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Entire functions sharing a small function with their two difference operators
- Feng Lü^{1}Email author,
- Yanfeng Wang^{1} and
- Junfeng Xu^{2}View ORCID ID profile
https://doi.org/10.1186/s13662-017-1281-4
© The Author(s) 2017
- Received: 12 December 2016
- Accepted: 17 July 2017
- Published: 1 August 2017
Abstract
In this article, we deduce a uniqueness result of entire functions that share a small entire function with their two difference operators, generalizing some previous theorems of (Farissi et al. in Complex Anal. Oper. Theory 10:1317-1327, 2015, Theorem 1.1) and (Chen and Li in Adv. Differ. Equ. 2014:311, 2014, Theorem 1.1) by omitting the assumption that the shared small entire function is periodic.
Keywords
- uniqueness
- entire functions
- difference operators
- Nevanlinna theory
MSC
- 30D35
- 30D30
- 39A10
1 Introduction and main result
Nevanlinna theory of value distributions is concerned with the density of points where a meromorphic function takes a certain value in the complex plane. Nowadays, there has been recent interest in connections between the Nevanlinna theory and the difference operator. In addition, many papers have been devoted to the investigation of the uniqueness problems related to meromorphic functions and their shifts or their difference operators and one got a lot of results (see, e.g., [3–8]).
A meromorphic function \(a(z)\) is said to be a small function with respect to \(f(z)\) if and only if \(T(r,a)=S(r,f)\), where \(S(r,f)=o(T(r,f))\), as \(r\rightarrow \infty \) outside of a possible exceptional set of finite logarithmic measure. Denote the set of all the small functions of \(f(z)\) by \(S(f)\). Let \(f(z)\) and \(g(z)\) be two meromorphic functions and let \(a(z)\) be a small entire function of \(f(z)\) and \(g(z)\). We say that \(f(z)\) and \(g(z)\) share \(a(z)\) IM, provided that \(f(z)-a(z)\) and \(g(z)-a(z)\) have the same zeros ignoring multiplicities. Similarly, we say that \(f(z)\) and \(g(z)\) share \(a(z)\) CM, provided that \(f(z)-a(z)\) and \(g(z)-a(z)\) have the same zeros counting multiplicities.
Recently, Chen et al. [2, 9] investigated two uniqueness problems on entire functions that share a small periodic entire function with their two difference operators as follows.
Theorem A
see [9], Theorem 1.1
Let \(f(z)\) be a nonconstant entire function of finite order, let \(a(z)(\not \equiv 0)\in S(f)\) be a periodic entire function with period c. If \(f(z)\), \(\Delta_{c}f(z)\), \(\Delta^{2}_{c}f(z)\) share \(a(z)\) CM, then \(\Delta^{2}_{c}f(z)\equiv \Delta_{c}f(z)\).
Theorem B
see [2], Theorem 1.2
Let \(f(z)\) be a nonconstant entire function of finite order. If \(f(z)\), \(\Delta_{c}f(z)\), \(\Delta^{2}_{c}f(z)\) share 0 CM, then \(\Delta^{2}_{c}f(z)\equiv C \Delta_{c}f(z)\), where C is a nonzero constant.
In 2015, El Farissi, Latreuch and Asiri further studied the above problem and obtained
Theorem C
see [1], Theorem 1.1
Let \(f(z)\) be a nonconstant entire function of finite order, let \(a(z)(\not \equiv 0) \in S(f)\) be a periodic entire function with period c. If \(f(z)\), \(\Delta_{c}f(z)\), \(\Delta^{2}_{c}f(z)\) share \(a(z)\) CM, then \(f(z)\equiv \Delta_{c} f(z)\).
Remark 1
It is necessary to point out that Theorems A and B have been generalized from \(\Delta^{2}_{c}f(z)\) to \(\Delta^{n}_{c}f(z)\) by Chen, Chen and Li in [9]. There are also some interesting results related the above theorems (see, e.g., [6, 10]).
In the previous results, we find that the shared small function \(a(z)\) is a periodic function with period c. So, it is natural to ask what will happen if the periodic condition of \(a(z)\) is omitted. In this paper, we focus on this problem and we obtain the following result.
Theorem 1
- (i)
If \(\Delta_{c}a(z)\equiv a(z)\), then \(\Delta^{2}_{c}f(z)-a(z)=C( \Delta_{c}f(z)-a(z))\), where C is a nonzero constant.
- (ii)
If \(\Delta_{c} a(z)\not \equiv a(z)\), then \(\Delta_{c}f(z)=f(z)\) or \(\Delta^{2}_{c}f(z)-a(z)=e^{\gamma }(\Delta_{c}f(z)-a(z))\), where γ is a polynomial with \(\deg \gamma < \rho (a)\).
Remark 2
We point out that Theorem 1 is a generalization of the previous theorems.
If \(a(z)\equiv 0\), then \(\Delta_{c}a(z)=a(z)\). Then it follows from (i) of Theorem 1 that \(\Delta^{2}_{c}f(z)=C \Delta_{c}f(z)\), where C is a nonzero constant.
If \(a(z)\not \equiv 0\) is a periodic function with period c, then \(\Delta_{c} a(z)\not \equiv a(z)\). It follows from (ii) of Theorem 1 that \(\Delta_{c}f(z)=f(z)\) or \(\Delta^{2}_{c}f(z)= \Delta_{c}f(z)\). Furthermore, by Theorem C we can deduce that \(\Delta_{c}f(z)=f(z)\).
As an application of Theorem 1, we can obtain an interesting result, where \(a(z)\) is a slow growth small function.
Theorem 2
Let \(f(z)\) be a nonconstant entire function of finite order, and let \(a(z)(\not \equiv 0)\in S(f)\) be an entire function with \(\rho (a)<1\). If \(f(z)\), \(\Delta_{c}f(z)\), \(\Delta^{2} _{c}f(z)\) share \(a(z)\) CM, then \(\Delta_{c}f(z)=f(z)\).
For convenience of the reader, we list here some notations. For a meromophic function f, we use the basic notations of the Nevanlinna theory of meromorphic functions such as \(T(r,f)\), \(m(r,f)\), \(N(r,f)\) and \(\overline{N}(r,f)\) as explained in [11–13].
2 Some lemmas
In this section, we state some results that we employ in our proofs.
Lemma 2.1
[4], Theorem 2.1
Lemma 2.2
[14], Lemma 3.3
Lemma 2.2 plays an important role in the proof of Theorem 2.
3 Proof of Theorem 1
Note that \(f(z)\) is a nonconstant entire function of finite order. Then \(\Delta_{c}f(z)\) and \(\Delta^{2}_{c}f(z)\) are also two entire functions of finite order.
Suppose that \(\Delta_{c} a(z)\equiv a(z)\). Obviously, we can get \(\Delta^{2}_{c} a(z)\equiv a(z)\). It is clear that \(g(z)\), \(\Delta _{c} g(z)\), and \(\Delta^{2}_{c} g(z)\) share 0 CM. By Theorem B, we can obtain \(\Delta^{2}_{c} g(z)\equiv C\Delta_{c}g(z)\), where C is a nonconstant. So \(\Delta^{2}_{c} f(z)-\Delta^{2}_{c}a(z)\equiv C(\Delta _{c}f(z)-\Delta_{c}a(z))\). That is, \(\Delta^{2}_{c} f(z)-a(z)\equiv C( \Delta_{c}f(z)-a(z))\).
In the following, we assume that \(\Delta_{c} a(z)\not \equiv a(z)\). We consider into two cases.
Case 1. \(\Delta^{2}_{c} a(z)\not \equiv a(z)\).
By (4) and Lemma 2.1, we deduce that \(\varphi (z)\in S(f)\).
Similarly, we get \(T(r,e^{Q})=S(r,f)\).
Note that \(a(z)\neq \Delta_{c}a(z)\), then \(e^{P(z)}\equiv 1\), so \(e^{Q(z)}\equiv 1\). Hence, we obtain \(\Delta_{c}f(z)=f(z)\).
Case 2. \(\Delta^{2}_{c}a(z)\equiv a(z)\).
By (2) and Lemma 2.1, we get \(e^{Q}\in S(f)\).
Suppose that \(n=\deg P\leq \deg Q\). Since \(e^{Q}\in S(f)\), we have \(e^{P}\in S(f)\).
Similar to the discussion of Subcase 1.1, we obtain \(\Delta_{c}f(z)=f(z)\).
Note that \(\deg P > \deg Q\), so \(e^{Q}\) is also a small function of h.
Suppose that \(z_{0}\) is a common zero of \(e^{P(z+c)}-1\) and \(e^{P(z)}e^{P(z+c)}+e^{P(z+c)}-e^{P(z)}-e^{Q(z)}\). It is easy to deduce that \(e^{Q(z_{0})}=1\). Note that \(e^{Q(z)}-1\in S(h)\). Thus, the claim holds.
Suppose that \(z_{0}\) is a zero of H. We claim \(\nu_{H}(z_{0})\leq \nu_{\Delta_{c} a-a}(z_{0})+\deg P\cdot \nu_{e^{Q}-1}(z_{0})\). Now we split into two cases.
Case A. \(z_{0}\) is not a zero of \(e^{P(z+c)}-1\). Then \(z_{0}\) must be a zero of \(\Delta_{c} a-a\). Hence \(\nu_{H}(z_{0})\leq \nu_{\Delta_{c} a-a}(z_{0})\).
Case B. \(z_{0}\) is a zero of \(e^{P(z+c)}-1\). Set \(P_{1}(z)=P(z+c)\). Obviously, \(\nu_{e^{P_{1}}-1}(z_{0})\leq \deg P\). Then \(H(z_{0})=0\) and \(e^{P(z_{0}+c)}-1=0\), which leads to \(e^{Q(z_{0})}=1\).
Hence, we finish the proof of Theorem 1.
4 Proof of Theorem 2
If \(a(z)\) is a constant, then it follows from Theorem C that \(f(z)\equiv \Delta_{c} f(z)\). In the following, we assume that \(a(z)\) is a nonconstant entire function.
Hence, we finish the proof of Theorem 2.
Declarations
Acknowledgements
The research was supported by NNSF of China Project (No. 11601521), the Fundamental Research Fund for Central Universities in China (Nos. 15CX05061A, 15CX05063A and 15CX08011), NSF of Guangdong Province (Nos. 2016A030313002, 2015A030313644) and Funds of Education Department of Guangdong (2016KTSCX145).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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