Abstract
In this paper, we investigate the existence and growth of solutions of the qdifference equation \(\prod_{i=1}^{n}f(q_{i}z)=R(z,f(z))\), where \(R(z,f(z))\) is an irreducible rational function in \(f(z)\). We also give an estimation of the growth of transcendental meromorphic solutions of the equation \(\prod_{i=1}^{n}f(q_{i}z)=f(z)^{m}\).
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1 Introduction and main results
A meromorphic function \(f(z)\) means meromorphic in the complex plane ℂ. If no poles occur, then \(f(z)\) reduces to an entire function. Assume that \(n(r, f)\) counts the number of poles of f in \(z\leq r\), each pole is counted according to its multiplicity, and that \(\overline{n}(r,f)\) counts the number of the distinct poles of f in \(z\leq r\), ignoring the multiplicity. The characteristic function of f is defined by
where
and
For more notations and definitions of the Nevanlinna value distribution theory of meromorphic functions, we refer to [1, 2].
A meromorphic function \(\alpha(z)\) is called a small function with respect to \(f(z)\), if \(T(r,\alpha)=S(r,f)\), where \(S(r,f)\) denotes any quantity satisfying \(S(r,f)=o(T(r,f))\) as \(r\rightarrow\infty\) outside a possible exceptional set E of logarithmic density 0. The order and the exponent of convergence of zeros of meromorphic function \(f(z)\) is, respectively, defined as
The difference operators for a meromorphic function \(f(z)\) are defined as
In the following, \(f(qz+c)\) is the qshift of \(f(z)\), \(f(qz)f(z)\) is the qdifference of \(f(z)\), where \(q\neq0,1\). If an equation includes qshifts or qdifferences of \(f(z)\), then the equation is called the qdifference equation. A Borel exceptional value of \(f(z)\) is any value a satisfying \(\lambda(fa)<\sigma(f)\).
In the last two decades, the existence and growth of meromorphic solutions of difference equations have been investigated in many papers [3–9]. Recently, with the development of the qdifference analog of Nevanlinna theory, there has been a renewed interest in studying meromorphic solutions of qdifference equations. For instance, Zheng and Chen [8] considered the growth problem of transcendental meromorphic solutions of some qdifference equations.
Theorem A
[8, Theorem 2]
Suppose that f is a transcendental meromorphic solution of the equation
where \(q\in\mathbb{C}\), \(q>1\), the coefficients \(a_{j}(z)\) are rational functions and P, Q are relatively prime polynomials in f over the rational functions satisfying \(p=\deg_{f}P\), \(t=\deg_{f}Q\), \(d=pt\geq 2\). If f has infinitely many poles, then for sufficiently large r, \(n(r,f)\geq Kd^{\frac{\log r}{n\logq}}\) holds for some constant \(K>0\). Thus, the lower order of f, which has infinitely many poles, satisfies \(\mu(f)\geq d^{\frac{\log d}{n\logq}}\).
In [10], Heittokangas et al. first considered meromorphic solutions with Borel exceptional zeros and poles of some type of difference equations and obtained the result as follows.
Theorem B
[10, Theorem 13]
Let \(c_{1}, \ldots,c_{n}\in\mathbb{C}\setminus\{0\}\) and suppose that f is a nonrational meromorphic solution of a difference equation of the form
with meromorphic coefficients \(a_{i}(z)\), \(b_{j}(z)\) of growth \(S(r,f)\) such that \(a_{p}(z)b_{t}(z)\not\equiv0\). If
then the above equation is of the form
where \(c(z)\) is meromorphic, \(T(r,c)=S(r,f)\), and \(k\in\mathbb{Z}\).
Recently, Zheng and Chen [9] considered a qdifference equation under a condition similar to Theorem B on meromorphic solution, and they obtained the following result.
Theorem C
[9, Theorem 1]
Suppose that f is a transcendental meromorphic solution of a qdifference equation of the form
where \(q_{i}\in\mathbb{C}\setminus\{0,1\}\), \(i=1, \ldots, n\), and \(R(z,f)\) is an irreducible rational function in f with meromorphic coefficients \(a_{i}(z)\) (\(i=0, \ldots, p\)) and \(b_{j}(z)\) (\(j=0, \ldots, t\)) of growth \(S(r,f)\) such that \(b_{t}(z)\equiv1\), \(a_{p}(z)\not\equiv0\). If
then the above equation is reduced to the form
or
For the qdifference equation (1.1), Theorem C only considered the case when solutions have Borel exceptional zeros and poles. But how about the existence and growth of meromorphic solutions of (1.1)? Theorem 1.1 considers under what conditions (1.1) will not have solutions with zero order.
Theorem 1.1
Let \(R(z,f)\) be an irreducible rational function in f with meromorphic coefficients \(a_{i}(z)\) (\(i=0,\ldots,p\)) and \(b_{j}(z)\) (\(j=0,\ldots,t\)) of growth \(S(r,f)\), \(d=\max\{p,t\}\), \(q_{i}\in \mathbb{C}\setminus\{0,1\}\), \(i=1,\ldots,n\).

(1)
If \(d>n\), then (1.1) has no transcendental meromorphic solution of zero order.

(2)
If \(d\neq n\), then (1.1) has no transcendental entire solution of zero order.
As many papers (see [8, 11]) obtained the lower bound of the order of solutions of difference equations. The natural question arises of the upper bound of the order of the solutions of (1.1). The following theorem answers this question partly.
Theorem 1.2
Suppose that \(f(z)\) is a transcendental meromorphic solution of (1.1), where \(R(z,f)\) is an irreducible rational function in f with meromorphic coefficients \(a_{i}(z)\) (\(i=0,\ldots,p\)) and \(b_{j}(z)\) (\(j=0,\ldots,t\)) of growth \(S(r,f)\), \(d=\max\{p,t\}\), \(q_{i}\in \mathbb{C}\), \(q_{i}>1\) (\(i=1, \ldots, n\)), \(q_{1}\leqq_{2}\leq\cdots \leqq_{n1}<q_{n}\). Then
When the function \(R(z,f)\) in (1.1) is reduced to \(f(z)^{m}\), we consider the qdifference equation
From the proof of Theorem 1.1, one can immediately get the following corollary.
Corollary 1.3
Let m, n be positive integers, \(q_{i}\in\mathbb{C}\setminus\{0,1\}\), \(i=1, \ldots, n\).

(1)
If \(m>n\), then (1.2) has no transcendental meromorphic solution of zero order.

(2)
If \(m\neq n\), then (1.2) has no transcendental entire solution of zero order.
The following theorem gives an estimation of the growth of the meromorphic solutions of (1.2), where \(q_{i}\in\mathbb{C}\), \(q_{i}>1 \) (\(i=1,\ldots,n\)).
Theorem 1.4
Suppose that \(f(z)\) is a transcendental meromorphic solution of (1.2), where \(m, n\) are positive integers, \(q_{i}>1\) (\(i=1, \ldots, n\)), \(q=\max\{q_{1},q_{2}, \ldots,q_{n}\}\). Then
Remark
If \(q_{k}=q\), we denote \(s=\max\{q_{i}, i=1,\ldots, n, i\neq k \} \), then \(s\leqq\). If \(s\neqq\), then from the proof of Theorem 1.2, we can immediately get
In the following, we consider transcendental entire solutions with \(\lambda(f)<\sigma(f)\) of (1.2).
Theorem 1.5
Suppose that \(f(z)\) is a transcendental entire solution of finite order of (1.2), where m, n are positive integers, \(q_{i}\in\mathbb {C}\setminus\{0,1\}\), \(i=1,\ldots,n\), if \(\lambda(f)<\sigma(f)\), then
The following example shows that the case of Theorem 1.5 can occur.
Example 1
Let \(m=20\), \(q_{1}=2\), \(q_{2}=4\), then the transcendental entire function \(f(z)=e^{z^{2}}\) satisfies the equation
Here, \(f(z)\) has finitely many zeros and satisfies \(q_{1}^{\sigma (f)}+q_{2}^{\sigma(f)}=m\).
2 Lemmas
To prove our results, we need some lemmas.
Lemma 2.1
[12] (ValironMohon‘ko)
Let \(f(z)\) be a meromorphic function, then for all irreducible rational functions in f,
with meromorphic coefficients \(a_{i}(z)\), \(b_{j}(z)\), the characteristic function of \(R(z,f(z))\) satisfies
where \(d=\max\{m,n\}\) and \(\psi(r)=\max\{T(r,a_{i}),T(r,b_{j})\}\).
Lemma 2.2 gives us the relationship of the characteristic function between \(f(z)\) and \(f(qz)\), provided that \(f(z)\) is a nonconstant meromorphic function of zero order.
Lemma 2.2
[7]
If \(f(z)\) is a nonconstant meromorphic function of zero order, and \(q\in\mathbb{C}\setminus\{0\}\), then
on a set of lower logarithmic density 1.
Lemma 2.3
[3]
Let \(f(z)\) be a nonconstant meromorphic function of zero order, and \(q\in\mathbb{C}\setminus\{0\}\), then
on a set of logarithmic density 1.
The next lemma is the relationship between \(T(r,f(qz))\) and \(T(qr,f(z))\).
Lemma 2.4
[4]
Let \(f(z)\) be a meromorphic function, and let \(q\in\mathbb{C}\setminus\{ 0\}\), then
Lemma 2.5
[13]
Let \(\phi: (1,\infty) \rightarrow(0,\infty)\) be a monotone increasing function, and let f be a nonconstant meromorphic function. If for some real constant \(\alpha\in(0,1)\), there exist real constants \(K_{1}>0\) and \(K_{2}\geq1\) such that
then
The following are the wellknown Weierstrass factorization theorem and Hadamard factorization theorem.
Lemma 2.6
[14]
If an entire function f has a finite exponent of convergence \(\lambda (f)\) for its zerosequence, then f has a representation in the form
satisfying \(\lambda(Q)=\sigma(Q)=\lambda(f)\). Further, if f is of finite order, then g in the above form is a polynomial of degree less than or equal to the order of f.
3 The proofs
3.1 Proof of Theorem 1.1
(1) Suppose that \(f(z)\) is a transcendental meromorphic solution of zero order of (1.1), it follows from Lemma 2.1 that
which implies
By Lemma 2.2, we obtain
on a set of lower logarithmic density 1. It is clear that (3.1) is a contradiction when \(d>n\).
Thus if \(d>n\), (1.1) has no transcendental meromorphic solution of zero order.
(2) Suppose that \(f(z)\) is a transcendental entire solution of zero order of (1.1), it follows from (1.1) that
Since \(f(z)\) is a transcendental entire solution of zero order, by Lemma 2.1 and Lemma 2.3, the above inequality can be reduced to
on a set of lower logarithmic density 1. It is clear that (3.2) is a contradiction when \(d< n\).
Using the same method of the proof of (1), we can get if \(d>n\), (1.1) has no transcendental entire solution of zero order.
Thus if \(d\neq n\), (1.1) has no transcendental entire solution of zero order.
3.2 Proof of Theorem 1.2
It follows from (1.2) and Lemma 2.1 that
Since \(q_{i}>1\) (\(i=1, \ldots, n\)), \(q_{1}\leqq_{2}\leq\cdots\leq q_{n1}<q_{n}\), by Lemma 2.4, we obtain
Setting \(R=q_{n}r\), \(\alpha=\frac{q_{n1}}{q_{n}}\), we have
By Lemma 2.5 and (3.3), we obtain
3.3 Proof of Theorem 1.4
We will divide the argument into two cases.
Case 1. If \(n\geq m\), it is easily to see \(\sigma(f)\geq \mu(f)\geq\frac{\log m\log n}{\logq}\) is obviously true.
Case 2. If \(n< m\), by Corollary 1.3, we get when \(n< m\), (1.2) has no transcendental meromorphic solution of zero order. So \(f(z)\) is a transcendental meromorphic solution of positive order. It follows from Lemma 2.4 that
Note that \(q=\max\{q_{1},q_{2},\ldots,q_{n}\}\), thus by (1.2), we obtain
By the above inequality, we have, for any given ϵ (\(0<\epsilon<1\)),
outside of a possible exceptional set of finite logarithmic measure. Now we can use [12, Lemma 1.1.1] to deal with the exceptional set here. It follows from (3.4) that for any given \(\alpha>1\), there exists an \(r_{0}>0\) such that
holds for all \(r\geq r_{0}\). This implies that
Inductively, for any \(k\in\mathbb{N}\), we obtain
for all \(r>r_{0}\). By (3.6), we can get for sufficiently large d, there exists a \(k\in\mathbb{N}\) such that \(d\in((\alphaq)^{k}r_{0},(\alphaq)^{k+1}r_{0})\), that is
Letting \(\epsilon\rightarrow0\) and \(\alpha\rightarrow1\), it follows from (3.8) that
where \(H= (\frac{m}{n} )^{\frac{\log(qr_{0})}{\log q}}T(r_{0},f)\) is a positive constant.
Thus we get
3.4 Proof of Theorem 1.5
Since \(f(z)\) is a transcendental entire function of finite order and \(\lambda(f)<\sigma(f)\), by Lemma 2.6, \(f(z)\) can be written as
where \(g(z)\) (≢0) is an entire function such that \(\sigma (g)=\lambda(g)=\lambda(f)<\sigma(f)\), \(h(z)\) is a polynomial. Set
where \(a_{d}, \ldots, a_{0}\) are constants. Since \(\sigma(f)<\infty\), it follows that
We obtain
where
On the other hand, we have
Since \(\sigma(g)< d\), by (1.2), (3.9) and (3.10), we obtain
which implies that
Since \(\sigma(f)=\deg(h(z))=d\), we have
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Acknowledgements
This research was partly supported by National Natural Science Foundation of China (11271304); the Natural Science Foundation of Fujian Province of China for Distinguished Young Scholars (2013J06001); and the Program for New Century Excellent Talents in University (NCET130510).
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Xu, N., Zhong, CP. Existence and properties of meromorphic solutions of some qdifference equations. Adv Differ Equ 2015, 16 (2015). https://doi.org/10.1186/s136620140346x
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DOI: https://doi.org/10.1186/s136620140346x