Issue 
Wuhan Univ. J. Nat. Sci.
Volume 29, Number 1, February 2024



Page(s)  7  12  
DOI  https://doi.org/10.1051/wujns/2024291007  
Published online  15 March 2024 
Mathematics
CLC number: O174.5
Meromorphic Functions Sharing Two Values
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China
^{†} Corresponding author. Email: alexehuang@sina.com
Received:
25
April
2023
Let and be two nonconstant meromorphic functions, and be a positive integer. If and share 1 CM (counting multiplicities), and share IM (ignoring multiplicities), and , then either or , where . As an application, shared values problems of a meromorphic function related to its shifts and difference operators are also investigated.
Key words: meromorphic function / shared values / difference operator / shift
Cite this article:: ZHAO Ying, HUANG Zhigang. Meromorphic Functions Sharing Two Values[J]. Wuhan Univ J of Nat Sci, 2024, 29(1): 712.
Biography: ZHAO Ying, female, Master candidate, research direction: complex analysis. Email: 1443751421@qq.com
Fundation item: Supported by the National Natural Science Foundation of China (11971344)
© Wuhan University 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
0 Introduction and Main Results
At the outset, it will be assumed that the standard definitions and terminologies of value distribution theory are known to the readers (see Ref.[1]). In the following, a meromorphic function always means meromorphic in the whole complex plane. By , we denote any quantity satisfying as outside of an exceptional set with finite logarithmic measure. A meromorphic function is said to be a small function of if it satisfies . We say two nonconstant meromrophic functions and share the valueCM (IM) if and have the same zeros counting multiplicities (ignoring multiplicities). Let be a positive integer, we denote by the counting function of a points of with multiplicity and by the counting function of a points of with multiplicity , where each a point is counted according to its multiplicity.
The following three theorems are classical and wellknown results in the study of the uniqueness problems of meromorphic functions.
Theorem 1^{[2]} If two nonconstant meromorphic functions and share five distinct values IM, then .
Theorem 2^{[2]} If two nonconstant meromorphic functions and share four distinct values CM, then or , where is a Mobiüs transformation.
Theorem 3^{ [3]} If two nonconstant meromorphic functions and share two values IM, and share two other values CM, then and share all values CM.
The assumption "4CM" in Theorem 2 has been improved to "2IM+2CM" by Gunderson^{ [3]}. However, Gunderson^{ [4]} gave an example to show that Theorem 2 is not true if the condition "4CM" is replaced by "4IM", and the question whether "4CM" in Theorem 2 can be replaced by "1CM+3IM" still remains open. Many researchers are devoted to studying the open problem. For partial progress on this, we refer the readers to survey ^{[5]}. Recently, Wang and Fang^{[6]} obtained some results from another direction.
Theorem 4 ^{[6]} Let and be nonconstant meromorphic functions, let be a finite nonzero value, and let be a positive integer satisfying . If and share CM, and share CM, and , then either or , where .
Naturally, a question arises:
Question 1 Can "2CM" be replaced by "1CM+1IM" in Theorem 4?
First, we give a partially answer to Question 1.
Theorem 5 Let and be nonconstant meromorphic functions, and let be a positive integer satisfying . Suppose that and share 1 CM, and share IM, and , then either or , where .
Remark 1 Actually, in the proof of Theorem 5, if we replace and with and , then the condition that and share 1 CM can be improved to and share CM, where is a nonzero finite value.
To reduce the number of shared values quickly, many authors began to consider the case that and have some special relationship. For example, when is the shift of , Heittokangas et al^{ [7]} obtained some uniqueness results in 2009. The background for these considerations lies in the development of the difference version to the usual Nevanlinna theory, especially the differencetype logarithmic derivative lemma, which starts in the papers^{ [810]}.
Theorem 6^{[7]} Let be a meromorphic function of finite order and , and let be three distinct periodic functions with period . If and share CM and IM, then .
Following that, several researchers began to study shared values problems related to a meromorphic function and its difference operators, see Refs.[1119]. Here, we recall the following results from Qi^{ [18]} and Chen et al^{[20]}.
Theorem 7^{ [1]} Let be a meromorphic function of finite order, let be two finite nonzero values, and let be a positive integer satisfying . If and share CM, and share CM, then , where .
Theorem 8^{ [20]} Let be a meromorphic function of finite order, , and let be a positive integer satisfying . If and share CM, and share CM, then , where , .
Theorem 7 and 8 both require that have finite order. As a result, we pose two questions:
Question 2 What can happen if is of infinite order in Theorem 78?
Question 3 Is it possible to widen the range of in Theorem 78?
In the following, as an application of Theorem 5, we give two uniqueness results about a meromorphic function related to its shifts and difference operators, which are answers to the two questions.
Theorem 9 Let be a meromorphic function, and let be a positive integer satisfying . If and share 1 CM, and share IM, and , then either , where .
Theorem 10 Let be a meromorphic function, and let be a positive integer satisfying and . If and share 1 CM, and share IM, , then , where .
1 Preliminary Lemmas
To prove our results of this paper, the following lemmas are required.
Lemma 1^{ [21]} Let be a nonconstant meromorphic function in the complex plane. Let and be distinct finite complex numbers. Then
where .
Lemma 2 Let and be nonconstant meromorphic functions with and . Suppose that and share 1 CM, and and share IM. If
Then .
Proof Set
From (2) and the lemma of logarithmic derivative, we have . If , then we see that the possible poles of can occur at the poles of and , and zeros of and . Therefore, we get
Since , we get
Then from the assumption that and share IM and , we have
If is a 1 point of with multiplicities , then by a short caculation with laurent series and (2) we see that is a zero of . Since and share 1 CM, we know the 1 points of and are not poles of . From (2) and Nevanlinna's first fundamental theorem, it is easy to deduce that
It is a contradiction.
Thus . From (2) we have
where is a nonconstant. By the assumption that , there exists such that , so .
Lemma 3 Let and be nonconstant meromorphic functions with and . Suppose that and share 1 CM, and and share IM. If is not a Mobiüs transformation of , then
where denotes the zeros of which are not zeros of .
Proof Set
If , then from (6), we have is a Mobiüs transformation of , a contradiction. Thus, , and . First, we know
From Lemma 2, we deduce that . If not, then is a transformation of , a contradiction. Moreover, by a short calculation with Laurent series, it is obvious that the simple 1 points of must be zeros of .Therefore, we can deduce from (7) that
Next, we need to estimate the poles of . By a calculation with Laurent series, and share 1 CM, and and share ∞IM, we get that the poles of can only occur at the poles of and , the zeros of and , the zeros of which are not zeros of , and the zeros of of which are not zeros of . Hence, we can get
From the assumption that and share IM, , and , so we have
This completes the proof of Lemma 3.
2 Proof of Theorem 5
Proof Set
Now we consider the following cases.
Case 1 . By (8) we get
where is a constant.
If , from (9) we get , and thus , where .
If , by (9) we obtain
Clearly, this leads to
By Lemma 1, (10) and the assumption that share IM, we get
Similarly, we have
Combining the above two inequalities, we have
which contradicts the condition .
Case 2 . If has a pole with multiplicity , is a pole of with multiplicity , then is a pole of with multiplicity , and it is also a pole of with multiplicity . By a short calculation, we deduce that is a zero of . Since and share IM, we have
Clearly, from (8), we see that . On the other hand, since and share 1 CM, by a calculation with Laurent series, we know the 1 points of and are not the poles of . Thus, the poles of can only occur at the zeros of and . Therefore, from (12) and the hypothesis that , we deduce
Next, we still discuss two cases.
Case 2.1 If is not a Mobiüs transformation of , then from Lemma 3, we get
and
Thus, by Lemma 1, we obtain
Similarly,
By (13)(17), we get
It is a contradiction.
Case 2.2 If is a Mobi's transformation of , that is
where are constants, and .
Next we discuss following two cases.
Case 2.2.1 . Thus . From (18), we have . If , , then . By Nevanlinna's second fundamental theorem and (13), we get
It is a contradiction.
Hence , so . If does not have 1 point, it is easy to get a contradiction by Nevanlinna's second fundamental theorem. So there exists such that . Thus we get , that is , , where .
Case 2.2.2 . If , then from (18) and share IM, we obtain , . By Nevanlinna's second fundamental theorem, we get a contradiction.
So , then . From (18), we have , . It is easy to get and . If , we get , which contradicts with Nevanlinna's second fundamental theorem. So , then . By similar reasoning as in Case 2.2.1, we can get . Thus we get , , where . This completes the proof of Theorem 5.
3 Proof of Theorem 9
By Theorem 5, we get or , where . If ,. From the assumption that , share IM, we obtain and . Otherwise, if or , then there exists such that or , then or . This is a contradiction. Similarly, we have . Thus , where be a nonconstant entire function, and hence . Since , we can deduce that , where is a nonconstant entire function. Hence, we have .
That is,
Therefore,
where A is a constant.
By calculation the derivative of (20) and , we get
We assert that . If , then , where is a constant. Hence, , where be a nonzero constant. By integrating, we can get , , Consequently, we have
Since is a constant, we can get . This is a contradiction.
Therefore, . From the lemma of logarithmic derivative,
Thus, is a small function of . From (21) we know , and it is easy to get is a constant by Nevanlinna's second fundamental theorem. Hence is a constant. We can derive a contradiction by similar discussion as above.
Therefore, we get , where .This completes the proof of Theorem 9.
4 Proof of Theorem 10
By Theorem 5, we get or , where . If , that is
From (24) and the fact , share IM, we get and . Thus , where be a nonconstant entire function. By (24) and , we obtain
That is
Since , we easily get , and obviously . Then we get , then , where are constants. This is a contradiction. Thus, we get , where . This completes the proof of Theorem 10.
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