Zeros and fixed points of the linear combination of shifts of a meromorphic function
- Weiwei Cui^{1},
- Lianzhong Yang^{1}Email author and
- Xiaoguang Qi^{2}
https://doi.org/10.1186/s13662-015-0632-2
© Cui et al. 2015
Received: 14 January 2015
Accepted: 2 September 2015
Published: 9 September 2015
Abstract
Let f be transcendental and meromorphic in the complex plane. In this article, we investigate the existences of zeros and fixed points of the linear combination and quotients of shifts of \(f(z)\) when \(f(z)\) is of order one. We also prove a result concerning the linear combination which extends a result of Bergweiler and Langley. Some results concerning the order of \(f(z) <1\) are also obtained.
Keywords
MSC
1 Introduction and main results
In this article, a function is called meromorphic if it is analytic in the whole complex plane except at possible isolated poles. We assume that readers are familiar with the basic results and notations of the Nevanlinna value distribution theory of meromorphic functions (see, e.g., [1–4]).
Let f be a transcendental meromorphic function in the plane. The forward differences \(\Delta^{n}(f)\) are defined in the standard way ([5], p.52) by \(\Delta f=f(z+1)-f(z)\), \(\Delta^{n+1}(f)=\Delta^{n}f(z+1)-\Delta ^{n}f(z)\), \(n=0,1,2,\ldots \) .
Recently, many excellent results concerning the Nevanlinna theory for difference operators have been obtained (see, e.g., [6–13]). For example, Halburd and Korhonen [11], respectively Chiang and Feng [7] obtained the difference analogue of logarithmic derivatives lemma. Their results became a starting point of investigating Nevanlinna theory on difference operators and difference equations. Another important result belongs to Bergweiler and Langley. In [6], Bergweiler and Langley firstly investigate the existence of zeros of Δf and \(\frac{\Delta f}{f}\). The results may be viewed as discrete analogue of the following existence theorem on the zeros of \(f'\).
Theorem A
Hurwitz theorem implies that under the hypotheses of Theorem A, \(f(z+c)-f(z)\) has a zero near \(z_{1}\) for all sufficiently small \(c\in C\setminus\{0\}\) if \(z_{1}\) is a zero of \(f'\). When f is a transcendental entire function of order less than one, then the first difference Δf, and by repetition of this argument each difference \(\Delta^{n}f\), for \(n\geq1\), is transcendental entire of order less than one and hence has infinitely many zeros. Bergweiler and Langley considered the divided difference case and obtained the following theorem.
Theorem B
([6])
From the proof of Theorem B, it can be seen that \(\delta_{0}\) is extremely small, so they conjectured that the conclusion of Theorem B still holds for \(\delta(f)<1\). Chen and Shon partly answered this question and obtained the following.
Theorem C
([17])
- (i)
at most finitely many zeros \(z_{j}\), \(z_{k}\) satisfy \(z_{j}-z_{k}=c\);
- (ii)
\(\underline{\lim}_{j \to + \infty} \vert \frac{z_{j+1}}{z_{j}}\vert =l>1\).
Bergweiler and Langley [6] also obtained the following results.
Theorem D
Let f be a function transcendental and meromorphic of lower order \(\lambda(f)<\lambda<1\) in the plane. Let \(c \in C\setminus\{0\}\) be such that at most finitely many poles \(z_{j}\), \(z_{k}\) of f satisfy \(z_{j}-z_{k}=c\). Then \(g(z)=f(z+c)-f(z)\) has infinitely many zeros.
In Theorem D, Bergweiler and Langley considered the existence of zeros of first difference operator when the transcendental meromorphic function is of lower order less than one. Chen and Shon [17] considered the case when the order of f is equal to one. They proved the following results.
Theorem E
Let \(c\in C\backslash\{0\}\) and f be a transcendental entire function of order \(\sigma(f)=\sigma=1\). If \(f(z)\) have infinitely many zeros with the exponent of convergence of zeros \(\lambda(f)<1\), then \(g(z)=\Delta f(z)=f(z+c)-f(z)\) has infinitely many zeros and infinitely many fixed points.
- (i)
at most finitely many zeros \(z_{j}\), \(z_{k}\) satisfy \(z_{j}-z_{k}=c\);
- (ii)
\(\underline{\lim}_{j \to + \infty} \vert \frac{z_{j+1}}{z_{j}}\vert =l>1\),
Hence, a natural question arises from Theorem E: What can be said about the existence of zeros and fixed points of Δf and \(\frac{\Delta f}{f}\), when \(f(z)\) is a meromorphic function with order \(\sigma(f)=1\)?
In [18], Cui and Yang considered and gave answers to this question. Now, since we introduce the general form of Δf and \(\frac{\Delta f}{f}\), the above question acquires its new forms:
Do the conclusions of Theorems B-E still hold for \(g(z)\) and \(g(z)/f(z)\), when \(f(z)\) is a meromorphic function with order \(\sigma(f)<1\) or even \(\sigma(f)\leq1\)? In the present article, we answer this question and obtain the following results.
Theorem 1.1
Remark 1.1
By Lemma 2.8 in the following part, we can easily have the following result concerning \(\sigma(f)<1\).
Theorem 1.2
Now we consider the existence of zeros and fixed points of \(G(z)\). From the proofs of Theorem 1.1 and Theorem 1.2, we notice that the property of \(g(z)\), i.e., whether it is transcendental or not, actually determines our conclusions. Therefore, in the following we still first consider the property of \(G(z)\) and then investigate zeros and fixed points of \(G(z)\) when f is of order equal to or less than one.
Theorem 1.3
- (i)f is transcendental entire and the order of growth of f satisfies$$\underline{\lim}_{r \to + \infty}\frac{T(r,f)}{r}=0; $$
- (ii)f is transcendental meromorphic and the order of growth of f satisfies$$\underline{\lim}_{r \to + \infty}\frac{T(r,f)}{r^{\frac{1}{2}}}=0. $$
Theorem F
([6])
In the following, we will investigate the existence of zeros and fixed points of \(G(z)\) and obtain some results related to this.
Theorem 1.4
The following theorem can be acquired by using Theorem 1.3 and using the same method as in the proof of Theorem 1.2.
Theorem 1.5
2 Some lemmas
Lemma 2.1
([15])
Remark 2.1
([6])
Lemma 2.2
([4])
- (i)
\(\sum_{j=1}^{n}f_{j}(z)e^{g_{j}(z)}\equiv0\);
- (ii)
when \(1\leq j< k\leq n\), \(g_{j}(z)-g_{k}(z)\) is not a constant;
- (iii)when \(1\leq j\leq n\), \(1\leq h< k\leq n\),where \(E\subset(1,\infty)\) is of finite linear measure or finite logarithmic measure.$$T(r,f_{j})=o\bigl({T\bigl(r,e^{g_{h}-g_{k}}\bigr)}\bigr)\quad (r \rightarrow\infty, r\notin E), $$
Lemma 2.3
([6])
Lemma 2.4
([6])
Lemma 2.5
([20])
Lemma 2.6
([7])
The following Lemma 2.7 can be easily obtained from Chiang and Feng (Theorem 9.4 [7]), here we omit its proof.
Lemma 2.7
Lemma 2.8
Proof
Suppose that \(g(z)\) is not transcendental. Then we divide our proof into two cases.
Case 2. If \(g(z)\) is rational, without loss of generality, we may suppose that \(g(z)\) is a nonzero polynomial.
Hence, \(g(z)\) is transcendental. □
3 Proofs of Theorem 1.1 and Theorem 1.2
Proof of Theorem 1.1
Set \(F(z)=p(z)f(z)\). Then, by the Hadamard factorization theorem and \(\lambda(f)<\sigma(f)\), we have \(F(z)=h(z)e^{az+b}\), where \(a\neq0\) and b are constants, \(h(z)\) is an entire function satisfying \(\sigma(h)=\lambda(h)=\lambda(f)<\sigma(f)=1\).
We affirm that \(a=d\).
Hence, \(g(z)\) has infinitely many fixed points. □
Proof of Theorem 1.2
According to Lemma 2.8, we know that \(g(z)\) is transcendental under conditions of Theorem 1.2 and the order \(\sigma(g)<1\), which means \(g(z)\) has either infinitely many zeros or infinitely many poles. Since \(f(z)\) has only finitely many poles, i.e., \(g(z)\) has finitely many poles. Hence \(g(z)\) must have infinitely many zeros.
The same discussion also holds when we consider the fixed points of \(g(z)\). Hence, \(g(z)\) has infinitely many fixed points. □
4 Proofs of Theorem 1.3 and Theorem 1.4
Proof of Theorem 1.3
Proof of Theorem 1.4
Now we prove that \(G(z)\) has infinitely many zeros. We continue to use symbols in the proofs of Theorem 1.1 and Theorem 1.2.
Theorem 1.4 follows. □
5 Proof of Theorem 1.5
When \(f(z)\) is transcendental and entire with \(\sigma(f)<1\), we can also obtain the same conclusion by the same discussion as above. Theorem 1.5 follows.
Declarations
Acknowledgements
The authors thank the referee for his/her valuable suggestions to improve the present article. The authors also thank Jun Wang for valuable suggestions and discussions. This work was supported by the NNSF of China (No. 11171013 & No. 11371225). The third author is also supported by the NNSF of China (No. 11301220), the Tianyuan Fund for Mathematics (No. 11226094), the NSF of Shandong Province, China (No. ZR2012AQ020) and the Fund of Doctoral Program Research of University of Jinan (XBS1211).
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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