Uniqueness Problems about Entire Functions with Their Difference Operator Sharing Sets

Abstract

In this paper, we study the uniqueness questions of finite order transcendental entire functions and their difference operators sharing a set consisting of two distinct entire functions of finite smaller order. Our results in this paper improve the corresponding results from Liu (2009) and Li (2012).

1. Introduction and Main Results

Before proceeding, we spare the reader for a moment and assume some familiarity with the basics of Nevanlinna theory of meromorphic functions in such as the first and second main theorems and the usual notations such as the characteristic function , the proximity function , and the counting function . denotes any quantity satisfying as , except possibly on a set of finite logarithmic measure not necessarily the same at each occurrence, see e.g., [1–3].

Let be a meromorphic functions on . Here, the order is defined byand the exponent of convergence of zeros is defined by

For a given , we say that two meromorphic functions and share CM (counting multiplicities) when and have the same a-points. Let be a finite set of some entire functions and an entire function. Then, a set is defined as

Assume that is another entire function. We say that and share a set , counting multiplicities (CM), provided that .

The uniqueness theory of meromorphic functions sharing sets generalizes that on sharing values and generally is more difficult. If meromorphic functions share a general set, it is not easy to determine these functions. In 1999, Li and Yang [4] deduced that if with contain two distinct constants, then must have special forms. Fang and Zalcman [5] used the theory of normal family to solve the above problem by proving that there exists a finite set containing three distinct elements such that if , then .

Recently years, Nevanlinna characteristic of , the value distribution theory for difference analogue, Nevanlinna theory of the difference operator, and the difference analogue of the lemma on the logarithmic derivative had been built, see e.g., [1–4, 6–14]. For meromorphic functions , we define its shift by and its difference operators by

By Nevanlinna theory of the difference operator, a natural question to ask whether the derivative can be replaced by the difference operator in the above question ?

In 2009, Liu [8] investigated the above question and proved the following result.

Theorem 1. Let be a nonzero complex number and be a transcendental entire function with finite order. If and share CM, then for all .

In 2012, from Theorem 1, considering the constant in set is replaced by the function, Li [9] proved the following.

Theorem 2. If and are two distinct entire functions, then is a nonconstant entire function whose and such that and . If and share CM, then for all .

After studying Theorem 2, we propose some questions as follows. Question 1: from Theorem 2, the condition seems more stronger. So, one may ask whether it can be weakened or moved? Question 2: what will happen if the shift be replaced by in Theorem 2?

Fortunately, we have recently given a positive answer for Question 1 (see [14]). In this work, we also discuss the above problems and especially for Question 2. Finally, we derive the following results.

Theorem 3. Suppose that are two distinct entire functions and is a nonconstant entire function of finite order with such that and . If and share CM, then must take one of the following conclusions:(1), where are two nonzero constants satisfying . Furthermore, .(2). Here, is an entire function and .Using the same method, we improve the above result from the shift to in above theorem and obtain the following result.

Theorem 4. Suppose that are two distinct entire functions and is a nonconstant entire function of finite order with such that and . If and share CM, then , where is an entire function such that .

2. Some Lemmas

We will introduce some lemmas for the proofs of our theorems in this section.

Lemma 1. (see [15]). Let be a meromorphic function of finite order and let be two arbitrary complex numbers such that . Assume that is the order of , then for each , we have

Lemma 2 (see [16]). Let be a function transcendental and meromorphic in the plane with order less than 1. Set . Then, there exists an -set such thatuniformly in for .

Lemma 3. (see [3]). Suppose that are meromorphic functions and are entire functions satisfying the following conditions:(1).(2).(3)For , , , . Then, .

3. Proof of Theorems

Proof of Theorem 1. Due to and share CM, so we setwhere is an entire function. And then it follows from (7) and max that is a polynomial.
Using Hadamard Factorization Theorem, we assume that , where is an entire function and is a polynomial which satisfiedSo,Put the forms of and into (7) to yieldTake . We assume that . Then,By Lemma 3, if , , and are not constants, then , a contradiction.
So, , , and , (where are three constants).
We can get , , and .
Hence, is a periodic function. We also know is a polynomial. So, we get (where and are two constants, and ). So, we obtain . Here, is an entire function and .
If below, obviously, is a small function of . Rewrite (10) asNote that . Without loss of generality, we set . Suppose that is a zero of , but not a zero of . From (12), we may easily obtain that is a zero of or . We denote by the reduced counting function of those common zeros of and . Similarly, we also denote the reduced counting function of those common zeros of and . Then,which implies that either or . We distinguish the two cases as follows: Case 1: . We may assume that is a common zero of and . It is obvious that is a zero of . If , then a contradiction. Hence, It deduces By Lemma 1, for any , We also get , where is a fixed positive constant. If , using and the above estimates of , It easily gets a contradiction. So, , this means that is a nonzero constant . Then, (16) changes to Also, noting that . Then, by Lemma 2, we get that there exists an -set E, as and such that So, and , and this also means that is a periodic function. If is a nonconstant function, then , a contradiction. Therefore, is a constant. Noting that and is a nonconstant entire function. Then, . Thus, we may set , where are two nonzero constants. Using the assumption of Case 1, one has and as common zeros, which are not zeros of . Suppose that is a common zero of and and not a zero of . Then, is a zero of . Moreover, this implies that . Finally, we deduce , which is the desired result. Case 2: . Suppose which is a common zero of and . Then, it is obvious that is a zero of . If , then a contradiction. Hence, If , then , a contradiction. Thus, . We may set that is a zero of , but not a zero of . It follows from (12) that is a zero of or . We take by the reduced counting function of those common zeros of and . Similarly, we denote by the reduced counting function of those common zeros of and . We obtainIt implies that either or . If , likewise with Case 1, we deduce the desired result. Hence, we set that below. Similarly with Case 2, we also get thatCombining (22) with (24), we deduce thatNote that . Thus, . Again using (24), we have . We rewrite it asThen,By Lemma 3, if , , and are not constants, then ; this is a contradiction.
So, , , and (where are three constants).
We can get , , and .
Hence, is a periodic function. We also know is a polynomial. So, we get (where and are two constants, and ). So, we obtain . Here, is an entire function and .
Therefore, the proof of the main Theorem 3 is finished.

Proof of Theorem 2. Note that and share CM. So, we also setwhere is an entire function. Furthermore, it deduces from (28) and max that is a polynomial.
Using Hadamard Factorization Theorem, we assume that , where is an entire function and is a polynomial satisfyingThen,where are constants. Substituting the forms of and into (28) yieldsSet . Suppose that . Then,By Lemma 3, if , are not constants, then ; a contradiction.
So, , , and (where are three constants).
We can get , , and .
Hence, is a periodic function. We also know is a polynomial. So, we get (where and are two constants, and ). So, we obtain . Here, is an entire function and .
If below, obviously, is a small function of . Rewrite (31) asDue to , without loss of generality, we set . Assume that is a zero of , but not a zero of . It deduces from (33) that is a zero of or . We also take the reduced counting function of those common zeros of and . Likewise, we denote by the reduced counting function of those common zeros of and . Then,this implies that either or . We may distinguish the following two cases. Case 1: . We set a common zero of and . Then, it is obvious that is a zero of . If , then a contradiction. Hence, It leads to By Lemma 3, if , , are not constants, then , a contradiction. So, , , (where are three constants). We can get , , . Here, is a periodic function. We also know is a polynomial. So, we get (where and are two constants, and ). Finally, we get , where is an entire function and . Case 2: . Suppose is a common zero of and . Then, it is easy to see that is a zero of . If , then a contradiction. Thus, If , then , a contradiction. Thus, . We assume that is a zero of , but not a zero of . It deduces from (33) that is a zero of or . We take by the reduced counting function of those common zeros of and . Similarly, we denote by the reduced counting function of those common zeros of and . Then,and this implies that either or . If . Similarly, as the same way in Case 1, we get the desired result. So, we assume that as follows. Similarly, as the way in Case 2, we can get thatIt follows from (39) and (41) thatNote that . Thus, . Again by (41), one has . We also rewrite it asBy Lemma 3, if , are not constants, then , a contradiction.
So, , (where are three constants).
We also get , , .
Hence, is a periodic function. We also know is a polynomial. So, we get (where and are two constants, and ). Finally, we get . Here, is an entire function and .
Proof of Theorem 4 is completed.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors typed, read, and approved the final manuscript.

Acknowledgments

The work presented in this paper was supported by the Plateau Disciplines in Shanghai, Leading Academic Discipline Project of Shanghai Dianji University (16JCXK02), and Philosophy and Social Sciences Planning Project of the Ministry of Education (Grant no. 18YJC630120).