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Uniqueness problems on entire functions that share a small function with their difference operators
Advances in Difference Equations volume 2014, Article number: 311 (2014)
In this paper, we consider uniqueness problems on entire functions that share a small periodic entire function with their two difference operators and obtain some results. Our first theorem provides a difference analogue of a result of Li and Yang (J. Math. Anal. Appl. 253(1):50-57, 2001).
1 Introduction and main results
Throughout this paper, a meromorphic function always means meromorphic in the whole complex plane, and c always means a nonzero constant. We use the basic notations of the Nevanlinna theory of meromorphic functions such as , , and as explained in [1–3]. In addition, we say that a meromorphic function is a small function of if , where , as outside of a possible exceptional set of finite logarithmic measure.
For a meromorphic function , we define its shift by , and define its difference operators by
In particular, for the case .
Let and be two meromorphic functions, and let be a small function of and . We say that and share IM, provided that and have the same zeros ignoring multiplicities. Similarly, we say that and share CM, provided that and have the same zeros counting multiplicities.
The problem on meromorphic functions sharing small functions with their derivatives is an important topic of uniqueness of meromorphic functions.
In 1986, Jank, Mues and Volkmann  proved the following result.
Theorem A ()
Let f be a nonconstant meromorphic function, and let be a finite constant. If f, , and share the value a CM, then .
Many authors have been considering about some related cases, and got some interesting results (see, e.g., [5, 6]). In 2001, Li and Yang  obtained the following result for a special case that is an entire function, and f, , and share one value.
Theorem B ()
Let be an entire function, let a be a finite nonzero constant, and let n (≥2) be a positive integer. If f, , and share the value a CM, then f assumes the form
where b, c are nonzero constants and .
Recently, a number of papers (including [7–12]) have focused on difference analogues of Nevanlinna theory. In addition, many papers have been devoted to the investigation of the uniqueness problems related to meromorphic functions and their shifts or their difference operators and got a lot of results (see, e.g., [13–15]).
Our aim in this paper is to investigate uniqueness problems on entire functions that share a small periodic entire function with their two difference operators and provide a difference analogue of Theorem B. We now state the following theorem, which is the main result of this paper.
Theorem 1.1 Let be a nonconstant entire function of finite order, and let be a periodic entire function with period c. If , , and () share CM, then .
Let , then , and hence , Δf, and share 1 CM, but . This example shows that the conclusion in Theorem 1.1 cannot be extended to in general.
Let , then , , and hence , Δf and share 0 CM, but (). This example shows that the restriction in Theorem 1.1 is necessary.
Remark In the above example (1), can be changed to , where is a periodic entire function with period 1, and the result still holds. This shows that the order of the function in Theorem 1.1 is not always one.
As a continuation of Theorem 1.1 and example (2) above, we prove the following result.
Theorem 1.2 Let be a nonconstant entire function of finite order. If , , and () share 0 CM, then , where C is a nonzero constant.
2 Proof of Theorem 1.1
Firstly, we present some lemmas which will be needed in the proof of Theorem 1.1.
Lemma 2.1 ()
Let , , and let be a meromorphic function of finite order. Then for any small periodic function with period c, with respect to ,
where the exceptional set associated with is of at most finite logarithmic measure.
Lemma 2.2 ()
Suppose that () and () () are entire functions satisfying
the orders of are less than that of for , ,
Proof of Theorem 1.1 Suppose on the contrary to the assertion that . Note that is a nonconstant entire function of finite order. By Lemma 2.1, for , we have
Since , , and share CM, we have
where and are polynomials.
From (2.1) and (2.2), we get . Then by supposition and (2.1), we see that . By Lemma 2.1, we deduce that
Note that . By using the second main theorem and (2.3), we have
Thus, by (2.3) and (2.4), we have . Similarly, .
Now we divide this proof into the following two steps.
Step 1. Suppose that is not a constant. Now we rewrite the second equation in (2.1) as
where , , .
We deduce that
Let be a finite set of n elements, and denote , where ∅ is an empty set. Then by an argument similar to the above, we deduce that
where A is any element of , , and , for , () are nonzero constants. In particular, , and
Moreover, is a polynomial of and its shifts .
Now set , where are constants satisfying and . Obviously, for , we have
By the above equation and (2.9), we obtain
Here , for , , are polynomials with degree less than m.
Rewrite the first equation in (2.1) as
This together with (2.8) gives
Notice that , , and . If , (2.12) yields
That is impossible.
Hence . This together with (2.11) gives
Now we distinguish three cases as follows:
Case (i). Suppose that . Then, for any , we see that
and for , we have
Since , for , , are polynomials with degree less than m, it is easy to see that, for ,
By Lemma 2.2, we have , which is impossible.
Case (ii). Suppose that . Then, by a similar argument to above, we can also get a contradiction.
Case (iii). Now suppose that . Set , with and . Rewrite (2.13) as
Subcase (i). If , for any , then we have
By this together with (2.14), (2.15), (2.16), and Lemma 2.2, we can get a contradiction.
Subcase (ii). If , for some . Without loss of generality, we assume that . Then we rewrite (2.16) as
By a similar method as the above, we can also get . That is impossible.
Subcase (iii). If , then we rewrite (2.16) as
By a similar argument to the above and Lemma 2.2, we can get
By (2.10) and (2.11), we get
Suppose that . Note that for , we have
where are polynomials with degree less than .
Rewrite (2.19) as
For any , we have
and for , we see that
By this, together with (2.20) and Lemma 2.2, we obtain , which is impossible.
Suppose that , then , with . It is easy to see that
This together with (2.19) gives
which yields . Therefore, for any ,
By the second equation in (2.1) and (2.21), we have
Rewriting (2.22), and combining it with the second equation in (2.1), we obtain
Substituting the first equation in (2.1) into (2.23), we get
If , (2.24) yields
We get a contradiction again.
Hence, . By (2.24), we see that , which implies . That is impossible.
Step 2. Suppose that is a constant. Now we rewrite the second equation in (2.1) as
Since is a periodic function with period c, we have
By Lemma 2.1 and the first equation in (2.1), we deduce that
From (2.27), we have
According to our assumption that and (2.25), it is easy to see that . Now suppose that is a zero of with multiplicity μ. Since , , and share CM, is a zero of and with multiplicity μ. Therefore, is a zero of with multiplicity at least μ. Then by (2.26) and (2.28), we see that
which implies . That is impossible.
Hence, we must have , and Theorem 1.1 is proved. □
3 Proof of Theorem 1.2
Note that is a nonconstant entire function of finite order. Since , , and () share 0 CM, we have
where and are polynomials.
If is a constant, then we can easily get from (3.1)
This completes our proof.
If is not a constant, by assuming that is not a constant, with a similar arguing as in the proof of Theorem 1.1, we can deduce that the case is impossible.
For the case , from (3.1), we can obtain that
Let be a zero of . Now we can find that (2.21) still holds here. Then from (2.21), we have , for all . Therefore, from (3.2), we see that is a zero of , then is a zero of , for all . Suppose that is a zero of of order , then we get from (2.21) and (3.2)
This indicates that is a zero of of order at least , which is impossible. Theorem 1.2 is thus proved.
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This work was supported by the NNSFC (No. 11301091) and the Guangdong Natural Science Foundation (No. S2013040014347) and the Foundation for Distinguished Young Talents in Higher Education of Guangdong (No. 2013LYM_0037).
The authors declare that they have no competing interests.
Both authors drafted the manuscript, and read and approved the final manuscript.