Abstract

The classical Valiron-Mohon'ko theorem has many applications in the study of complex equations. In this paper, we investigate rational functions in f(z) and the shifts of f(z). We get some results on their characteristic functions. These results may be viewed as difference analogues of Valiron-Mohon'ko theorem.

1. Introduction and Results

We use the basic notions of Nevanlinna’s theory in this work (see [1, 2]). Let be a meromorphic function. We say that a meromorphic function is a small function of if , where outside a possible exceptional set of finite logarithmic measure.

The Valiron-Mohon’ko theorem has been proved to be an extremely useful tool in the study of meromorphic solutions of differential, difference, and functional equations. It is stated as follows.

Theorem A (see [3, page 29]). Let be a meromorphic function. Then for all irreducible rational functions in with meromorphic coefficients , such that the characteristic function of satisfies

Recently, a number of papers have focused on difference analogues of Nevanlinna’s theory; see, for instance, [4–12]. Among these papers, difference polynomials are investigated extensively (see [5, 9–11]). But the difference analogues of Valiron-Mohon’ko theorem have not been established. In this paper, we are devoted to this work.

A difference polynomial of is an expression of the form where is an index set, are complex constants, and are nonnegative integers. In what follows, we assume that the coefficients of difference polynomials are, unless otherwise stated, small functions. The maximal total degree of in and the shifts of is defined by

First, we investigate the rational function where is an arbitrary complex number, and and are small functions of with or . Our result is stated as follows.

Theorem 1. Let be a meromorphic function of finite order such that . Suppose that is a difference polynomial in and that is of the form (6). Then

In many papers (see, for instance, [7, 13, 14]), linear difference expressions often appear. Concerning their characteristic functions, we have the following corollary, which is obtained easily from Theorem 1.

Corollary 2. Let be a meromorphic function of finite order such that . Suppose that is a linear combination in and the shifts of . Then

Next we consider the rational function where are different complex constants. We get the following result.

Theorem 3. Let be a meromorphic function of finite order such that . Suppose that is a difference polynomial in and that is of the form (9). Then

As for the general rational function in and the shifts of , we get the following two results.

Theorem 4. Let be a meromorphic function of finite order such that . Suppose that and are difference polynomials in and that is of the form (11).(i)If and contains just one term of maximal total degree, then (ii)If and contains just one term of maximal total degree, then

Theorem 5. Let be a meromorphic function of finite order such that . Suppose that and are difference polynomials in and that is of the form (11). Then

The following two examples show that the results in Theorems 1–5 are sharp; that is, “≤” and “≥” cannot be replaced by “<”, “>” or “”.

Example 6. Let and Let Then and . Clearly, Therefore,

Example 7. Let and Let Then and . Clearly, Therefore,

2. Proof of Theorem 1

We need the following lemmas for the proof of Theorem 1.

The difference analogue of the logarithmic derivative lemma was given by Halburd-Korhonen [8, Corollary 2.2] and Chiang-Feng [7, Corollary 2.6], independently. The following Lemma 8 is a variant of [8, Corollary 2.2].

Lemma 8. Let be a nonconstant meromorphic function of finite order, and let be two arbitrary complex numbers. Then,

In the remark of [15, page 15], it is pointed out that the following lemma holds.

Lemma 9. Let be a nonconstant finite order meromorphic function and let be an arbitrary complex number. Then,

Let be a meromorphic function. It is shown in [16, page 66] that for an arbitrary , the following inequalities: hold as . From its proof we see that the above relations are also true for counting functions. So by these relations and Lemma 9, we get the following lemma.

Lemma 10. Let be a nonconstant finite order meromorphic function and let be an arbitrary complex number. Then,

Remark 11. In [7], Chiang and Feng proved a similar result. Let be a meromorphic function with , and let be fixed; then for each , we have

Proof of Theorem 1. Let and .
Rearranging the expression of by collecting together all terms having the same total degree, we get where, for ,
Since the coefficients of are small functions of , we have So by Lemma 8, we have, for all the estimates
Without loss of generality, we may assume in (6). Otherwise, substituting for , we get By Lemma 10, we see that So, in the following discussion, we only discuss the form
Assume first that . Clearly, we may assume that . By (29), we get If , then . So by (32), we get If , then rewrite in the form So we have By (39) and the inductive argument, we have
To estimate , we use the form Clearly, So by (31), , and Lemma 10, we get Combining this equality with (40), we get and we have completed the case .
We now proceed to the case . Clearly, in this case we may assume that . By (29), we see that (6) becomes By (45), we get where By (32), we get, for , the estimates By (46), using the same method as in (36)–(40), we get
To estimate , we use the form By (31), , and Lemma 10, we get Combining this equality with (49), we get Theorem 1 is proved.

3. Proof of Theorem 3

Proof. Let be of the form (28) and . Rearranging the expression of , we get (29) and (30). We only discuss the case since the case is easier.
Rewrite in the form where By Lemma 8, we get By (29) and (53), we get By (32) and (55), we have, for all , the estimates By (57), using the same method as in (36)–(40), we get Combining the above two inequalities with (56), we get
To estimate , we use the form By (31), , and Lemma 10, we get Combining this inequality with (59), we get Theorem 3 is proved.

4. Proof of Theorem 4

We need the following lemma for the proof of Theorem 4.

Lemma 12 (see [11]). Let be a meromorphic function of finite order such that . Suppose that is a difference polynomial in and contains just one term of maximal total degree. Then,

Proof of Theorem 4. We have the following.
Case  1. Suppose that and contains just one term of maximal total degree.
Let and . By Lemma 12, we get By Theorem 1, we get By (11), we get By (64)–(66), we get So we have,
Case  2. Suppose that and contains just one term of maximal total degree.
In this case, we consider . Using the same method as in Case 1, we can easily get Theorem 4 is proved.

5. Proof of Theorem 5

Proof. Let be of the form (28) and . Let and .
Rearranging the expression of , we get (29) and (30).
Similarly, rearranging the expression of , we get where, for , By (29) and (71), we get Since , by Lemma 10, we have, for an arbitrary , By (74) and Lemma 8, we have, for an arbitrary , Since the coefficients and of and are small functions of , by (30), (72), and (75), we get
By (73), we are not clear whether is an irreducible rational function in . So by Theorem A, we get Theorem 5 is proved.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11226090 and 11171119) and Guangdong Natural Science Foundation (S2012040006865).