for any positive integer there exists a set that depends on and has finite logarithmic measure, such that

3. Proof of Theorems

Proof. Since
then
Applying the first fundamental theorem, we get
Combining (14)-(15) we have
Since
then, we can get
Therefore,
Thus from Lemma 3 and (20), we deduce
By Lemmas 5 and 6, we know that is a transcendental meromorphic function of order of growth . It follows from Lemma 8 and (19) that there exists a set that has finite logarithmic measure, such that for any we have
From (16) and (21)-(22), we have
Denoting by (23) we derive,

3.1. The Rest of the Proof of Theorem 1

By Lemma 6, we know that . If , by and , there exists a number , such that for any sufficient   we have

Combining (24) and (25), we can get a contradiction. Therefore, we have .

3.2. The Rest of the Proof of Theorem 3

Since , then . By (24), we can get

Since , then there is a positive number such that

If has only a finite number of fixed points, then from (26) and (27) we would have

This contradicts being transcendental. Therefore, 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

This research was partly supported by the National Natural Science Foundation of China (Grant nos. 11201395 and 61202313), by the Natural Science Foundation of Jiang-Xi Province in China (Grant nos. 20132BAB211001 and 20122BAB201044) and by the Science Foundation of Educational Commission of Hubei Province (Grant nos. Q20132801 and D20132804).