Abstract
In this paper, the 3IM+1CM theorem with a general difference polynomial will be established by using new methods and technologies. Note that the obtained result is valid when the sum of the coefficient of is equal to zero or not. Thus, the theorem with the condition that the sum of the coefficient of is equal to zero is also a good extension for recent results. However, it is new for the case that the sum of the coefficient of is not equal to zero. In fact, the main difficulty of proof is also from this case, which causes the traditional theorem invalid. On the other hand, it is more interesting that the nonconstant finite-order meromorphic function can be exactly expressed for the case . Furthermore, the sharpness of our conditions and the existence of the main result are illustrated by examples. In particular, the main result is also valid for the discrete analytic functions.
1. Introduction and Results
It is well known that any polynomial is uniquely determined by its zero points (the set on which the polynomial takes zeros) except for a nonconstant factor, but it is not true for transcendental entire or meromorphic functions. For example, functions and have the same and points. Thus, it is interesting and complex to uniquely determine a meromorphic function. It is well known that the classical uniqueness results of the value distribution theory of meromorphic functions are the 5IM theorem (or five-point theorem) and the 4CM theorem (or four-point theorem) which had been obtained in the study by Nevanlinna [1], where IM means ignoring multiplicities and CM is counting multiplicities.
Let and be two nonconstant meromorphic functions on the complex plane and be a finite or infinite complex number in the whole complex plane . If and have the same zeros with the same multiplicities, we say that and share CM (counting multiplicities). If not considering the multiplicities, we say that and share IM (ignoring multiplicities). Clearly, they also share IM if and share CM. In this case, the famous five-point theorem and four-points theorem mean that if two nonconstant meromorphic functions and share five distinct values IM, then , and if two nonconstant meromorphic functions and share four distinct values CM, then is a factional linear transformation of , respectively.
The 4CM theorem in the study by Nevanlinna [1] can also be understood to be the 4CM+0 IM theorem which motivates us to consider the other cases as the 3CM+1IM theorem, the 2CM+2IM theorem, the 3IM+1CM theorem, or the 0CM+4IM theorem. Obviously, it is very good if the 0CM+4IM theorem is a fact. Unfortunately, Gundersen [2] has constructed counterexamples which show that the conclusion of the four-point theorem is invalid if CM is replaced by IM; that is, the 0CM+4IM theorem is false. However, the 2CM+2IM theorem is valid; please see the study by Gundersen [3]. It is known that the 2CM+2IM theorem implies that the 3CM+1IM theorem holds (see Gundersen [2, 3]). Thus, in [4], Gundersen proposed the following well-known question.
Problem 1. (3IM+1CM open problem [4]). If two nonconstant meromorphic functions share three values IM and share a fourth value CM, then do the functions necessarily share all four values CM?
It is very difficult to completely solve Problem 1 (see Gundersen [4]). In the past years, some researchers established 3IM+1CM theorems with some conditions, for example, the restriction of zero points for (see Mues [5–7], Reinders [8, 9], Wang [10], Ueda [11, 12], Yi and Zhou [13], Czubiak and Gundersen [14], Qiu [15], Wang [16, 17], Huang [18, 19], Huang and Du [20], Ishizaki [21], Li [22], Yao [23], and Yang and Yi [24]), the periodicity hypothesis (see Lin and Ishizaki [25] and Banerjee and Ahamed [26]), or the differential limitation (see Lahiri and Sinha [27], Gundersen [28, 29], Mues and Steinmetz [30, 31], Mues and Reinders [32], and Frank and Hua [33]).
In recent years, the difference variant of the Nevanlinna theory has been established in [34–37]. Using these theories, some mathematicians began to consider the uniqueness of meromorphic functions sharing values with their shifts or difference operators and produced many fine works; for example, see Banerjee and Bhattacharyya [38], Ahamed [39, 40], Ma et al. [41], Jiang et al. [42], Charak et al. [43], Lin et al. [44, 45], and Li et al. [46, 47].
In this paper, we will establish the 3IM+1CM theorem with the difference condition. Actually, Li et al. [46] obtained the following uniqueness result when and fulfill the condition 3IM+1CM, wherefor .
Theorem 1 (see [46]). Let be a nonconstant meromorphic function with finite order, and let be a nonzero complex number. Suppose that and share IM and share CM, where are three distinct finite values. Then, for all .
Noting thatnaturally, a more general caseshould be considered, where is a positive integer, are distinct finite values, and are nonzero constants. In the case, Problem 1 turns to be the following problem.
Problem 2. When a nonconstant meromorphic function and its general linear difference polynomial satisfy the condition 3IM+1CM, does the uniqueness result still hold?
In the following, we will give our main result, some corollaries, and remarks.
Theorem 2. Let be a nonconstant meromorphic function with finite order; let be defined as (3). If and share IM and share CM, where are three distinct finite values, then for all , one of the following results holds:(i);(ii); furthermore,and , where , and are constants.
To explain a nonconstant meromorphic function with finite order, we need the following definition.
Definition 1 (see [24, 48]). For a nonconstant meromorphic function , the order of , denoted by , is defined aswhereand denotes the number of poles of (counting multiplicities) in .
A nonconstant meromorphic function with finite order means that .
Remark 1. The following example shows that Theorem 2 (ii) can happen. Letwhere , , and are nonzero constants satisfying , and then and share CM, and .
Remark 2. The conditions of Theorem 2 are sharp. From the example in the following, we know that the assumption 3IM+1CM cannot be relaxed to 4IM in Theorem 2. For example, see Steinmetz [49]. Let denote the Weierstrass -function with a pair of primitive periods and , and setThen, the functions and share the values and some values and IM, where . We note that it is showed that three meromorphic functions , and share the four values , and IM in [49]. However, or .
Corollary 1. Let be a nonconstant meromorphic function with finite order; let be defined as (3). Assume that and share IM and share CM, where are three distinct finite values. If , then for all .
Corollary 2. Let be a nonconstant meromorphic function with finite order; let be defined as (3). Assume that and share IM and share CM, where are three distinct finite values. If , then for all .
Remark 3. Note thatand the sum of its coefficients is zero. So, even if in , is still a special case of . Thus, Theorem 2 is a large extension of Theorem 1.2 in Li et al. [46]; naturally, the corresponding results in the study by Heittokangas et al. [50], Li et al. [46, 47], and so on are also the special cases of Theorem 2. However, our result is new when .
Remark 4. Choosing some special and for , we can obtain some difference equations of the form . Thus, the main result is also valid for the discrete analytic functions (see [51–55]). For example, letwhich is the special case of (3). Thus, Theorem 2 is also valid for (10) or (11). When and , equation (10) is the definition of discrete analytic and harmonic functions (see Hundhausen [56]). At this time, we can obtain the uniqueness of (10) or (11). Another definition of discrete analytic and harmonic functions:can be seen in Harman [57]. Clearly, our theorem is also valid for (12) or (13).
We note that the main tool of proof for Theorem 1.2 in Li et al. [46] is the traditional Theorem 1.62 in [24]; however, it is invalid for Theorem 2 because can be nonzero. We specially thank Corollary 1.105 in [58] which will be our key tool and listed in Section 2 as Lemma 8. On the other hand, our methods and technologies are also different with Li et al. [46]; for example, the proof is divided into the six cases in Li et al. [46], but it is only two cases in the proof of Theorem 2. For the other detail cases, please see the proof in Section 3. However, it is worth mentioning that the meromorphic function can be exactly expressed byin some special cases. Certainly, some preliminaries of the difference value distribution theory [34–37] are also used in the proof of Theorem 2.
In the final, the present paper will be organized as follows. In Section 2, we will give some preliminaries which can be seen in the listed references. And the main result will be proved in the final section.
2. Some Lemmas
In this paper, the value distribution theory established by R. Nevanlinna is the main tool for the studies. For convenience of the reader who might not be familiar with Nevanlinna theory, we list here some results from Nevanlinna theory (see, e.g., [24, 48]).
The following known results are important in the value distribution theory (see, e.g., [24, 48]).(i)The arithmetic properties of and are as follows: The same inequalities hold for .(ii)The Nevanlinna first fundamental theorem is as follows: .(iii)The logarithmic derivative lemma is as follows: , if the order is finite.
In the following, we present some lemmas, which will be needed in the sequel.
Lemma 1 (see Theorem 3 in [59]). Let and be two nonconstant rational functions. If and share IM, where are four distinct values in the extended complex plane, then .
Lemma 2 (see Lemma 1 in [4]). Let and be two distinct nonconstant meromorphic functions, and let be four distinct values in the extended complex. If and share IM, then for a set with finite linear measure ,(i), as and (ii), as as and , where
Remark 5. Under the assumptions of Lemma 2, if and in Lemma 2 are of finite order, by the context of paper [26] and the proof of Lemma 1 in [26], we can find that the conclusion of Lemma 2 can be changed to(i)(ii)
Lemma 3 (see Lemma 2 and Corollary 1 in [3]). Let and be distinct nonconstant meromorphic functions that share four values , and IM, where . Then, for a set with finite linear measure , the following statements hold:(i) and as , , and is a set with finite linear measure, where and count, respectively, only those points in and which do not occur when for some .(ii)For , we next let refer only to those -points that are multiple for both and and count each such point the number of times of the smaller of the two multiplicities. Then, as and .
Remark 6. Under the assumptions of Lemma 3, if and in Lemma 3 are of finite order, by the context of paper [38] and the proof of Lemma 2 and Corollary 1 in [38], we can find that the conclusion of Lemma 3 can be changed into(i) and (ii)
Lemma 4 (see Corollary 2.5 in [34]). Let be a nonconstant meromorphic function of finite order and be a fixed nonzero complex number. Then, for each ,
Lemma 5 (see Theorem 1 in [4]). Let and be two nonconstant meromorphic functions that share IM and CM, where are four distinct values in the extended complex plane. Suppose that there exists some real constant and some set that has infinite linear measure such thatfor all , where , if . Then, and share all four values CM.
Lemma 6 (see Theorem 4.3 in [24]). Let and be two distinct nonconstant meromorphic functions and be four distinct values in the extend complex plane. If and share CM, then , where is a Mobius transformation such that two of the four values are fixed points and another two (are Picard exceptional values of and ) exchange each other under .
Lemma 7 (see Theorem 1.62 in [24]). Let be nonconstant meromorphic functions, and let be a meromorphic function such that . Suppose that there exists a subset whose linear measure is such thatAs and , where , then .
Lemma 8 (see Corollary 1.105(iii) in [58]). Assume that entire functions vanish nowhere on such thatPartition the index set into subsets , , putting two indices and in the same subset if and only if is a constant. Then, we have
3. Proof of Theorem 2
If is a constant, by the assumption that and share IM and share CM, we know that has more than two Picard values. And it is impossible, so is nonconstant. Assume that is a nonconstant rational function; according to the definition of , we can find that is also a nonconstant rational function, and then using Lemma 1, we deduce that . Next, we suppose that is a nonconstant transcendental meromorphic function. Since and and share IM and share CM, applying Remark 5, we obtain
Then, we can see that is also a transcendental meromorphic function. Next, we assume .
Under the assumption that and share CM, Remark 5 (ii) leads towhere denotes the counting function of the multiple poles of in , and each point in is counted according to its multiplicity as a pole of . is similar with . Hence,where is the counting function of the common simple poles of and . By the definition of and Lemma 4, we deduce
From (23), (24), Remark 5 (ii), and the assumption that and share IM and CM, we deduce
Hence, we getwhere with logarithmic measure . In view of Lemma 5, we deduce that and share and CM. Then, by Lemma 6, we obtain that is a Möbius transformation of . We setwhere , and are constants and . We consider the following two cases.
Case 1. Suppose that . Then, from (27), we getwhere and are two constants, and , and . If two of are not the Picard exceptional value of , we get , which is a contradiction, so two of are Picard exceptional values of . Without loss of generality, we set and as Picard exceptional values of , and by the assumption of Theorem 2, we get and as Picard exceptional values of . By rewriting (28), we obtainand noting , we deduce or . Similarly, we getand or .
and imply that , which is impossible. By and , we obtain and , that is, , which is a contradiction to the assumption. From and , we have which is also impossible. Combining and , we get and , and thenSince , we setwhere , , are constants, and . By a calculation, for , we deduce thatSet ; then, when , we obtainand while , we knowEquation (32) impliesSo,Substituting (36) and (37) to (31), we getWhen , from (33), (34), and (38), we knowand when , we getCombining (39) and (40), we deduce thatThen, by (31), we obtainSubstituting (41) to (36) and (37), we haveSubstituting (43) and (44) to (42), we deduceDenoting , , and , where , thenwhere . If there exist some , , such that , we denote and ; then, we get a similar equation with the above one.
So,Then, by a calculation, we getwhere denotes the sum for any numbers in . Applying Lemma 8, we get that there exits nonnegative integer and such that is a constant. From (33), we know that is a polynomial of . We setwhere are constants. We substitute into (43) and getwhere are constants.
Case 2. Suppose that . By (27), we getwhere , , , and .
Since and share CM, from (52), we know is a Picard exceptional value of and . If can take every one of , we get at least two of are equal, which is a contradiction, so one of is a Picard exceptional value of and . Without loss of generality, we set is a Picard exceptional value of ; by assumption, is also a Picard exceptional value of . Noting that is Picard exceptional values of and they share CM, by (52), we have is a Picard exceptional value of , so . Since is Picard exceptional values of and they share CM; by (52), we see that is a Picard exceptional value of . Then, we have . Thus, (52) turns to bewhere and are not all zero.
Noting that and share and CM and neither of them is a Picard exceptional value of and , from (53), we getSo, and . Then, by (53), we haveBy rewriting it, we obtainNoting that both and are Picard exceptional values of and and , we setwhere , , are constants, and . Then, by the definition of , we getSubstituting (57) and (58) to the above equation, we haveThen, using Lemma 7, we get a contradiction. Thus, we complete the proof of Theorem 2.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The study was supported by the National Natural Science Foundation of China (Grant no. 11871314) and the Scientific Research Project of Shanxi Datong University (Grant no. 2020K19).