Abstract
We mainly investigate the unicity of meromorphic functions sharing two or three sets with their linear difference polynomials and prove some results.
1. Introduction and Main Results
In this paper, we assume the reader is familiar with the fundamental results and the basic notations of the Nevanlinna theory of meromorphic functions (see, e.g., [1–3]). Let be meromorphic in the whole plane. We use the notation to denote the order of growth of the meromorphic function . In addition, we denote by any quantity satisfying , as outside of a possible exceptional set of finite logarithmic measure. We say that a meromorphic function is a small function of provided that . Let be the set of all small functions of .
For a set , we define the following:
Let and be meromorphic functions. If and , respectively, then we say that and share a set CM and IM, respectively.
Furthermore, let be a nonzero complex constant. We define the shift of by , and define the difference operators of by
The unicity theory of meromorphic functions sharing sets is an important topic of the uniqueness theory. First of all, we recall the following theorem given by Li and Yang in [4].
Theorem A (see [4]). Let and let with and having no common factors. Let and be two nonzero constants such that the equation has no multiple roots. Let . Then, for any two nonconstant meromorphic functions and , the conditions and imply .
Yi and Lin considered the case with the condition that two meromorphic functions share three sets and got the result as follows.
Theorem B (see [5]). Let , , and , where , are nonzero constants such that has no repeated root and is an integer. If, for two nonconstant meromorphic functions and , for , and , then .
Recently, a number of papers have focused on difference analogues of the Nevanlinna theory (see, e.g., [6–9]). In particular, there has been an increasing interest in studying the uniqueness problems related to meromorphic functions and their shifts or their difference operators (see, e.g., [10–16]).
In 2010, Zhang considered a meromorphic function sharing sets with its shift and proved the following result.
Theorem C (see [16]). Let and let with and having no common factors. Let and be two nonzero constants such that the equation has no multiple roots. Let . Suppose that is a nonconstant meromorphic function of finite order. Then and imply .
For an analogue result in difference operator, B. Chen and Z. Chen proved the following theorem in [10].
Theorem D (see [10]). Let and let with and having no common factors. Let and be two nonzero constants such that the equation has no multiple roots. Let . Suppose that is a nonconstant meromorphic function of finite order satisfying and . If then .
It is natural to ask what happens if the shift or difference operator is replaced by a general expression of , such as a linear difference polynomial of .
Here, a linear difference polynomial of is an expression of the form where , are small functions of , are complex constants, and is a nonnegative integer.
In this paper, our aim is to investigate the uniqueness problems of linear difference polynomials of . In particular, we primarily consider the linear difference polynomial which satisfies one of the following conditions:
Corresponding to the above question, we obtain the following results.
Theorem 1. Let and let with and having no common factors. Let and be two nonzero constants such that the equation has no multiple roots. Let . Suppose that is a nonconstant meromorphic function of finite order and is of the form (4) satisfying the condition in (5). If and , then .
Corollary 2. Let , , and be given as in Theorem 1. Suppose that is a nonconstant meromorphic function of finite order satisfying the following: If and , then .
With an additional restriction on the order of growth of , we prove the following fact.
Theorem 3. Let , , and be given as in Theorem 1. Suppose that is a nonconstant meromorphic function of finite order such that . If and , then .
Remark 4. Note that, in Theorem 3, we do not assume that the linear polynomial satisfies the condition in (5). In fact, since , by (19), we can easily get , which implies . Then using a similar method as in the proof of Theorem 1, we can complete the proof of Theorem 3.
Now we may ask what happens if the condition in Theorem 1 is replaced by a weaker condition containing the case or even . By considering three sets, we get the following theorem.
Theorem 5. Let , be nonnegative integers such that . Let and be nonzero constants such that has no multiple roots. Let , , and . Suppose that is a nonconstant meromorphic function of finite order, is of the form (4) satisfying the condition in (5), and for . Then one has the following.(i)If , then , where .(ii)If and are coprime, then .
Remark 6. Taking in Theorem 5, we can obtain an analogue result of Theorem B related to linear difference polynomials.
Furthermore, the following result is a corollary of Theorem 5 related to difference operators.
Corollary 7. Let , , and , , be given as in Theorem 5. Suppose that is a nonconstant meromorphic function of finite order satisfying and for . Then one has the following. (i)If , then , where .(ii)If and are coprime, then .
Finally, we give some examples for our results.
Examples. In the following, let be an entire function with period 1 such that (see [17]).(1)For the case (i) of condition (5), let , , and let . Then for , and the sum of the coefficients of is equal to . These examples satisfy Theorems 1 and 5 but do not satisfy Theorem D.(2)For the case (ii) of condition (5), let and let . Then , the sum of the coefficients of equals , and This example satisfies Theorems 1 and 5 and Corollaries 2 and 7.(3)For Theorem 3, let and let . Then and the sum of the coefficients of equals . This example satisfies Theorem 3 but does not satisfy Theorem D and Theorems 1 and 5.
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 [7] and Chiang-Feng [6] independently. We recall the following lemmas.
Lemma 8 (see [7]). Let be a nonconstant meromorphic function of finite order, and . Then for all outside of a possible exceptional set with finite logarithmic measure.
Lemma 9 (see [8]). Let , let , and let be a meromorphic function of finite order. Then for any small periodic function with period , consider the following: where the exceptional set associated with is of at most finite logarithmic measure.
Let be a meromorphic function of finite order. Notice that if (≢0) is of the form (4) such that , then, for any given complex constant , . This indicates that and hence With this, one can easily prove Lemma 10 below by a similar reasoning as in the proof of the difference analogue of the second main theorem of the Nevanlinna theory in [8] by Halburd and Korhonen. We omit those details.
Lemma 10. Let , let be a meromorphic function of finite order, and let be of the form (4) such that . Let and let be distinct complex constants. Then where and the exceptional set associated with is of at most finite logarithmic measure.
Remark 11. If the linear difference polynomial is replaced by Lemma 10 also holds even if the distinct complex constants are replaced by which are distinct meromorphic periodic functions with period such that for all .
The following is the standard Valiron-Mohon’ko theorem; (see Theorem 2.2.5 in the book of Laine [2]).
Lemma 12 (see [2]). Let be a meromorphic function. Then, for all irreducible rational functions in , with meromorphic coefficients , such that The characteristic function of satisfies where .
Proof of Theorem 1. Since and share CM, we see that and . Then by Lemma 8, we have
Since , where and the equation has no multiple roots, we know that and share 0 CM. Then from this and the condition , there exists a polynomial such that
Suppose that . Note that and the equation has no multiple roots. Let denote all different roots of the equation .
Next we prove that . We know that
(i)If , we see that
Then we deduce from this, (19), and Lemma 9 that
(ii)If , we have
From this, (19), and Lemma 9, we get
Applying Lemma 10 to , we get
Then the assumptions in (5), (24), and (25) yield the following:
To sum up, we now prove that . Rewriting (19), we get
Denote . It follows from Lemma 12 and that
Hence, .
By (18) and (27) and applying the second main theorem for three small target functions, we deduce the following:
By combining (28) and (29), we have
which contradicts with .
Now we turn to consider the case . Equation (19) yields the following:
Set , and we have
If is not a constant, (32) can be rewritten as
where and .
By the assumption that and have no common factors, we see that , are different. Assume that is a -point of of multiplicity , where . Notice that
is a constant. Then (33) implies that is a pole of . Thus, . This yields the following, for :
Then by (35), we get
which is impossible with and .
Hence, is a constant. Since is a nonconstant meromorphic function, we deduce from (32) that . This yields , which completes the proof of Theorem 1.
3. Proof of Theorem 5
Since is a nonconstant meromorphic function of finite order, for , , , and , we have , , and , and we also get (18) and (19).
Since and share , CM, there exists a polynomial such that
By Lemma 8, we see that
As in the proof of Theorem 1, we see that still holds in both cases (i) and (ii).
Rewriting (19), we have Combining this and (37), we get
Suppose that . If , (40) becomes By the condition that , implies .
It follows from (38), (41), and that which is a contradiction, since .
If , it follows from (38), (41), and that That is impossible.
Therefore, . Notice that . Using a similar method, we can prove that . Then (40) implies that .
If , we have . Obviously, is a constant. Set . Thus, by (37), we get , where .
If and are coprime, and imply that . Thus, by (37), we get . Thus, Theorem 5 is proved.
Conflict of Interests
The authors declare that they have no conflict of interests.
Authors’ Contribution
Both authors drafted the paper and read and approved the final paper.
Acknowledgments
The authors are grateful to the editor and referees for their valuable suggestions. This work was supported by the NNSFC (no. 11171119, 11301091), the Guangdong Natural Science Foundation (no. S2013040014347), and the Foundation for Distinguished Young Talents in Higher Education of Guangdong (no. 2013LYM_0037).