Entire functions sharing a small function with their two difference operators.

In this article, we deduce a uniqueness result of entire functions that share a small entire function with their two difference operators, generalizing some previous theorems of (Farissi et al. in Complex Anal. Oper. Theory 10:1317-1327, 2015, Theorem 1.1) and (Chen and Li in Adv. Differ. Equ. 2014:311, 2014, Theorem 1.1) by omitting the assumption that the shared small entire function is periodic.

Introduction and main result

Nevanlinna theory of value distributions is concerned with the density of points where a meromorphic function takes a certain value in the complex plane. Nowadays, there has been recent interest in connections between the Nevanlinna theory and the difference operator. 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 one got a lot of results (see, e.g., [3-8]).

In order to state the main result, we give the following definition. For a meromorphic function , we define its shift by and its difference operators by

A meromorphic function is said to be a small function with respect to if and only if , where , as outside of a possible exceptional set of finite logarithmic measure. Denote the set of all the small functions of by . Let and be two meromorphic functions and let be a small entire 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.

Recently, Chen et al. [2, 9] investigated two uniqueness problems on entire functions that share a small periodic entire function with their two difference operators as follows.

Theorem A

see [9], Theorem 1.1

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

Theorem B

see [2], Theorem 1.2

Let be a nonconstant entire function of finite order. If , , share 0 CM, then , where C is a nonzero constant.

In 2015, El Farissi, Latreuch and Asiri further studied the above problem and obtained

Theorem C

see [1], Theorem 1.1

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

Remark 1

It is necessary to point out that Theorems A and B have been generalized from to by Chen, Chen and Li in [9]. There are also some interesting results related the above theorems (see, e.g., [6, 10]).

In the previous results, we find that the shared small function is a periodic function with period c. So, it is natural to ask what will happen if the periodic condition of is omitted. In this paper, we focus on this problem and we obtain the following result.

Theorem 1

Let be a nonconstant entire function of finite order, and let be an entire function. If , , share CM, then one of the following assertions holds:

If , then , where C is a nonzero constant.

If , then or , where γ is a polynomial with .

Remark 2

We point out that Theorem 1 is a generalization of the previous theorems.

If , then . Then it follows from (i) of Theorem 1 that , where C is a nonzero constant.

If is a periodic function with period c, then . It follows from (ii) of Theorem 1 that or . Furthermore, by Theorem C we can deduce that .

As an application of Theorem 1, we can obtain an interesting result, where is a slow growth small function.

Theorem 2

Let be a nonconstant entire function of finite order, and let be an entire function with . If , , share CM, then .

For convenience of the reader, we list here some notations. For a meromophic function f, we use the basic notations of the Nevanlinna theory of meromorphic functions such as , , and as explained in [11-13].

Some lemmas

In this section, we state some results that we employ in our proofs.

Lemma 2.1

[4], Theorem 2.1

Let , and let f be a meromorphic function with a finite order. Then for all small periodic functions where for all r outside of a possible exceptional set E with finite logarithmic measure.

Lemma 2.2

[14], Lemma 3.3

Let g be a nonconstant meromorphic function in the plane of order less than 1, and let . Then there exists a ϵ-set E such that uniformly in η for .

Lemma 2.2 plays an important role in the proof of Theorem 2.

Proof of Theorem 1

Note that is a nonconstant entire function of finite order. Then and are also two entire functions of finite order.

Set . Then

Since , , share CM, we have where and are two polynomials.

Suppose that . Obviously, we can get . It is clear that , , and share 0 CM. By Theorem B, we can obtain , where C is a nonconstant. So . That is, .

In the following, we assume that . We consider into two cases.

Case 1. .

Set

From (1) and (2), we can rewrite the above function as which implies that φ is an entire function.

By (4) and Lemma 2.1, we deduce that .

Subcase 1.1. We assume that . Rewrite (5) as By the second main theorem, we deduce

Hence

Similarly, we get .

Rewrite equation (1) as which implies that . Then we deduce By (2), we have Combining equations (7) and (8) yields From (9), we have Suppose that is a nonconstant function. Denote . Obviously, η is a small function of . It follows from (11) that have no zeros. Note that has only one Picard value, say 0. Then , which implies . Again by (11), we have If is a constant, then . By (11) we also get .

Furthermore, it follows from (10), and that

Note that , then , so . Hence, we obtain .

Subcase 1.2. We assume that . Then it follows from (3) and (5) that where is a polynomial and . From this, we also have .

Case 2. .

By (2) and Lemma 2.1, we get .

Rewrite (1) and (2) as Then Rewrite (12) as .

Now, substitute the form of into (14) yields

Suppose that . Since , we have .

Similar to the discussion of Subcase 1.1, we obtain .

Now, we assume that . Suppose that Set . Then , , where , are two small functions of h.

Note that , so is also a small function of h.

From (15), we have

We claim that is irreducible except the factor which is a small function of h.

Suppose that is a common zero of and . It is easy to deduce that . Note that . Thus, the claim holds.

Rewrite (16) as Denote , where , . By the above equation, we see that , which implies that . Next we can prove that In fact, from (17) we have Note that is an entire function. All the zeros of H come from the zeros of and . By we denote the multiplicity of the zero of meromorphic function F at the point z.

Suppose that is a zero of H. We claim . Now we split into two cases.

Case A. is not a zero of . Then must be a zero of . Hence .

Case B. is a zero of . Set . Obviously, . Then and , which leads to .

Assume that . Then we can rewrite (17) as Further, we know is a small function of and Thus, it follows from (19) that is not an entire function, a contradiction. So . The above discussion yields Thus the claim holds.

By the claim and , we get

On the other hand, we have a contradiction. Thus, the case cannot occur.

Hence, we finish the proof of Theorem 1.

Proof of Theorem 2

If is a constant, then it follows from Theorem C that . In the following, we assume that is a nonconstant entire function.

Suppose that . Then we have Note that . Now, we apply Lemma 2.2 to this case.

Then there exists a ϵ-set E of finite logarithmic measure, so that for all in . It is absurd. Thus, . It follows from (ii) of Theorem 1 that or where γ is a polynomial.

Suppose that Note that . We get which implies that is a nonzero constant, say C. Furthermore, we get Rewrite it as Applying Lemma 2.2 again, we deduce that , which implies that . Then the above difference equation reduces to Set . Then the above equation can be rewritten as which implies that is a periodic function. If is a nonconstant function, then . It contradicts . Thus, b is a constant. Hence Then applying Lemma 2.2 again, there exists a ϵ-set E of finite logarithmic measure, so that for all in . Rewrite (20) as Then choose a sequence such that , and , as . Substituting into the above function yields a contradiction by Lemma 2.2. Thus, this case cannot occur. Then we deduce the desired result.

Hence, we finish the proof of Theorem 2.