Uniqueness on entire functions and their nth order exact differences with two shared values

Abstract Let f(z) be an entire function of hyper order strictly less than 1. We prove that if f(z) and its nth exact difference Δ c n f ( z ) {\Delta }_{c}^{n}f(z) share 0 CM and 1 IM, then Δ c n f ( z ) ≡ f ( z ) {\Delta }_{c}^{n}f(z)\equiv f(z) . Our result improves the related results of Zhang and Liao [Sci. China A, 2014] and Gao et al. [Anal. Math., 2019] by using a simple method.


Introduction and main results
We assume the reader is familiar with the fundamental results and standard notations of Nevanlinna's theory, as found in [1,2], such as the characteristic function T(r, f ) of a meromorphic function f(z). Notation S (r, f ) means any quantity such that S(r, f ) = o(T(r, f )) as r → ∞ outside of a possible set of finite logarithmic measures. Moreover, the order ρ( f ) and hyper order ρ 2 ( f ) of f(z) are defined as usual as follows: For a value ∈ ∪ {∞} a , we say that two meromorphic functions f(z) and g(z) share a CM (IM) provided that f and g have the same apoints counting multiplicities (ignoring multiplicities).
About 10 years ago, Halburd and Korhonen [3,4] and Chiang and Feng [5] established the difference analogue of Nevanlinna's theory for finite-order meromorphic functions, independently. Later, Halburd et al. [6] showed in 2014 that it is still valid for meromorphic functions of hyper-order strictly less than 1. So far, it has been a most useful tool to study the uniqueness problems between meromorphic functions f(z) and their shifts f(z + c) or nth exact differences ( ) Δ f z c n (n ≥ 1). For some related results in this topic, we refer the reader to [7][8][9][10][11][12] and so on. In 2013, Chen and Yi first proved an uniqueness theorem for a meromorphic function f(z) and its first order exact difference Δ c f(z) with three distinct shared values CM in [13], which had been improved by Zhang and Liao [14] in 2014 as follows. . Furthermore, f(z) must be of the following form f(z) = 2 z h(z), where h(z) is a periodic entire function with period 1.
Theorem A had been improved by Lü and Lü [15] from "entire function" to "meromorphic function" in 2016. More recently, Gao et al. [16] obtained the following uniqueness theorem concerning the nth exact difference.

Theorem B.
[16] Let f be a transcendental meromorphic function of hyper order strictly less than 1 such that It is obvious that both Theorems A and B require three shared values CM. So, a nature question is: could those conditions of sharing values be weaken?
In this study, we shall prove a uniqueness theorem for entire functions that share two finite values "1 CM + 1 IM" with their nth exact differences, by using a simple method which is very differential to the proof of Theorems A and B. In fact, we obtain the following result. Here, we shall only give an example to illustrate it as follows.
a π log 2 and c = π. Then, the nth exact difference of f(z) is as follows: strictly less than the degree of f in this case for n ≥ 1.
Remark 3. As per Theorems A and B, we all hope that the restriction on the growth of f can be dropped. But it seems not to be easy. However, we can also find out many entire functions satisfying the difference equation with hyper order greater than 1. Those discussions are arranged in Section 4.

Some lemmas
To prove our result, we need the following auxiliary results. To estimate N(r, f(z + c)) and T(r, f(z + c)), we need the next result.  On the other hand, by the assumptions of Theorem 1.1, we know that  where h is some entire function. In addition, by using Lemma 2. Next, by the assumption that f and Δ f c n share 1 IM, we can deduce from (3.4) and (3.5) that Rewrite formula (3.4) as Finally, by using the second main theorem for three small functions (Lemma 2.2), we deduce from (3.6), (3.7) and (3.9) that which is impossible. And this completes the proof of Theorem 1.1.

Examples and discussions
To construct the proper examples for Theorem 1.1, we recall a result obtained by Ozawa [17]. That is, for an arbitrary number σ ∈ [1,∞), there exists a periodic entire function D(z) with period c ≠ 0 such that ρ(D) = σ. Throughout this section, the notation D(z) always means such an entire function.