Results on the growth of meromorphic solutions of some linear difference equations with meromorphic coefficients
© Yuan and Ling; licensee Springer. 2014
Received: 9 July 2014
Accepted: 14 November 2014
Published: 3 December 2014
In this paper, we investigate the growth of meromorphic solutions of some linear difference equations. We obtain some new results on the growth of meromorphic solutions when most coefficients in such equations are meromorphic functions, which are supplements of previous results due to Li and Chen (Adv. Differ. Equ. 2012:203, 2012) and Liu and Mao (Adv. Differ. Equ. 2013:133, 2013).
1 Introduction and main results
Recently, meromorphic solutions of complex difference equations have become a subject of great interest from the viewpoint of Nevanlinna theory due to the apparent role of the existence of such solutions of finite order for the integrability of discrete difference equations.
About the growth of meromorphic solutions of some linear difference equations, some results can be found in [4–20]. Laine and Yang  considered the entire functions coefficients case and got the following.
Theorem A 
we have .
then we have .
Recently in , Peng and Chen investigated the order and the hyper-order of solutions of some second-order linear differential equations and proved the following results.
Theorem C 
has infinite order and .
Moreover, Xu and Zhang  extended the above result from entire coefficients to meromorphic coefficients.
It is well known that is regarded as the difference counterpart of . Thus a natural question is: Can we change the above second-order linear differential equation to the linear difference equation? What conditions will guarantee that every meromorphic solution will have infinite order when most coefficients in such equations are meromorphic functions?
Li and Chen  considered the following difference equation and obtained the following theorem.
Theorem D 
where , , (≢0) () are all entire functions and , then we have .
The main purpose of this paper is to investigate the growth of meromorphic solutions of certain linear difference equations with meromorphic coefficients. The remainder of the paper studies the properties of meromorphic solutions of a nonhomogeneous linear difference equation. In fact, we prove the following results, in which there are still some coefficients dominating in some angles.
Theorem 1.3 Under the assumption for the coefficients of (1.1) in Theorem 1.1, if is a finite order meromorphic solution to (1.1), then . What is more, either or .
Theorem 1.4 Under the assumption for the coefficients of (1.2) in Theorem 1.2, if is a finite order meromorphic solution to (1.2), then . What is more, either or .
one of their results can be stated as follows.
Theorem E 
Let (), where are polynomials with degree n (≥1), (≢0) are entire functions of . If () are distinct complex numbers, then every meromorphic solution f (≢0) of Eq. (1.3) satisfies .
In this paper, we extend and improve the above result from entire coefficients to meromorphic coefficients in the case where the polynomials are of degree 1.
Theorem 1.6 Let (), are distinct complex constants, suppose that (≢0), (≢0) are meromorphic functions and , , then every meromorphic solution f (≢0) of Eq. (1.4) satisfies .
where (≢0) is a meromorphic function.
Theorem 1.7 Let () satisfy the hypothesis of Theorem 1.5 or Theorem 1.6, and let be a meromorphic function of , then at most one meromorphic solution of Eq. (1.5) satisfies and , the other solutions f satisfy .
2 Some lemmas
In this section, we present some lemmas which will be needed in the sequel.
Lemma 2.1 
Lemma 2.2 
Lemma 2.3 
- (i)if , then
- (ii)if , then
where is a finite set.
Lemma 2.3 applies in Theorem 1.1 where (≢0) is an entire function.
Lemma 2.4 
where is a finite set, which has linear measure zero.
In Lemma 2.4, if is replaced by , then we have the same result.
Lemma 2.5 
- (i)if , then
- (ii)if , then
Lemma 2.5 applies in Theorem 1.2 where (≢0) is a meromorphic function.
Lemma 2.6 
are not constant functions for ,
(, ), where E is an exceptional set of finite linear measure, and .
Lemma 2.7 
Let , where are polynomials with degree n (≥1), (≢0) are meromorphic functions of order . If () are distinct complex numbers, then .
3 Proofs of the results
3.1 The proof of Theorem 1.1
for all r outside of a possible exceptional set with finite logarithmic measure.
for all r outside of a possible exceptional set with finite linear measure.
Setting , ().
Case 1. , which is .
Subcase 1.1. Assume that . By Lemma 2.4, for the above ϵ, there is a ray such that (where and are defined as in Lemma 2.4, is of linear measure zero) satisfying , or , for a sufficiently large r.
By , and , we know that (3.9) is a contradiction.
When , , using a proof similar to the above, we can get a contradiction.
Subcase 1.2. Assume that . By Lemma 2.4, for the above ϵ, there is a ray such that (where and are defined as in Lemma 2.4, is of linear measure zero) satisfying .
Since , we see that , then , .
By , and , we know that (3.14) is a contradiction.
Case 2. , which is .
Using the same reasoning as in Subcase 1.1, we can get a contradiction.
Subcase 2.2. Assume that , then . By Lemma 2.4, for the above ϵ, there is a ray such that (where and are defined as in Lemma 2.4, is of linear measure zero), then , , .
Since , , and , then , thus . For a sufficiently large r, we get (3.10), (3.11) and (3.12) hold.
Using the same reasoning as in Subcase 1.2, we can get a contradiction. Thus we have .
3.2 The proof of Theorem 1.2
Then, using a similar argument to that of Theorem 1.1, we obtain a contradiction.
3.3 The proof of Theorems 1.3 and 1.4
Since , we have .
Next, we assert that either or . If the assertion does not hold, we have .
where , are entire functions such that , and is a polynomial such that deg .
Notice that deg for and . Thus, Lemma 2.6 is valid for (3.20), hence we get that for , a contradiction to our assumption. This completes our proof.
The proof of Theorem 1.4 is similar to that of Theorem 1.3.
3.4 The proof of Theorem 1.5
Set , and (). Then is a set of linear measure zero.
- (i)if , then(3.22)
- (ii)if , then(3.23)
Set , then is a set of linear measure zero.
Since are distinct complex constants, then there exists only one such that for any . Now we take a ray such that .
Let , , then . We discuss the following two cases.
This is impossible.
This is a contradiction. Hence we get .
3.5 The proof of Theorem 1.6
- (i)if , then(3.26)
- (ii)if , then(3.27)
Then, using a similar argument to that of Theorem 1.5 and Theorem 1.1 and only replacing (3.22) (or (3.23)) by (3.26) (or (3.27)), we can prove Theorem 1.6.
3.6 The proof of Theorem 1.7
Let be a meromorphic solution of (1.5). Suppose that , then by Lemma 2.7, we obtain . This contradicts , therefore we have .
Suppose that there exist two distinct meromorphic solutions , of Eq. (1.5) such that , then is a meromorphic solution of the homogeneous linear difference equation corresponding to (1.5) and . By Theorem 1.5 or Theorem 1.6, we get a contradiction. So Eq. (1.5) has at most one meromorphic solution satisfying .
where , are entire functions such that , and is a polynomial of degree 1.
Since are distinct complex numbers, by Lemma 2.7, we obtain that the order of the left-hand side of (3.29) is 1. This contradicts .
For , by using an argument similar to the above, we also obtain a contradiction.
It is obvious that provided that . Therefore we have .
The author thanks the referee for his/her valuable suggestions to improve the present article. The research was supported by the Beijing Natural Science Foundation (No. 1132013) and the Foundation of Beijing University of Technology (No. 006000514313002).
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