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Some results on entire functions that share one value with their difference operators
- BaoQin Chen^{1},
- Sheng Li^{1}Email authorView ORCID ID profile and
- Fujie Chai^{2}
https://doi.org/10.1186/s13662-018-1653-4
© The Author(s) 2018
- Received: 6 December 2017
- Accepted: 28 May 2018
- Published: 7 June 2018
Abstract
Keywords
- Difference operators
- Entire functions
- Shared values
MSC
- 30D35
- 39B32
1 Introduction and main results
Let \(S(r,f)\) denote any quantity that satisfies \(S(r,f)=o(T(r,f(z)))\) as \(r\to\infty\) possibly outside of an exceptional set of finite logarithmic measure. A meromorphic function \(h(z)\) is said to be a small function of \(f(z)\) if \(T(r,h(z))=S(r,f)\).
For two meromorphic functions \(f(z)\) and \(g(z)\), and a finite constant a, let \(z_{k}\) (\(k=1,2,\ldots\)) be zeros of \(f(z)-a\), \(\tau(k)\) be the multiplicity of the zero \(z_{k}\), and we write \(f(z)=a\Rightarrow g(z)=a\), provided that \(z_{k}\) (\(k=1,2,\ldots\)) are also zeros of \(g(z)-a\) (ignoring multiplicities); and \(f(z)=a\rightarrow g(z)=a\), provided that \(z_{k}\) (\(k=1,2,\ldots\)) are also zeros of \(g(z)-a\) with multiplicity at least \(\tau(k)\). Then we say that \(f(z)\) and \(g(z)\) share a IM if \(f(z)=a\Leftrightarrow g(z)=a\). Similarly, we say that \(f(z)\) and \(g(z)\) share a CM if \(f(z)=a\rightleftharpoons g(z)=a\).
The uniqueness theory of meromorphic functions is an important part of Nevanlinna theory. The classical results in the uniqueness theory of meromorphic functions are the five-value theorem and four-value theorem due to Nevanlinna [18]. He proved that if two meromorphic functions \(f(z)\), \(g(z)\) share five distinct values in the extended complex plane IM, then \(f(z)\equiv g(z)\), and similarly, if two meromorphic functions \(f(z)\), \(g(z)\) share four distinct values in the extended complex plane CM, then \(f(z) =T (g(z))\), where T is a Mobius transformation. In the past ninety years, many analysts have been devoted to improving the Nevanlinna’s results mentioned above by reducing the number of shared values. It is well known that the assumption 4 CM in the four-value theorem has been improved to 2 CM + 2 IM by Gundersen [6] and cannot be improved to 4 IM [5], while 1 CM + 3 IM remains an open problem.
To reduce the number of shared values quickly, many authors began to consider the case that \(f(z)\) and \(g(z)\) have some special relationship. One of successful attempts in this direction was created by Rubel and Yang [19]. In 1977, they proved that: for a non-constant entire function \(f(z)\), if \(f(z)\) and \(f'(z)\) share two distinct finite values \(a,b\) CM, then \(f(z)\equiv f'(z)\). Then many authors began to investigate the uniqueness of meromorphic functions sharing values with their derivatives (see e.g. [10, 13, 20, 24]) Here we recall two results relative to our main results in this paper. The first is the following result proved by Jank, Mues, and Volkmann in 1986.
Theorem A
([10])
Let \(f(z)\) be a non-constant entire function, let \(a \neq0\) be a finite constant. If \(f(z)\) and \(f'(z)\) share the value a IM, and if \(f''(z)=a\) whenever \(f(z)=a\), then \(f(z)\equiv f'(z)\).
The second is the following result, an improvement of Theorem A by considering higher order derivatives, proved by Wang and Yi in 2003.
Theorem B
([20])
Recently, lots of papers (including [1–4, 7–9, 12, 14, 15, 17, 23]) have focused on difference analogues of Nevanlinna theory and uniqueness of meromorphic functions and their shifts or their difference operators. Many classical results of the uniqueness theory have been extended to the difference field. For instance, Heittokangas et al. [9] considered the uniqueness problems on the meromorphic functions sharing values with their shifts and proved some original results corresponding to Nevanlinna’s five-value theorem and four-value theorem; Chen and Yi [3], Li and Gao [14], and Liu and Yang [16] studied uniqueness of entire functions sharing values with their difference operators and proved some meaningful results.
In this paper, we consider the following question: what happens if we replace the derivatives of non-constant entire function \(f(z)\) with its difference operators in Theorem A and Theorem B? Then we prove three results as follows, including Theorem 1.2, which can be regarded as a difference analogue of Theorem B to some extent.
Theorem 1.1
Theorem 1.2
Example
Let \(f(z)=e^{\frac{1}{4}{ (\frac{\pi}{2}i+\ln2 )z}}-1+i\), then \(\Delta_{2}f\equiv\Delta_{2}^{5}f\equiv\Delta_{2}^{9}f=ie^{\frac{1}{4}{ (\frac{\pi}{2}i+\ln2 )z}}\). Here \(f(z)=i\rightleftharpoons\Delta_{2}f=i\) and \(f(z)=i\rightarrow\Delta_{2}^{5}f=\Delta_{2}^{9}f=i\), but \(f(z)\not\equiv\Delta_{2}f\equiv\Delta_{2}^{5}f\equiv\Delta_{2}^{9}f\). This example shows that the conclusion \(\Delta^{n}_{c}f\equiv\Delta^{m}_{c}f\) in Theorem 1.2 cannot be extended to \(f(z)\equiv\Delta_{c}f\) in general.
Remark
- (i)In the above example, we find thatwhere \(i^{4}=i^{8}=1\). This shows that the conclusion of Theorem 1.1 also holds here. However, \(m(r,1/(f(z)-i))\neq S(r,f)\). We conjecture that Theorem 1.1 is still valid even if condition (1.1) is changed by a less restrictive one. In view of this, we give Theorem 1.3 in the following.$$\Delta_{2}^{4}f\equiv\Delta_{2}^{8}f=e^{\frac{1}{4}{ (\frac{\pi}{2}i+\ln 2 )z}}=f(z)-i+1=f(z)-a+ \frac{a}{i}, $$
- (ii)
In the above example, we also find that \(\Delta_{2}f\equiv\Delta_{2}^{5}f\equiv\Delta_{2}^{9}f\). We wonder whether \(\Delta^{n}_{c}f\equiv\Delta^{m}_{c}f\) in Theorem 1.2 can be extended to \(\Delta^{n}_{c}f\equiv\Delta^{m}_{c}f\equiv\Delta_{c}f\) or not.
Theorem 1.3
2 Proof of Theorem 1.1
Lemma 2.1
([12])
Lemma 2.2
Lemma 2.3
([7])
Remark
By the recent results of Halburd, Korhonen, and Tohge [8], we can easily find that Lemmas 2.1–2.3 still hold for the meromorphic functions with hyper-order less than one.
Proof of Theorem 1.1
3 Proof of Theorem 1.2
Now \(\alpha(z)\equiv0\), and we have \(\beta(z)\not\equiv0\) since \(\alpha(z)\not\equiv\beta(z)\). Using the similar reasoning as above, we can also get \(\Delta_{c}^{m}f(z)\equiv\Delta_{c}f(z)\), which contradicts the fact \(\beta(z)\not\equiv0\). Therefore, we prove that \(\Delta_{c}^{n}f(z)\equiv\Delta_{c}^{m}f(z)\).
4 Proof of Theorem 1.3
Declarations
Acknowledgements
The authors are very appreciative of the editors and reviewers for their constructive suggestions and comments for the readability of our paper.
Funding
This work was supported by the Natural Science Foundation of Guangdong (2015A030313620), Excellent Young Teachers Training Program of Guangdong High Education (YQ2015089), Excellent Young Teachers Training Program of Guangdong Ocean University(2014007, HDYQ2015006), Project of Enhancing School with Innovation of Guangdong Ocean University (GDOU2016 050206, GDOU2016050209).
Authors’ contributions
All authors have drafted the manuscript, read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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