A Positive Answer for 3IM+1CM Problem with a General Difference Polynomial

Huicai Xu,Qingcai Zhang,Shugui Kang,Li Li

doi:10.1155/2021/7113065

Huicai Xu, Qingcai Zhang + Show 2 more

Open Access

https://doi.org/10.1155/2021/7113065

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Abstract

In this paper, the 3IM+1CM theorem with a general difference polynomial L z , f will be established by using new methods and technologies. Note that the obtained result is valid when the sum of the coefficient of L z , f is equal to zero or not. Thus, the theorem with the condition that the sum of the coefficient of L z , f 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 L z , f 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 f can be exactly expressed for the case f ≡ − L z , f . 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.

Highlights

Introduction and ResultsIt is well known that any polynomial is uniquely determined by its zero points except for a nonconstant factor, but it is not true for transcendental entire or meromorphic functions
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]
We present some lemmas, which will be needed in the sequel

Summary

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. It is well known that the classical uniqueness results of the value distribution theory of meromorphic functions are the 5IM theorem (or fivepoint 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. The difference variant of the Nevanlinna theory has been established in [34,35,36,37] 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]. The main result will be proved in the final section

Some Lemmas

Proof of Theorem 2