Growth of the solutions of some q-difference differential equations
- Hong-Yan Xu^{1}Email author,
- Lian-Zhong Yang^{2} and
- Hua Wang^{1}
https://doi.org/10.1186/s13662-015-0520-9
© Xu et al. 2015
Received: 29 December 2014
Accepted: 7 April 2015
Published: 10 June 2015
Abstract
In view of Nevanlinna theory, we study the growth and poles of solutions of some complex q-difference differential equations. We obtain the estimates on the Nevalinna order, the lower order, and the counting function of poles for meromorphic solutions of such equations.
Keywords
MSC
1 Introduction and main results
In this paper, the fundamental theorems and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used (see Hayman [1], Yang [2] and Yi and Yang [3]). For a meromorphic function \(f(z)\), we also use \(S(r,f)\) to denote any quantity satisfying \(S(r,f)=o(T(r,f))\) for all r outside a possible exceptional set E of finite logarithmic measure \(\lim_{r\rightarrow\infty}\int_{[1,r)\cap E}\frac{dt}{t}<\infty\), and a meromorphic function \(a(z)\) is called a small function with respect to f, if \(T(r,a)=S(r,f)=o(T(r,f))\).
In 2002, Gundersen et al. [8] studied the growth of meromorphic solutions of q-difference equations and obtained results as follows.
Theorem 1.1
([8], Theorem 3.2)
Theorem 1.2
([8], Theorem 3.4)
Theorem 1.3
([9])
Theorem 1.4
([9])
Recently, Zhang [10] further studied the growth of solutions of (1) and obtained the following theorem.
Theorem 1.5
([10], Theorem 1.1)
Regarding Theorem 1.5, Zhang [10] asked the following question: Is the order of transcendental solutions of (1) exactly \(\rho(f)\leq \frac{\log2}{\log|q|}\)?
In this paper, we further investigate the growth of solution of some class of q-difference differential equation and obtain the following results.
Theorem 1.6
The following example shows that (2) has non-transcendental entire function solution.
Example 1.1
The following example shows that (2) also has a transcendental entire function solution.
Example 1.2
Remark 1.1
Thus, a question arises naturally: Does (2) have a transcendental meromorphic solution?
When the constant q of the right of (2) is replaced by a function, the following example shows that the equation has a transcendental meromorphic solution.
Example 1.3
Thus, we have the following theorems.
Theorem 1.7
Theorem 1.8
The following example shows that (4) has a transcendental meromorphic solution f with the order \(\rho(f)=\frac{\log(s+1)}{\log|q|}\).
Example 1.4
Let \(p(z)=p_{k}z^{k}+p_{k-1}z^{k-1}+\cdots+p_{1}z+p_{0}\), where \(p_{k}(\not\equiv0), \ldots, p_{0}\) are complex constants. Now, we investigate the growth of solutions of such equations, where qz is replaced by \(p(z)\) in (2)-(4), and we obtain the following result.
Theorem 1.9
Recently, there were many results on meromorphic solutions of complex functional equations (see [11–20]). In 2007, Barnett et al. [21] firstly established an analog of the logarithmic derivative lemma on q-difference operators. In 2010, by applying their theorems, Zheng and Chen [22] considered the growth of meromorphic solutions of q-difference equations and obtained results which extended some theorems given by Heittokangas et al. [23].
Theorem 1.10
([22], Theorem 2)
From Theorem 1.10, we further study the growth of the solutions of a class of q-difference differential equation and obtain a result as follows.
Theorem 1.11
Remark 1.2
The following example shows that (6) has a non-transcendental solution.
Example 1.5
The following examples show that (6) has transcendental entire and meromorphic solutions.
Example 1.6
Example 1.7
Example 1.8
2 Some lemmas
Lemma 2.1
(Valiron-Mohon’ko [24])
Lemma 2.2
Lemma 2.3
([8])
Lemma 2.4
([25])
Lemma 2.5
Let \(g: (0,+\infty)\rightarrow R\), \(h: (0,+\infty)\rightarrow R\) be monotone increasing functions such that \(g(r)\leq h(r)\) outside of an exceptional set E with finite linear measure, or \(g(r)\leq h(r)\), \(r\notin H\cup(0,1]\), where \(H\subset(1,+\infty)\) is a set of finite logarithmic measure. Then, for any \(\alpha>1\), there exists \(r_{0}\) such that \(g(r)\leq h(\alpha r)\) for all \(r\geq r_{0}\).
Lemma 2.6
([29])
- (i)
Suppose that \(\psi(\mu r^{m})\leq A\psi(r)+B\) (\(r\geq r_{0}\)), where μ (\(\mu>0\)), m (\(m>1\)), A (\(A\geq1\)), B are constants. Then \(\psi(r)=O((\log r)^{\alpha})\) with \(\alpha=\frac{\log A}{\log m}\), unless \(A=1\) and \(B>0\); and if \(A=1\) and \(B>0\), then for any \(\varepsilon>0\), \(\psi(r)=O((\log r)^{\varepsilon})\).
- (ii)
Suppose that (with the notation of (i)) \(\psi(\mu r^{m})\geq A\psi(r)\) (\(r\geq r_{0}\)). Then for all sufficiently large values of r, \(\psi(r)\geq K(\log r)^{\alpha}\) with \(\alpha=\frac{\log A}{\log m}\), for some positive constant K.
Lemma 2.7
(see [12])
3 Proofs of Theorems 1.6-1.8
3.1 The proof of Theorem 1.6
3.2 The proof of Theorem 1.7
Since \(\varphi_{1}(z)\) is a rational function, we have \(T(r,\varphi_{1}(z))=O(\log r)\). If f is a transcendental entire function, similar to the argument as in Theorem 1.6, we easily get \(\rho(f)\leq\frac{\log(s+1)-\log n}{\log|q|}\).
Thus, this completes the proof of Theorem 1.7.
3.3 The proof of Theorem 1.8
Hence, from (14) and (15), we complete the proof of Theorem 1.8.
4 The proof of Theorem 1.9
Thus, this completes the proof of Theorem 1.9.
5 The proof of Theorem 1.11
Thus, this completes the proof of Theorem 1.11.
Declarations
Acknowledgements
The authors thank the referee for his/her valuable suggestions to improve the present article.
The first author was supported by the NSF of China (11301233, 61202313), the Natural Science Foundation of Jiangxi Province in China (20132BAB211001, 20151BAB201008), and the Foundation of Education Department of Jiangxi (GJJ14644) of China. The second author was supported by the NSF of China (11371225, 11171013, and 11041005).
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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