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On Malmquist type theorem of systems of complex difference equations
Advances in Difference Equations volume 2015, Article number: 131 (2015)
Abstract
The main purpose of this paper is to give the Malmquist type result of the meromorphic solutions of a system of complex difference equations of the following form:
where \(c_{1}, c_{2}, \ldots, c_{n}\) are distinct, nonzero complex numbers, the coefficients \(\alpha_{\lambda_{1}, \mu_{1}}(z)\) (\(\lambda _{1} \in I_{1}\), \(\mu_{1}\in J_{1}\)), \(\beta_{\lambda_{2}, \mu_{2}}(z)\) (\(\lambda_{2} \in I_{2}\), \(\mu_{2}\in J_{2}\)), \(a_{i}(z)\) (\(i=0,1,\ldots, p\)), \(b_{j}(z)\) (\(j=0,1,\ldots, q\)), \(d_{k}(z)\) (\(k=0,1,\ldots, s\)), and \(e_{l}(z)\) (\(l=0,1,\ldots, t\)) are small functions relative to \(f(z)\) and \(g(z)\), \(I_{i} = \{\lambda _{i}=(l_{\lambda_{i}, 1}, l_{\lambda_{i}, 2}, \ldots, l_{\lambda_{i},n})| l_{\lambda_{i}, \nu}\in{N} \cup\{0\},\nu= 1,2,\ldots,n\}\) (\(i=1,2\)) and \(J_{j} = \{\mu_{j}=(m_{\mu_{j}, 1}, m_{\mu_{j}, 2}, \ldots, m_{\mu_{j},n})| m_{\mu_{j}, \nu}\in{N} \cup\{0\},\nu=1,2, \ldots,n\}\) (\(j=1, 2\)) are finite index sets. The growth of meromorphic solutions of a related system of complex functional equations is also investigated.
1 Introduction and main results
Let \(f(z)\) be a meromorphic function in the complex plane C. We assume that the reader is familiar with the standard notations and results in Nevanlinna’s value distribution theory of meromorphic functions (see e.g. [1–3]). We use \(\rho(f)\) to denote the growth order of a meromorphic function \(f(z)\). The notation \(S(r,f)\) denotes any quantity that satisfies the condition \(S(r,f)= o(T(r,f))\) as \(r \rightarrow\infty\) possibly outside an exceptional set of r of finite logarithmic measure. A meromorphic function \(a(z)\) is called a small function of \(f(z)\) if and only if \(T(r,a(z))=S(r,f)\).
In the last ten years, there has been a great deal of interest in studying the properties of complex difference equations (see e.g. [4–20]). Especially, a number of papers (see e.g. [4, 6, 10, 11, 15, 17–19]) focusing on a Malmquist type theorem of the complex difference equations emerged. In 2000, Ablowitz et al. [4] proved some results on the Malmquist theorem of the complex difference equations by utilizing Nevanlinna theory. They obtained the following two results.
Theorem A
If the second-order difference equation
with polynomial coefficients \(a_{i}\) (\(i=1,2,\ldots,p\)) and \(b_{j}\) (\(j=1,2,\ldots,q\)), admits a transcendental meromorphic solution of finite order, then \(d=\max\{p,q\}\leq2\).
Theorem B
If the second-order difference equation
with polynomial coefficients \(a_{i}\) (\(i=1,2,\ldots,p\)) and \(b_{j}\) (\(j=1,2,\ldots,q\)), admits a transcendental meromorphic solution of finite order, then \(d=\max\{p,q\}\leq2\).
Subsequently, Heittokangas et al. [10], Laine et al. [15] and Huang et al. [11], respectively, gave some generalizations of the above two results. In 2010, the first author in this paper and Liao [18] obtained the following more general result.
Theorem C
Let \(c_{1}, c_{2}, \ldots, c_{n}\) be distinct, nonzero complex numbers, and suppose that \(f(z)\) is a transcendental meromorphic solution of the difference equation
with coefficients \(\alpha_{\lambda}(z)\) (\(\lambda\in I\)), \(a_{i}(z)\) (\(i=0,1,\ldots, p\)), and \(b_{j}(z)\) (\(j=0,1,\ldots, q\)), which are small functions relative to \(f(z)\), where \(I = \{\lambda=(l_{\lambda, 1}, l_{\lambda, 2}, \ldots, l_{\lambda,n})| l_{\lambda, \nu}\in{N} \cup\{0\}, \nu=1,2, \ldots,n\}\) is a finite index set, and denote
If the order \(\rho(f)\) is finite, then \(d=\max\{p,q\}\leq \sigma\).
If all the coefficients in the complex difference equation (1) are rational functions, then in [19], we have the following Malmquist type result, which is reminiscent of the classical Malmquist theorem in complex differential equations.
Theorem D
Let \(c_{1}, c_{2}, \ldots, c_{n}\) be distinct, nonzero complex numbers and suppose that \(f(z)\) is a transcendental meromorphic solution of the equation
where \(I = \{\lambda=(l_{\lambda, 1}, l_{\lambda, 2}, \ldots, l_{\lambda,n})| l_{\lambda, \nu}\in{N} \cup\{0\}, \nu=1,2, \ldots,n\}\) is a finite index set, P and Q are relatively prime polynomials in f over the field of rational functions, the coefficients \(\alpha_{\lambda }\) (\(\lambda\in I \)) are rational functions. Denoting the degree of \(P[z,f]\) by
If \(f(z)\) is finite order and has at most finitely many poles, then \(R(z,f)\) reduces to a polynomial in f of degree \(d \leq\gamma_{P}\).
More recently, people began to study the properties of meromorphic solutions of systems of complex difference equations. In [21], Gao discussed the proximity function and counting function of meromorphic solutions of some classes of systems of complex difference equations. In 2013, Wang et al. [16] investigated the growth of meromorphic solutions of systems of complex difference equations.
Now, we give the Malmquist type result of a system of complex difference equations as follows.
Theorem 1
Let \(c_{1}, c_{2}, \ldots, c_{n}\) be distinct, nonzero complex numbers and suppose that \((f(z), g(z))\) is a transcendental meromorphic solution of a system of complex difference equations of the form
with coefficients \(\alpha_{\lambda_{1}, \mu_{1}}(z)\) (\(\lambda_{1} \in I_{1}\), \(\mu_{1}\in J_{1}\)), \(\beta_{\lambda_{2}, \mu_{2}}(z)\) (\(\lambda_{2} \in I_{2}\), \(\mu_{2}\in J_{2}\)), \(a_{i}(z)\) (\(i=0,1,\ldots, p\)), \(b_{j}(z)\) (\(j=0,1,\ldots, q\)), \(d_{k}(z)\) (\(k=0,1,\ldots, s\)), and \(e_{l}(z)\) (\(l=0,1,\ldots, t\)) are small functions relative to \(f(z)\) and \(g(z)\), \(a_{p}(z), b_{q}(z), d_{s}(z), e_{t}(z)\not\equiv0\), where \(I_{i} = \{\lambda_{i}=(l_{\lambda_{i}, 1}, l_{\lambda_{i}, 2}, \ldots, l_{\lambda_{i},n})| l_{\lambda_{i}, \nu} \in{N} \cup\{0\}, \nu=1,2, \ldots,n\}\) (\(i=1,2\)), and \(J_{j} = \{\mu_{j}=(m_{\mu_{j}, 1}, m_{\mu _{j}, 2}, \ldots, m_{\mu_{j},n})| m_{\mu_{j}, \nu}\in{N} \cup\{0\},\nu=1,2, \ldots,n\}\) (\(j=1, 2\)) are finite index sets, and denote
(\(\nu=1,2,\ldots,n\)), and
If \(\max\{p,q\}>\sigma_{12}\), \(\max\{s,t\}>\sigma_{21}\), and \(\max\{\rho (f), \rho(g)\}<+\infty\), then \(\rho(f)= \rho(g)\) and \((\max\{p,q\} -\sigma_{12})\cdot(\max\{s,t\}-\sigma_{21})\leq \sigma_{11}\sigma_{22}\).
Example 1
It is easy to check that \((f(z), g(z))=(\tan z, \cot z)\) satisfies the following system of difference equations:
In Example 1, we have \(\max\{p,q\}=4\), \(\max\{s,t\}=4\), \(\sigma_{11}=1\), \(\sigma_{12}=3\), \(\sigma_{21}=2\), \(\sigma_{22}=2\), \(\rho(f)=\rho(g)=1<+\infty\), and \((\max\{p,q\}-\sigma_{12})\cdot(\max\{ s,t\}-\sigma_{21})=(4-3)(4-2)=2=1\times2= \sigma_{11}\sigma_{22}\). Therefore, the estimation in Theorem 1 is sharp.
Remark 1
Obviously, if the condition \(\max\{p,q\}>\sigma_{12}\), \(\max\{s,t\} >\sigma_{21}\) in Theorem 1 is replaced by \((\max\{p,q\}-\sigma _{12})(\max\{s,t\}-\sigma_{21})=0\) or \((\max\{p,q\}-\sigma_{12})(\max\{ s,t\}-\sigma_{21})<0\), the estimation \((\max\{p,q\}-\sigma_{12})\cdot (\max\{s,t\}-\sigma_{21})\leq \sigma_{11}\sigma_{22}\) is still correct. If \(\sigma_{11}=0\) or \(\sigma _{22}=0\), then the first or second equation in (2) gets the form of (1). For some results as regards (1), the reader may refer to the paper [18].
Remark 2
If the condition \(\max\{p,q\}>\sigma_{12}\), \(\max\{s,t\}>\sigma _{21}\) in Theorem 1 is replaced by \(\max\{p,q\}<\sigma_{12}\), \(\max\{s,t\}<\sigma_{21}\), then the estimation \((\max\{p,q\}-\sigma_{12})\cdot(\max\{s,t\}-\sigma_{21})\leq \sigma_{11}\sigma_{22}\) is not true generally. For example, \((f(z), g(z))=(\tan z, \cot z)\) satisfies the following system of complex difference equations:
where \(\max\{p,q\}=2\), \(\max\{s,t\}=2\), \(\sigma_{11}=2\), \(\sigma_{12}=6\), \(\sigma_{21}=5\), \(\sigma_{22}=1\), \(\max\{p,q\}<\sigma_{12}\), \(\max\{s,t\}<\sigma_{21}\). However, \((\max\{ p,q\}-\sigma_{12})\cdot(\max\{s,t\}-\sigma_{21})=(-4)\times(-3)=12>2= \sigma_{11}\sigma_{22}\).
If \(\sigma_{12}=\sigma_{21}=0\), then we have the following simpler result.
Corollary 1
Let \(c_{1}, c_{2}, \ldots, c_{n}\) be distinct, nonzero complex numbers, and suppose that \((f(z), g(z))\) is a transcendental meromorphic solution of a system of complex difference equations of the form
with coefficients \(\alpha_{\lambda}(z)\) (\(\lambda\in I\)), \(\beta_{\mu}(z)\) (\(\mu\in J\)), \(a_{i}(z)\) (\(i=0,1,\ldots, p\)), \(b_{j}(z)\) (\(j=0,1,\ldots, q\)), \(d_{k}(z)\) (\(k=0,1,\ldots, s\)), and \(e_{l}(z)\) (\(l=0,1,\ldots, t\)) are small functions relative to \(f(z)\) and \(g(z)\), \(a_{p}(z), b_{q}(z), d_{s}(z), e_{t}(z)\not\equiv0\), where \(I = \{\lambda=(l_{\lambda, 1}, l_{\lambda, 2}, \ldots, l_{\lambda,n})| l_{\lambda, \nu}\in{N} \cup\{0\}, \nu=1,2, \ldots,n\}\) and \(J = \{\mu=(m_{\mu, 1}, m_{\mu, 2}, \ldots, m_{\mu,n})|m_{\mu, \nu}\in{N} \cup\{0\},\nu=1,2, \ldots,n\}\) are two finite index sets, and denote
and
If \(\rho(f)<+\infty\) or \(\rho(g)<+\infty\), then \(\rho(f)=\rho(g)\) and \(\max\{p,q\}\cdot\max\{s,t\}\leq \sigma_{1} \sigma_{2}\).
Example 2
Let \(c_{1}=\arctan2\), \(c_{2}=\arctan(-2)\). It is easy to check that \((f(z), g(z))=(\tan z, \cot z)\) satisfies the following system of difference equations:
In Example 2, we have \(\max\{p,q\}=4\), \(\max\{s,t\}=3\), \(\sigma_{1}=4\), \(\sigma_{2}=3\), \(\rho(f)=\rho(g)=1<+\infty\), and \(\max\{p,q\}\cdot\max\{s,t\}=\sigma _{1}\cdot\sigma_{2}=12\). Therefore, the estimation in Corollary 1 is sharp.
In [15], Laine et al. also considered the growth of meromorphic solutions of some classes of complex difference functional equations and obtained the following result.
Theorem E
Suppose that f is a transcendental meromorphic solution of the equation
where \(p(z)\) is a polynomial of degree \(k \geq2\), \(\{J\}\) is the collection of all subsets of \(\{1,2,\ldots,n\}\). Moreover, we assume that the coefficients \(\alpha_{J}(z)\) are small functions relative to f and that \(n\geq k\). Then
where \(\alpha=\frac{\log n}{\log k}\).
In 2010, Zhang et al. [18] got a more generalized result than Theorem E. Next we give the growth of meromorphic solutions of a system of complex functional equations as follows.
Theorem 2
Let \(c_{1}, c_{2}, \ldots, c_{n}\) be distinct, nonzero complex numbers and suppose that \((f(z), g(z))\) is a transcendental meromorphic solution of a system of complex functional equations of the form
where \(p(z)\) is a polynomial of degree \(k\geq2\), \(I_{i} = \{\lambda_{i}=(l_{\lambda_{i}, 1}, l_{\lambda_{i}, 2}, \ldots, l_{\lambda_{i},n})| l_{\lambda_{i}, \nu} \in{N} \cup\{0\}, \nu=1,2, \ldots,n\}\) (\(i=1,2\)) and \(J_{j} = \{\mu_{j}=(m_{\mu_{j}, 1}, m_{\mu_{j}, 2}, \ldots, m_{\mu_{j},n})| m_{\mu_{j}, \nu}\in{N} \cup\{0\},\nu=1,2, \ldots,n\}\) (\(j=1, 2\)) are finite index sets, and denote
(\(\nu=1,2,\ldots,n\)),
and
Moreover, we assume that the coefficients \(\alpha_{\lambda_{1}, \mu _{1}}(z)\) (\(\lambda_{1} \in I_{1}\), \(\mu_{1}\in J_{1}\)), \(\beta_{\lambda_{2}, \mu_{2}}(z)\) (\(\lambda_{2} \in I_{2}\), \(\mu_{2}\in J_{2}\)) are small functions relative to \(f(z)\) and \(g(z)\), and that \(2{\sigma}\geq k\). Then
where \(\alpha=\frac{\log{2\sigma}}{\log k}\).
If \(\sigma_{12}=\sigma_{21}=0\), then we can obtain the following result easily.
Corollary 2
Let \(c_{1}, c_{2}, \ldots, c_{n}\) be distinct, nonzero complex numbers and suppose that \((f(z), g(z))\) is a transcendental meromorphic solution of a system of complex functional equations of the form
where \(p(z)\) is a polynomial of degree \(k\geq2\), \(I = \{\lambda=(l_{\lambda, 1}, l_{\lambda, 2}, \ldots, l_{\lambda,n})| l_{\lambda, \nu}\in{N} \cup\{0\}, \nu=1,2, \ldots,n\}\) and \(J = \{\mu=(m_{\mu, 1}, m_{\mu, 2}, \ldots, m_{\mu,n})| m_{\mu, \nu}\in{N} \cup\{0\},\nu=1,2, \ldots,n\}\) are two finite index sets, and denote
and
Moreover, we assume that the coefficients \(\alpha_{\lambda}(z)\) (\(\lambda \in I\)), \(\beta_{\mu}(z)\) (\(\mu\in J\)) are small functions relative to \(f(z)\) and \(g(z)\), and that \(2{\sigma}\geq k\). Then
where \(\alpha=\frac{\log{2\sigma}}{\log k}\).
2 Some lemmas
In order to prove our results, we need the following lemmas.
Lemma 1
(see [3])
Let \(f(z)\) be a meromorphic function. Then for all irreducible rational functions in f,
such that the meromorphic coefficients \(a_{i}(z)\), \(b_{j}(z)\) satisfy
we have
In [22], AZ Mokhon’ko and VD Mokhon’ko gave an estimation of Nevanlinna’s characteristic function of
where \(f_{1},f_{2},\ldots,f_{n}\) are distinct meromorphic functions, \(I=\{\lambda=(l_{\lambda, 1}, l_{\lambda , 2}, \ldots, l_{\lambda, n})| l_{\lambda, \nu}\in{N} \cup\{0\}, \nu=1,2,\ldots,n\}\) and \(J=\{\mu=(m_{\mu, 1}, m_{\mu, 2}, \ldots, m_{\mu, n})| m_{\mu, \nu}\in{N} \cup\{0\}, \nu=1,2, \ldots,n\}\) are two finite index sets. However, the method of the proof was too complex. For \(F(z)\) of the form \(\sum_{\lambda\in I}f_{1}^{l_{\lambda, 1}}f_{2}^{l_{\lambda, 2}}\cdots \, f_{n}^{l_{\lambda, n}}\), Zheng et al. [20] gave a simpler proof, but the estimation of \(T(r,F)\) was not sharp. For completeness, we give the proof of the following lemma.
Lemma 2
Let \(f_{1},f_{2},\ldots,f_{n}\) be distinct meromorphic functions. Then
where \(I=\{(l_{\lambda, 1}, l_{\lambda, 2}, \ldots, l_{\lambda, n})| l_{\lambda, j}\in{N} \cup\{0\}, j=1,2, \ldots,n\}\) is an finite index set consisting of t elements and \(\sigma_{j} = \max_{\lambda\in I}\{l_{\lambda, j}\}\) (\(j=1,2,\ldots,n\)).
Proof
All the poles of the function \(\sum_{\lambda\in I} f_{1}^{l_{\lambda, 1}}f_{2}^{l_{\lambda, 2}}\cdots\, f_{n}^{l_{\lambda, n}}\) are generated by the poles of the functions \(f_{j}\) (\(j=1,2,\ldots,n\)), and every pole of multiplicity k of \(f_{j}\) (\(j=1,2,\ldots,n\)) has order at most \(k\sigma_{j}\). This implies that
Thus we obtain
We next prove that
and we define
for \(j=1,2,\ldots,n\). Thus we have
By the definition of \(m(r,f)\), we immediately conclude that
By (7) and (8), the assertion follows. □
Remark 3
If we suppose that \(\alpha_{\lambda}(z)=o(T(r,f_{j}))\) (\(\lambda\in I\)) hold for all \(j\in\{1,2,\ldots,n\}\), and denote \(T(r,a_{\lambda})=S(r,f)\) (\(\lambda \in I\)), then we have the following estimation:
Lemma 3
(see [6])
Let \(f(z)\) be a meromorphic function with order \(\rho= \rho(f)\), \(\rho< + \infty\), and c be a fixed non zero complex number, then for each \(\varepsilon> 0\), we have
Lemma 4
(see [3])
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 of finite linear measure. Then, for any \(\alpha>1\), there exists \(r_{0}>0\) such that \(g(r)\leq h(\alpha r)\) for all \(r_{0}\).
Lemma 5
(see [23])
Let f be a transcendental meromorphic function, and \(p(z)=a_{k}z^{k}+a_{k-1}z^{k-1}+\cdots+a_{1}z+a_{0}\), \(a_{k}\neq0\), be a nonconstant polynomial of degree k. Given \(0<\delta<|a_{k}|\), denote \(\lambda= |a_{k}|+\delta\) and \(\mu=|a_{k}|-\delta\). Then given \(\varepsilon>0\) and \(a\in{C}\cup\{\infty\}\), we have
for all r large enough.
3 Proofs of theorems
Proof of Theorem 1
We assume that \((f(z), g(z))\) is a transcendental meromorphic solution of the system of complex difference equations (2). By the first equation in (2), Lemma 1, Lemma 2, and Lemma 3, we have, for each \(\varepsilon>0\),
By the above inequality, we get, for each \(\varepsilon>0\),
Since \(\max\{p,q\}>\sigma_{12}\) by the assumption, we have, for each \(\varepsilon>0\),
Similarly, by the second equation in (2), we obtain, for each \(\varepsilon>0\),
By (12) and \(\max\{s,t\}>\sigma_{21}\), we have, for each \(\varepsilon>0\),
and
Using (11), we can obtain \(\rho(g)\leq\rho(f)\). Similarly, we can get \(\rho(f)\leq\rho (g)\) from (14). Therefore, we have \(\rho(f)=\rho(g)\).
It follows from (10) and (13) that
From (15), we conclude that
This yields the asserted result. □
Proof of Theorem 2
We assume that \((f(z), g(z))\) is a transcendental meromorphic solution of a system of complex functional equations (5). Let \(C=\max\{|c_{1}|,|c_{2}|,\ldots,|c_{n}|\}\). According to the first equation in (5), Lemma 2, Lemma 3, and the last assertion of Lemma 5, we get
Since \(T(r+C,f) \leq T(\beta r,f)\) and \(T(r+C,g) \leq T(\beta r,g)\) hold for r large enough for \(\beta>1\), we may assume r to be large enough to satisfy
outside a possible exceptional set of finite linear measure. By Lemma 4, we know that, whenever \(\gamma>1\),
holds for all r large enough. Let \(t=\gamma\beta r\), then the inequality (16) may be written in the form
Similarly, by the second equation in (5), for all r large enough and \(\beta>1\), \(\gamma>1\), we have
Let \(t=\gamma\beta r\), we have
Letting \(s=\log t + \frac{\log\frac{\mu}{(\gamma\beta)^{k}}}{k-1}\), then \(t=e^{s} (\frac{\mu}{(\gamma\beta)^{k}} )^{\frac{1}{1-k}}\) and
So
Let \(T(t,f)=T (e^{s} (\frac{\mu}{(\gamma\beta)^{k}} )^{\frac {1}{1-k}},f )=\Phi(s,f)\), \(T(t,g)=T (e^{s} (\frac{\mu}{(\gamma\beta)^{k}} )^{\frac {1}{1-k}},g )=\Phi(s,g)\), \(M= \max \{\frac{\sigma_{11} (1+\varepsilon)}{1-\varepsilon},\frac{\sigma_{12} (1+\varepsilon)}{1-\varepsilon},\frac{\sigma_{21} (1+\varepsilon)}{1-\varepsilon},\frac{\sigma_{22} (1+\varepsilon)}{1-\varepsilon} \}\), then from (17) and (19), we have
Similarly, from (18), we can get
The inequalities (20) and (21) hold for all s large enough.
Letting now \(\alpha=\frac{\log2M}{\log k}\), namely, \(2M=k^{\alpha}\). Write \(\Psi(s,f)=\frac{\Phi(s,f)}{s^{\alpha}}\) and \(\Psi(s,g)=\frac{\Phi(s,g)}{s^{\alpha}}\), thus we have
and
The inequalities (22) and (23) hold for all s large enough, we may assume that (22) and (23) hold for all \(s\geq s_{0}\).
Let \(M_{1}=\sup_{s_{0}\leq s \leq k s_{0}}\Psi(s,f)\) and \(M_{2}=\sup_{s_{0}\leq s \leq k s_{0}}\Psi(s,g)\), then by (22) and (23) we can obtain
Similarly, we have
Thus, we deduce that
Therefore, \(\Psi(s,f)\) and \(\Psi(s,g)\) are bounded for all \(s\geq s_{0}\). There exist some constants \(K_{1}\), \(K_{2}\), \(K_{3}\), \(K_{4}\), such that, for any \(\varepsilon>0\),
and
Therefore, we have
and
where
Letting now \(\alpha= \frac{\log2\sigma}{\log k}\), we obtain the required form. Theorem 2 is proved. □
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Acknowledgements
The author would like to thank Professor Liangwen Liao and the anonymous referees for their valuable comments and suggestions. The research was supported by NSF of China (11426118), Natural Science Foundation of Jiangsu Province (BK20140767), Natural Science Foundation of the Jiangsu Higher Education Institutions (14KJB110004), Qing Lan Project of Jiangsu Province and Colonel-level topics (JSNU-ZY-01).
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Zhang, J. On Malmquist type theorem of systems of complex difference equations. Adv Differ Equ 2015, 131 (2015). https://doi.org/10.1186/s13662-015-0475-x
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DOI: https://doi.org/10.1186/s13662-015-0475-x