Abstract

We study the uniqueness problems on entire functions and their difference operators or shifts. Our main result is a difference analogue of a result of Jank-Mues-Volkmann, which is concerned with the uniqueness of the entire function sharing one finite value with its derivatives. Two relative results are proved, and examples are provided for our results.

1. Introduction and Main Results

Throughout this paper, we assume the reader is familiar with the standard notations and fundamental results of Nevanlinna theory of meromorphic functions (see, e.g., [1–3]). In what follows, a meromorphic function always means meromorphic in the whole complex plane, and always means a nonzero complex constant. For a meromorphic function , we define its shift by and its difference operators by

For a meromorphic function , we use to denote the family of all meromorphic functions that satisfy , where , as outside of a possible exceptional set of finite logarithmic measure. Functions in the set are called small functions with respect to .

Let and be two meromorphic functions, and let be a small function with respect to and . We say that and share IM, provided that and have the same zeros (ignoring multiplicities), and we say that and share CM, provided that and have the same zeros with the same multiplicities.

Uniqueness theory of meromorphic functions is an important part of the Nevanlinna theory. In the past 40 years, a very active subject is the investigation on the uniqueness of the entire function sharing values with its derivatives, which was initiated by Rubel and Yang [4]. We first recall the following result by Jank et al. [5].

Theorem A (see [5]). Let be a nonconstant meromorphic function, and let be a finite constant. If , , and share the value CM, then .

Recently, value distribution in difference analogues of meromorphic functions has become a subject of some interest (see, e.g., [6–11]). In particular, a few authors started to consider the uniqueness of meromorphic functions sharing small functions with their shifts or difference operators (see, e.g., [12, 13]).

In this paper, we consider difference analogues of Theorem A.

Theorem 1.1. Let   be a nonconstant entire function of finite order, and let   be a periodic entire function with period . If , , and   share   CM, then .

Example 1.2. Let and . Then, for any , we notice that , , and share CM and can easily see that . This example satisfies Theorem 1.1.

Remark 1.3. In Example 1.2, we have . However, it remains open whether the claim in Theorem 1.1 can be replaced by in general. In fact, the next example resulted from our efforts to find an entire function satisfying Theorem 1.1, while .

Example 1.4. Let , , , and . Then we observe that , , and share 0 CM. Here, we also get .
From this example, it is natural to ask what happens if , , and share 0 CM, where and are two (not necessarily distinct) small periodic entire functions. Considering this question, we prove the following Theorem 1.5, whose proof is omitted as it is similar to the proof of Theorem 1.1.

Theorem 1.5. Let be a nonconstant entire function of finite order, and let , be periodic entire functions with period . If , , and share 0 CM, then .

Now it would be interesting to know what happens if the difference operators of are replaced by shifts of in Theorem 1.5. We prove the following result concerning this question.

Theorem 1.6. Let   be a nonconstant entire function of finite order, let ,   be two distinct periodic entire functions with period , and let and be positive integers satisfying . If , , and   share 0 CM, then for all .

Example 1.7. Let , , , and . Then we notice that , , and share 0 CM and can easily see that for all . This example satisfies Theorem 1.6.

Example 1.8. Let , where is a periodic entire function with period 1. Then , , and share 0 CM, while . This example shows that the condition that and are distinct in Theorem 1.6 cannot be deleted.

2. Proof of Theorem 1.1

Lemma 2.1 (see [8, Theorem  2.1]). Let   be a meromorphic function of finite order and let be a nonzero complex constant. Then, for each ,

Lemma 2.2 (see [10, Lemma  2.3]). Let , , and let be a meromorphic function of finite order. Then for any small periodic function with period , with respect to , where the exceptional set associated with is of at most finite logarithmic measure.

Proof of Theorem 1.1. Suppose, on the contrary, the assertion that . Note that is a nonconstant entire function of finite order. By Lemma 2.1, and are entire functions of finite order.
Since , , and share CM, then we have where and are polynomials.
Set
From (2.3), we get . Then by supposition and (2.4), we see that . By Lemma 2.2, we deduce that
Note that . By using the second main theorem and (2.5), we have Thus, by (2.5) and (2.6), we have . Similarly, .
By Lemma 2.2 and the first equation in (2.3), we deduce that and
From (2.7), we see that
Now we rewrite the second equation in (2.3) as and deduce that This together with the first equation in (2.3) gives that is,
Thus, (2.11) can be rewritten as where which satisfy and .
Now we rewrite as
Suppose that . Let be a zero of with multiplicity . Since , share CM, then is a zero of with multiplicity . Thus, is a zero of with multiplicity at least . Then, by (2.8) and (2.14), we see that
On the other hand, we have Then, by (2.15) and (2.16), we get , which is a contradiction.
Thus, . Noting that , we deduce that . So, , since is a polynomial.
By the second equation in (2.3), we obtain , which leads to . This is a contradiction. The proof is thus completed.

3. Proof of Theorem 1.6

Lemma 3.1 (see [9, Corollary  2.2]). Let be a nonconstant meromorphic function of finite order, and . Then for all outside of a possible exceptional set with finite logarithmic measure.

Proof of Theorem 1.6. Suppose, on the contrary, the assertion that . Since , , and share 0 CM, then we have where and are polynomials.
By (3.2), we obtain
Set . Then by supposition, we see that . By Lemma 3.1, we deduce that
Note that . Thus, using a similar method as in the proof of Theorem 1.1, we get and .
By Lemma 3.1 and the first equation in (3.2), we deduce that
From (3.5), we see that
Now we rewrite the second equation in (3.2) as and deduce that This together with the first equation in (3.2) gives that is, Now we rewrite (3.9) as
Suppose that ; then, by (3.9), we get , which is a contradiction.
Now we have . Then using a similar method as in the proof of Theorem 1.1, we can also get a contradiction and obtain that for all . Thus, Theorem 1.6 is proved.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11171119). The authors would like to thank the referee for valuable suggestions.