# Growth of solutions to two systems of q-difference differential equations

## Abstract

This paper is devoted to studying the growth of entire or meromorphic solutions to two systems of q-difference differential equations. The estimates on the growth order of meromorphic solutions are obtained, which are extensions of previous results due to Xu et al . Examples are given to illustrate the existence of solutions of such systems.

## Introduction and main results

Let $$f(z)$$ be a non-constant meromorphic function in the complex plane $$\mathbb{C}$$. We use $$\rho(f)$$ and $$\mu(f)$$ to denote the order and the lower order of $$f(z)$$, and use $$\lambda(\frac{1}{f})$$ and $$\overline{\lambda}(\frac{1}{f})$$ to denote the exponent of convergence of poles and that of the distinct poles of $$f(z)$$, respectively. In addition, we say a meromorphic function $$\alpha(z)$$ ($$\not\equiv0,\infty$$) is a small function of $$f(z)$$ provided that $$T(r,\alpha)=S(r,f)$$, where $$S(r,f)$$ denotes any quantity that satisfies the condition $$S(r,f)=o(T(r,f))$$ as $$r\rightarrow\infty$$, possibly outside a set of r with finite logarithmic measure. Nevanlinna theory is an important tool in this paper, its standard symbols and fundamental results come mainly from [12, 21].

As we all know, it is an interesting problem to consider the Malmquist theorem  for differential equations. Laine  gave the following result.

### Theorem A

()

Let

$$\bigl(w'(z) \bigr)^{n}=R(z,w),$$
(1)

where the right-hand side

\begin{aligned} R(z,w)=\frac{\sum_{i=0}^{k}a_{i}(z)w^{i}}{\sum_{j=0}^{l}b_{j}(z)w^{j}} \end{aligned}

is rational in both arguments. If equation (1) has a transcendental meromorphic solution, then$$l=0$$and$$k\leq2n$$.

With the establishment of the difference analog of Nevanlinna theory, many studies [1, 8, 13, 20] about the Malmquist-type theorem of complex difference equations or systems have appeared. Gundersen et al.  considered the growth of meromorphic solutions to a certain type of complex q-difference equation and proved the following result.

### Theorem B

()

Let$$w(z)$$be a transcendental meromorphic solution of the equation

$$w(qz)=R(z,w),$$

where$$q\in\mathbb{C}$$, $$|q|>1$$, $$R(z,w)$$is irreducible inw, which is defined as in Theorem A, and the coefficients$$a_{i}(z)$$and$$b_{j}(z)$$are small functions ofwand$$a_{k}(z)b_{l}(z)\not\equiv0$$. If$$m: =\max\{k,l\}\geq1$$, then$$\rho(w)=\frac{\log m}{\log|q|}$$.

After these results, many scholars studied a series of complex q-difference differential equations and systems about the Malmquist-type theorem [5, 6, 19]. Xu et al.  investigated the following system:

$$\textstyle\begin{cases} [w_{1}'(q_{1}z) ]^{n_{1}}=R_{2} (z,w_{2}(z) ), \\ [w_{2}'(q_{2}z) ]^{n_{2}}=R_{1} (z,w_{1}(z) ), \end{cases}$$
(2)

where $$q_{1}, q_{2}\in\mathbb{C}\setminus\{0\}$$, $$n_{1}, n_{2}\in{\mathbb {Z}_{+}}$$, and

\begin{aligned} R_{1} \bigl(z,w_{1}(z) \bigr)= \frac{\sum_{i=0}^{k_{1}}a_{i}(z)w_{1}(z)^{i}}{\sum_{j=0}^{l_{1}}b_{j}(z)w_{1}(z)^{j}}, \qquad R_{2} \bigl(z,w_{2}(z) \bigr)= \frac{\sum_{i=0}^{k_{2}}c_{i}(z)w_{2}(z)^{i}}{\sum_{j=0}^{l_{2}}d_{j}(z)w_{2}(z)^{j}} \end{aligned}
(3)

are irreducible rational functions, and $$a_{i}(z)$$, $$b_{j}(z)$$ are small functions with respect to $$w_{1}$$, and $$c_{i}(z)$$, $$d_{j}(z)$$ are small functions with respect to $$w_{2}$$. They obtained the estimates on the growth order for meromorphic solutions of system (2).

We consider the question of what happens if system (2) is more general, for example,

$$\textstyle\begin{cases} \varOmega_{1} (z,w_{1}^{(h_{1})}(q_{1}z) )=R_{2} (z,w_{2}(z) ), \\ \varOmega_{2} (z,w_{2}^{(h_{2})}(q_{2}z) )=R_{1} (z,w_{1}(z) ), \end{cases}$$
(4)

where $$q_{1}, q_{2}\in\mathbb{C}\setminus\{0\}$$, $$h_{1}, h_{2}\in{\mathbb{Z}_{+}}$$, and $$R_{1}(z,w_{1}(z))$$, $$R_{2}(z,w_{2}(z))$$ are defined as in (3), and

$$\begin{gathered} \varOmega_{1} \bigl(z,w_{1}^{(h_{1})}(q_{1}z) \bigr)= \frac{\sum_{m_{1}=0}^{p_{1}}u_{m_{1}}^{1}(z) [w_{1}^{(h_{1})}(q_{1}z) ]^{m_{1}}}{\sum_{n_{1}=0}^{s_{1}}v_{n_{1}}^{1}(z) [w_{1}^{(h_{1})}(q_{1}z) ]^{n_{1}}}, \\ \varOmega_{2} \bigl(z,w_{2}^{(h_{2})}(q_{2}z) \bigr)= \frac{\sum_{m_{2}=0}^{p_{2}}u_{m_{2}}^{2}(z) [w_{2}^{(h_{2})}(q_{2}z) ]^{m_{2}}}{\sum_{n_{2}=0}^{s_{2}}v_{n_{2}}^{2}(z) [w_{2}^{(h_{2})}(q_{2}z) ]^{n_{2}}} \end{gathered}$$
(5)

are irreducible rational functions in $$w_{1}^{(h_{1})}(q_{1}z)$$, $$w_{2}^{(h_{2})}(q_{2}z)$$, respectively, and the meromorphic coefficients $$u_{m_{t}}^{t}(z)$$ ($$m_{t}=0,\ldots,p_{t}$$), $$v_{n_{t}}^{t}(z)$$ ($$n_{t}=0,\ldots,s_{t}$$) are of growth $$S(r,w_{t})$$, $$t=1,2$$, and $$u_{p_{t}}^{t}(z)v_{s_{t}}^{t}(z)\not\equiv0$$, $$t=1,2$$.

For the question above, we study the growth of solutions to the system of q-difference differential equations (4). Further, set

\begin{aligned} \tau_{t}=\max\{p_{t},s_{t}\} \quad \text{and} \quad \sigma_{t}=\max\{k_{t},l_{t}\}, \quad t=1,2. \end{aligned}

Clearly, $$\tau_{t}\geq1$$ and $$\sigma_{t}\geq1$$. Also set

\begin{aligned} \tau=\tau_{1}\tau_{2}, \qquad \sigma=\sigma_{1} \sigma_{2}, \qquad q=q_{1}q_{2}, \end{aligned}

and

\begin{aligned} \kappa_{1}=\tau(h_{1}+1), \qquad \kappa_{2}= \tau(h_{2}+1), \qquad \kappa=\tau (h_{1}+1) (h_{2}+1). \end{aligned}

Now, we state the first result in this paper.

### Theorem 1.1

Let$$(w_{1},w_{2})$$be a pair of transcendental solutions of system (4). Then one of the following cases holds.

1. (i)

For$$|q_{1}|>1$$, $$|q_{2}|>1$$, if$$w_{1}$$, $$w_{2}$$are meromorphic and$$\sigma>\kappa$$, then$$\mu(w_{t})\geq\frac{\log\sigma-\log\kappa}{\log|q|}$$, $$t=1,2$$; if$$w_{t}$$is meromorphic and$$\sigma>\kappa_{t}$$, $$t=1$$or$$t=2$$, and the other is entire, then$$\mu(w_{t})\geq\frac{\log\sigma-\log\kappa_{t}}{\log|q|}$$, $$t=1,2$$; if$$w_{1}$$, $$w_{2}$$are entire and$$\sigma>\tau$$, then$$\mu(w_{t})\geq\frac{\log\sigma-\log\tau}{\log|q|}$$, $$t=1,2$$.

2. (ii)

For$$|q_{1}|<1$$, $$|q_{2}|<1$$, if$$w_{1}$$, $$w_{2}$$are meromorphic and$$\sigma\leq\kappa$$, then$$\rho(w_{t})\leq\frac{\log\sigma-\log\kappa}{\log|q|}$$, $$t=1,2$$; if$$w_{t}$$is meromorphic and$$\sigma\leq\kappa_{t}$$, $$t=1$$or$$t=2$$, and the other is entire, then$$\rho(w_{t})\leq\frac{\log\sigma-\log\kappa_{t}}{\log|q|}$$, $$t=1,2$$; if$$w_{1}$$, $$w_{2}$$are entire and$$\sigma\leq\tau$$, then$$\rho(w_{t})\leq\frac{\log\sigma-\log\tau}{\log|q|}$$, $$t=1,2$$.

3. (iii)

For$$|q_{1}|=|q_{2}|=1$$, if$$w_{1}$$, $$w_{2}$$are meromorphic, then$$\sigma\leq\kappa$$, furthermore, if$$\kappa_{t}<\sigma\leq\kappa$$, $$t=1,2$$, then$$\overline{\lambda} (\frac{1}{w_{t}} )=\lambda (\frac {1}{w_{t}} )=\rho(w_{t})$$, $$t=1,2$$; if$$w_{t}$$is meromorphic, $$t=1$$or$$t=2$$, and the other is entire, then$$\sigma\leq\kappa_{t}$$, $$t=1$$or$$t=2$$, furthermore, if$$\tau<\sigma\leq\kappa_{t}$$, $$t=1$$or$$t=2$$, then$$\overline{\lambda} (\frac{1}{w_{t}} )=\lambda (\frac {1}{w_{t}} )=\rho(w_{t})$$, $$t=1$$or$$t=2$$; if$$w_{1}$$, $$w_{2}$$are entire, then$$\sigma\leq\tau$$.

In the past few decades, meromorphic solutions of complex functional equations were studied by Bergweiler et al. [3, 4], Heittokangas et al. , and Rieppo . Silvennoinen  investigated the existence and growth of solutions to an equation of the form $$w(g(z))=R(z,w)$$ and proved the following result.

### Theorem C

()

Let

$$w \bigl(g(z) \bigr)=R(z,w),$$
(6)

where the right-hand side$$R(z,w)$$is defined as in Theorem A, $$a_{i}(z)$$, $$b_{j}(z)$$are of growth$$S(r,w)$$, and$$g(z)$$is entire. If equation (6) has a non-constant meromorphic solutionw, then$$g(z)$$is a polynomial.

Gao  considered the system of functional equations

$$\textstyle\begin{cases} w_{1} (g(z) )=R_{2} (z,w_{2}(z) ), \\ w_{2} (g(z) )=R_{1} (z,w_{1}(z) ), \end{cases}$$
(7)

where $$g(z)$$ is an entire function, $$R_{1}(z,w_{1}(z))$$, $$R_{2}(z,w_{2}(z))$$ are defined as in (3), and obtained the following result.

### Theorem D

()

If system (7) has a pair of non-constant meromorphic solutions$$(w_{1},w_{2})$$, then$$g(z)$$is a polynomial.

There are some results about the existence and growth of meromorphic solutions of several systems of complex functional equations [8, 19, 20]. Xu et al.  studied the problem when $$R_{1}(z,w_{1}(z))$$, $$R_{2}(z,w_{2}(z))$$ in (2) are replaced by $$R_{1}(z,w_{1}(g_{1}(z)))$$, $$R_{2}(z,w_{2}(g_{2}(z)))$$, respectively, and (2) is turned into the following system:

$$\textstyle\begin{cases} [w_{1}'(q_{1}z) ]^{n_{1}}=R_{2} (z,w_{2} (g_{2}(z) ) ), \\ [w_{2}'(q_{2}z) ]^{n_{2}}=R_{1} (z,w_{1} (g_{1}(z) ) ), \end{cases}$$
(8)

where $$g_{1}(z)$$, $$g_{2}(z)$$ are polynomials, and obtained the estimates of the growth order of meromorphic solutions of system (8).

A similar question to ask is what happens if system (8) is more general, for example,

$$\textstyle\begin{cases}\varOmega_{1} (z,w_{1}^{(h_{1})}(q_{1}z) )=R_{2} (z,w_{2} (g_{2}(z) ) ), \\ \varOmega_{2} (z,w_{2}^{(h_{2})}(q_{2}z) )=R_{1} (z,w_{1} (g_{1}(z) ) ), \end{cases}$$
(9)

where $$R_{1}(z,w_{1}(z))$$, $$R_{2}(z,w_{2}(z))$$, $$\varOmega_{1} (z,w_{1}^{(h_{1})}(q_{1}z) )$$, and $$\varOmega_{2} (z,w_{2}^{(h_{2})}(q_{2}z) )$$ are defined as in (3), (5), respectively. Further, set

\begin{aligned} g_{1}(z)=\alpha_{\gamma_{1}}z^{\gamma_{1}}+\alpha_{\gamma_{1}-1}z^{\gamma _{1}-1}+ \cdots+\alpha_{0} \end{aligned}

and

\begin{aligned} g_{2}(z)=\beta_{\gamma_{2}}z^{\gamma_{2}}+\beta_{\gamma_{2}-1}z^{\gamma _{2}-1}+ \cdots+\beta_{0} \end{aligned}

be two polynomials, where $$\alpha_{\gamma_{1}},\alpha_{\gamma_{1}-1},\ldots ,\alpha_{0}$$, $$\beta_{\gamma_{2}},\beta_{\gamma_{2}-1},\ldots,\beta_{0}$$ are complex constants, and $$\gamma_{t}\geq2$$ ($$t=1,2$$) are two positive integers.

The second result in this paper concerns the growth of solutions to the system of functional equations (9).

### Theorem 1.2

Let$$(w_{1},w_{2})$$be a pair of transcendental solutions of system (9). Then one of the following cases holds.

1. (i)

If$$w_{1}$$, $$w_{2}$$are meromorphic and$$\sigma\leq\kappa$$, then

\begin{aligned} T \bigl(r,w_{t}(z) \bigr)=O \bigl((\log r)^{\alpha_{1}} \bigr), \quad t=1,2, \end{aligned}

where$$\alpha_{1}=\frac{\log\kappa-\log\sigma}{\log(\gamma_{1}\gamma_{2})}$$.

2. (ii)

If$$w_{t}$$is meromorphic and$$\sigma\leq\kappa_{t}$$, $$t=1$$or$$t=2$$, and the other is entire, then

\begin{aligned} T \bigl(r,w_{t}(z) \bigr)=O \bigl((\log r)^{\alpha_{2}} \bigr), \quad t=1 \textit{ or } t=2, \end{aligned}

where$$\alpha_{2}=\frac{\log\kappa_{t}-\log\sigma}{\log(\gamma_{1}\gamma _{2})}$$, $$t=1$$or$$t=2$$.

3. (iii)

If$$w_{1}$$, $$w_{2}$$are entire and$$\sigma\leq\tau$$, then

\begin{aligned} T \bigl(r,w_{t}(z) \bigr)=O \bigl((\log r)^{\alpha_{3}} \bigr), \quad t=1,2, \end{aligned}

where$$\alpha_{3}=\frac{\log\tau-\log\sigma}{\log(\gamma_{1}\gamma_{2})}$$.

## Examples

In this section, we give examples to illustrate that the cases can occur in Theorem 1.1.

The following Examples 2.12.4 are about case (i) of Theorem 1.1.

### Example 2.1

Let $$q_{1}=2$$, $$q_{2}=3$$. Then $$(w_{1},w_{2})= (\frac{e^{z}}{z},\frac {e^{z}}{z^{2}} )$$ satisfies the system

$$\textstyle\begin{cases} \frac{[w_{1}'(2z)]^{2}+w_{1}'(2z)+1}{[w_{1}'(2z)]^{3}+w_{1}'(2z)+2} =\frac {2z^{4}(2z-1)^{2}w_{2}(z)^{4}+4z^{2}(2z-1)w_{2}(z)^{2}+8}{z^{6}(2z-1)^{3}w_{2}(z)^{6}+4z^{2}(2z-1)w_{2}(z)^{2}+16}, \\ \frac{[w_{2}'(3z)]^{2}+w_{2}'(3z)+2}{w_{2}'(3z)+1} =\frac{(3z-2)^{2}w_{1}(z)^{6}+9(3z-2)w_{1}(z)^{3}+162}{9(3z-2)w_{1}(z)^{3}+81}, \end{cases}$$

where $$h_{1}=h_{2}=1$$, $$\tau_{1}=3$$, $$\tau_{2}=2$$, and $$\sigma_{1}=\sigma_{2}=6$$. Here $$\sigma=36>\kappa=24$$ and $$\mu(w_{t})=1\geq\frac{\log36-\log24}{\log6}=\frac{\log36-\log24}{\log 36-\log6}$$, $$t=1,2$$.

### Example 2.2

Let $$q_{1}=2$$, $$q_{2}=3$$. Then $$(w_{1},w_{2})= (e^{z},\frac{e^{z}}{z} )$$ satisfies the system

$$\textstyle\begin{cases} \frac{z[w_{1}''(2z)]^{3}+2w_{1}''(2z)+z^{2}}{z^{2}[w_{1}''(2z)]^{2}+w_{1}''(2z)+z} =\frac{64z^{6}w_{2}(z)^{6}+8zw_{2}(z)^{2}+z}{16z^{5}w_{2}(z)^{4}+4zw_{2}(z)^{2}+1}, \\ \frac{9z^{4}[w_{2}'(3z)]^{2}+3z^{2}w_{2}'(3z)}{w_{2}'(3z)+2} =\frac{3z^{2}(3z-1)^{2}w_{1}(z)^{6}+3z^{2}(3z-1)w_{1}(z)^{3}}{(3z-1)w_{1}(z)^{3}+6z^{2}}, \end{cases}$$

where $$h_{1}=2$$, $$h_{2}=1$$, $$\tau_{1}=3$$, $$\tau_{2}=2$$, and $$\sigma_{1}=\sigma_{2}=6$$. Then we have $$\sigma=36>\kappa_{2}=12$$ and $$\mu(w_{t})=1\geq\frac{\log36-\log12}{\log6}=\frac{\log3}{\log6}$$, $$t=1,2$$.

### Example 2.3

Let $$q_{1}=3$$, $$q_{2}=2$$. Then $$(w_{1},w_{2})= (\frac{e^{z}}{z^{2}},e^{z} )$$ satisfies the system

$$\textstyle\begin{cases} \frac{[w_{1}'(3z)]^{3}+z[w_{1}'(3z)]^{2}}{[w_{1}'(3z)]^{2}+w_{1}'(3z)} =\frac {(3z-2)^{2}w_{2}(z)^{6}+9z^{4}(3z-2)w_{2}(z)^{3}}{9z^{3}(3z-2)w_{2}(z)^{3}+81z^{6}}, \\ \frac{(z+1)w_{2}''(2z)+1}{[w_{2}''(2z)]^{2}+w_{2}''(2z)+z} =\frac{4z^{4}(z+1)w_{1}(z)^{2}+1}{16z^{8}w_{1}(z)^{4}+4z^{4}w_{1}(z)^{2}+z}, \end{cases}$$

where $$h_{1}=1$$, $$h_{2}=2$$, $$\tau_{1}=3$$, $$\tau_{2}=2$$, $$\sigma_{1}=4$$, and $$\sigma_{2}=6$$. It is known that $$\sigma=24>\kappa_{1}=12$$ and $$\mu(w_{t})=1\geq\frac{\log24-\log12}{\log6}=\frac{\log2}{\log6}$$, $$t=1,2$$.

### Example 2.4

Let $$q_{1}=2$$, $$q_{2}=3$$. Then $$(w_{1},w_{2})=(e^{z},ze^{z})$$ satisfies the system

$$\textstyle\begin{cases} \frac{w_{1}''(2z)+z}{[w_{1}''(2z)]^{3}+z^{2}w_{1}''(2z)+1} =\frac{4z^{4}w_{2}(z)^{2}+z^{7}}{64w_{2}(z)^{6}+4z^{6}w_{2}(z)^{2}+z^{6}}, \\ \frac{[w_{2}''(3z)]^{2}+zw_{2}''(3z)+1}{w_{2}''(3z)+z} =\frac{81(3z+2)^{2}w_{1}(z)^{6}+9z(3z+2)w_{1}(z)^{3}+1}{9(3z+2)w_{1}(z)^{3}+z}, \end{cases}$$

where $$h_{1}=h_{2}=2$$, $$\tau_{1}=3$$, $$\tau_{2}=2$$, and $$\sigma_{1}=\sigma_{2}=6$$. Thus, $$\sigma=36>\tau=6$$ and $$\mu(w_{t})=1\geq\frac{\log36-\log6}{\log6}=1$$, $$t=1,2$$.

The following Examples 2.52.8 are about case (ii) of Theorem 1.1.

### Example 2.5

Let $$q_{1}=\frac{1}{2}$$, $$q_{2}=\frac{1}{3}$$. Then $$(w_{1},w_{2})= (\frac {e^{z}}{z},\frac{e^{z}}{z-1} )$$ satisfies the system

$$\textstyle\begin{cases} \frac{[w_{1}'(\frac{1}{2}z)]^{2}+1}{[w_{1}'(\frac{1}{2}z)]^{4}+1} =\frac{z^{4}(z-1)(z-2)^{2}w_{2}(z)+z^{8}}{(z-1)^{2}(z-2)^{4}w_{2}(z)^{2}+z^{8}}, \\ \frac{[w_{2}'(\frac{1}{3}z)]^{3}+1}{[w_{2}'(\frac{1}{3}z)]^{6}+1} =\frac{z(z-6)^{3}(z-3)^{6}w_{1}(z)+(z-3)^{12}}{z^{2}(z-6)^{6}w_{1}(z)^{2}+(z-3)^{12}}, \end{cases}$$

where $$h_{1}=h_{2}=1$$, $$\tau_{1}=4$$, $$\tau_{2}=6$$, and $$\sigma_{1}=\sigma_{2}=2$$. Clearly, $$\sigma=4\leq\kappa=96$$ and $$\rho(w_{t})=1\leq\frac{\log4-\log96}{\log\frac{1}{6}}=\frac{\log24}{\log 6}$$, $$t=1,2$$.

### Example 2.6

Let $$q_{1}=\frac{1}{3}$$, $$q_{2}=\frac{1}{2}$$. Then $$(w_{1},w_{2})= (e^{z},\frac {e^{z}}{z-1} )$$ satisfies the system

$$\textstyle\begin{cases} \frac{[w_{1}'(\frac{1}{3}z)]^{6}+1}{[w_{1}'(\frac{1}{3}z)]^{3}+1} =\frac{(z-1)^{2}w_{2}(z)^{2}+729}{27(z-1)^{2}w_{2}(z)^{2}+729}, \\ \frac{[w_{2}'(\frac{1}{2}z)]^{4}+1}{[w_{2}'(\frac{1}{2}z)]^{2}+1} =\frac{(z-4)^{4}w_{1}(z)^{2}+(z-2)^{8}}{(z-4)^{2}(z-2)^{4}w_{1}(z)+(z-2)^{8}}, \end{cases}$$

where $$h_{1}=h_{2}=1$$, $$\tau_{1}=6$$, $$\tau_{2}=4$$, and $$\sigma_{1}=\sigma_{2}=2$$. Then we have $$\sigma=4\leq\kappa_{2}=48$$ and $$\rho(w_{t})=1\leq\frac{\log4-\log48}{\log\frac{1}{6}}=\frac{\log12}{\log 6}$$, $$t=1,2$$.

### Example 2.7

Let $$q_{1}=\frac{1}{2}$$, $$q_{2}=\frac{1}{4}$$. Then $$(w_{1},w_{2})= (\frac {ze^{z}}{z-1},ze^{z} )$$ satisfies the system

$$\textstyle\begin{cases} [w_{1}'' (\frac{z}{2} ) ]^{2} =\frac{(z^{3}-4z^{2}-4z+32)^{2}w_{2}(z)}{16z(z-2)^{6}}, \\ {}[w_{2}' (\frac{z}{4} ) ]^{8} =\frac{(z-1)^{2}(z+4)^{8}w_{1}(z)^{2}}{16^{8}z^{2}}, \end{cases}$$

where $$h_{1}=2$$, $$h_{2}=1$$, $$\tau_{1}=2$$, $$\tau_{2}=8$$, $$\sigma_{1}=2$$, and $$\sigma_{2}=1$$. It is known that $$\sigma=2\leq\kappa_{1}=48$$ and $$\rho(w_{t})=1\leq\frac{\log2-\log48}{\log\frac{1}{8}}=\frac{\log24}{\log 8}$$, $$t=1,2$$.

### Example 2.8

Let $$q_{1}=\frac{1}{2}$$, $$q_{2}=\frac{1}{4}$$. Then $$(w_{1},w_{2})=(ze^{z},e^{2z})$$ satisfies the system

$$\textstyle\begin{cases} 4096 [w_{1}'' (\frac{z}{2} ) ]^{4} =(z+4)^{4}w_{2}(z), \\ 4z [w_{2}' (\frac{z}{4} ) ]^{2} =w_{1}(z), \end{cases}$$

where $$h_{1}=2$$, $$h_{2}=1$$, $$\tau_{1}=4$$, $$\tau_{2}=2$$, and $$\sigma_{1}=\sigma_{2}=1$$. Here $$\sigma=1\leq\tau=8$$ and $$\rho(w_{t})=1\leq\frac{\log1-\log8}{\log\frac{1}{8}}=1$$, $$t=1,2$$.

The following Examples 2.92.12 are about case (iii) of Theorem 1.1.

### Example 2.9

Let $$q_{1}=q_{2}=1$$. Then $$(w_{1},w_{2})= (\frac{e^{z}}{e^{z}-1},\frac {ze^{z}}{e^{z}-1} )$$ satisfies the system

$$\textstyle\begin{cases} z^{2} [w_{1}'(z) ]^{2} =w_{2}(z)^{2}, \\ w_{2}'(z) =-zw_{1}(z)^{2}+(1+z)w_{1}(z), \end{cases}$$

where $$h_{1}=h_{2}=1$$, $$\tau_{1}=2$$, $$\tau_{2}=1$$, and $$\sigma_{1}=\sigma_{2}=2$$. Clearly, $$\sigma=4\leq\kappa=8$$.

### Example 2.10

Let $$q_{1}=1$$, $$q_{2}=-1$$. Then $$(w_{1},w_{2})= (\frac{1}{e^{z}-1},\frac {1}{1-e^{z}} )$$ satisfies the system

$$\textstyle\begin{cases} w_{1}''(z) =-2w_{2}(z)^{3}+3w_{2}(z)^{2}-w_{2}(z), \\ w_{2}'(-z) =-w_{1}(z)^{2}-w_{1}(z), \end{cases}$$

where $$h_{1}=2$$, $$h_{2}=1$$, $$\tau_{1}=\tau_{2}=1$$, $$\sigma_{1}=2$$, and $$\sigma_{2}=3$$. Then we have $$\kappa_{t}<\sigma=6\leq\kappa=6$$, $$t=1,2$$, and $$\overline{\lambda} (\frac{1}{w_{t}} )=\lambda (\frac {1}{w_{t}} )=\rho(w_{t})=1$$, $$t=1,2$$.

### Example 2.11

Let $$q_{1}=q_{2}=1$$. Then $$(w_{1},w_{2})= (e^{z},\frac{1}{e^{z}-1} )$$ satisfies the system

$$\textstyle\begin{cases} w_{1}''(z)^{2} =\frac{[w_{2}(z)+1]^{2}}{w_{2}(z)^{2}}, \\ w_{2}'(z)^{4} =\frac{w_{1}(z)^{4}}{[w_{1}(z)-1]^{8}}, \end{cases}$$

where $$h_{1}=2$$, $$h_{2}=1$$, $$\tau_{1}=2$$, $$\tau_{2}=4$$, $$\sigma_{1}=8$$, and $$\sigma_{2}=2$$. It is known that $$\tau=8<\sigma=16\leq\kappa_{2}=16$$ and $$\overline{\lambda} (\frac{1}{w_{2}} )=\lambda (\frac {1}{w_{2}} )=\rho(w_{2})=1$$.

### Example 2.12

Let $$q_{1}=q_{2}=1$$. Then $$(w_{1},w_{2})=(e^{z},ze^{2z})$$ satisfies the system

$$\textstyle\begin{cases} \frac{[w_{1}''(z)]^{2}+1}{[w_{1}''(z)]^{4}+1} =\frac{zw_{2}(z)+z^{2}}{w_{2}(z)+z^{2}}, \\ \frac{[w_{2}'(z)]^{2}+1}{w_{2}'(z)+1} =\frac{(2z+1)^{2}w_{1}(z)^{4}+1}{(2z+1)w_{1}(z)^{2}+1}, \end{cases}$$

where $$h_{1}=2$$, $$h_{2}=1$$, $$\tau_{1}=4$$, $$\tau_{2}=2$$, $$\sigma_{1}=4$$, and $$\sigma_{2}=1$$. Then we have $$\sigma=4\leq\tau=8$$.

## Lemmas

To prove Theorems 1.1 and 1.2, we need the following lemmas. Yang and Yi  showed the value distribution of a meromorphic function and its derivative.

### Lemma 3.1

()

\begin{aligned} N \bigl(r,f^{(k)} \bigr)=N(r,f)+k\overline{N}(r,f), \qquad T \bigl(r,f^{(k)} \bigr)\leq T(r,f)+k\overline{N}(r,f)+S(r,f). \end{aligned}

The following lemma is to compare the Nevanlinna functions of $$f(z)$$ and $$f(cz)$$.

### Lemma 3.2

()

\begin{aligned} \overline{N} \bigl(r,f(cz) \bigr)=\overline{N} \bigl( \vert c \vert r,f(z) \bigr)+O(1), \qquad T \bigl(r,f(cz) \bigr)=T \bigl( \vert c \vert r,f(z) \bigr)+O(1). \end{aligned}

In 1972, Bank  established the following lemma.

### Lemma 3.3

()

Let$$g(r)$$and$$h(r)$$be monotone non-decreasing functions on$$(0,+\infty)$$such that$$g(r)\leq h(r)$$, possibly outside a set ofrwith finite logarithmic measure. Then, for any real number$$a>1$$, there exists$$r_{0}>0$$such that$$g(r)\leq h(ar)$$for all$$r>r_{0}$$.

Gundersen et al.  showed a method to obtain an upper bound for the growth order of a meromorphic function.

### Lemma 3.4

()

Let$$f(z)$$be a non-constant meromorphic function, and let$$\varPsi: (1,\infty)\rightarrow(0,\infty)$$be a monotone non-decreasing function. If for some real number$$a\in(0,1)$$, there exist real numbers$$K_{1}>0$$and$$K_{2}\geq1$$such that

\begin{aligned} T(r,f)\leq K_{1}\varPsi(ar)+K_{2}T(ar,f)+S(ar,f), \end{aligned}

then

\begin{aligned} \rho(f)\leq\frac{\log K_{2}}{-\log a}+\limsup_{r\rightarrow\infty}\frac {\log\varPsi(r)}{\log{r}}. \end{aligned}

The following lemma gives us a method to have a lower bound for the lower order of a meromorphic function.

### Lemma 3.5

()

Let$$\varPsi: (r_{0},\infty)\rightarrow(1,\infty)$$be a monotone non-decreasing function, where$$r_{0}\geq1$$. If for some real number$$a>1$$, there exists a real number$$b>1$$such that$$\varPsi(ar)\geq b\varPsi(r)$$, then

\begin{aligned} \liminf_{r\rightarrow\infty}\frac{\log\varPsi(r)}{\log{r}}\geq\frac {\log b}{\log a}. \end{aligned}

The following result about estimate of the Nevanlinna characteristic function of a meromorphic function composed with polynomials is given by Goldstein.

### Lemma 3.6

()

Let$$f(z)$$be a transcendental meromorphic function and$$g(z)=a_{m}z^{m}+a_{m-1}z^{m-1}+\cdots+a_{0}$$be a polynomial with degreem (≥1). For given$$\delta\in(0,|a_{m}|)$$, let$$\lambda=|a_{m}|+\delta$$, $$\mu =|a_{m}|-\delta$$, then

\begin{aligned} (1-\varepsilon)T \bigl(\mu r^{m},f \bigr)\leq T(r,f\circ g)\leq(1+ \varepsilon )T \bigl(\lambda r^{m},f \bigr) \end{aligned}

for any given$$\varepsilon>0$$and sufficiently larger.

Goldstein  showed the following lemma.

### Lemma 3.7

()

Let$$\phi(r)$$be a positive function defined on$$[r_{0},\infty)$$and bounded in every finite interval. Assume that$$\phi(\mu r^{k})\leq a\phi(r)+b$$ ($$r\geq r_{0}$$), whereμ (>0), k (>1), a (≥1), andbare constants. Then$$\phi(r)=O((\log r)^{\alpha})$$with$$\alpha =\frac{\log a}{\log k}$$, unless$$a=1$$and$$b>0$$; and if$$a=1$$and$$b>0$$, then for any$$\varepsilon>0$$, $$\phi(r)=O((\log r)^{\varepsilon})$$.

## Proofs of the results

### Proof of Theorem 1.1

Suppose first that $$(w_{1},w_{2})$$ is a pair of transcendental solutions of system (4). In the following, we consider three cases.

Case (i): $$|q_{1}|>1$$ and $$|q_{2}|>1$$. Suppose that both $$w_{1}$$ and $$w_{2}$$ are meromorphic. It follows from Valiron–Mohon’ko theorem [15, Theorem 2.2.5], Lemma 3.1, and Lemma 3.2 that

\begin{aligned} T \bigl(r,R_{2}(z,w_{2}) \bigr) & = \sigma_{2}T(r,w_{2})+S(r,w_{2}) \\ & = T \bigl(r,\varOmega_{1} \bigl(z,w_{1}^{(h_{1})}(q_{1}z) \bigr) \bigr) \\ & = \tau_{1}T \bigl(r,w_{1}^{(h_{1})}(q_{1}z) \bigr)+S \bigl(r,w_{1}^{(h_{1})}(q_{1}z) \bigr) \\ & \leq\tau_{1} \bigl[T \bigl(r,w_{1}(q_{1}z) \bigr)+h_{1}\overline {N} \bigl(r,w_{1}(q_{1}z) \bigr)+S \bigl(r,w_{1}(q_{1}z) \bigr) \bigr] \\ &\quad + S \bigl(r,w_{1}^{(h_{1})}(q_{1}z) \bigr) \\ & \leq\tau_{1}(h_{1}+1)T \bigl( \vert q_{1} \vert r,w_{1} \bigr)+S \bigl( \vert q_{1} \vert r,w_{1} \bigr), \end{aligned}

that is,

\begin{aligned} \sigma_{2}T(r,w_{2})+S(r,w_{2})\leq \tau_{1}(h_{1}+1)T \bigl( \vert q_{1} \vert r,w_{1} \bigr)+S \bigl( \vert q_{1} \vert r,w_{1} \bigr). \end{aligned}
(10)

Similarly, we have

\begin{aligned} \sigma_{1}T(r,w_{1})+S(r,w_{1})\leq \tau_{2}(h_{2}+1)T \bigl( \vert q_{2} \vert r,w_{2} \bigr)+S \bigl( \vert q_{2} \vert r,w_{2} \bigr). \end{aligned}
(11)

Thus, from (10) and (11), we obtain

\begin{aligned} \sigma T(r,w_{t})+S(r,w_{t})\leq\kappa T \bigl( \vert q \vert r,w_{t} \bigr)+S \bigl( \vert q \vert r,w_{t} \bigr),\quad t=1,2. \end{aligned}
(12)

Now $$\sigma>\kappa$$, and for any given $$\varepsilon>0$$,

\begin{aligned} \sigma(1-\varepsilon)T(r,w_{t})\leq\kappa(1+ \varepsilon)T \bigl( \vert q \vert r,w_{t} \bigr),\quad t=1,2, \end{aligned}
(13)

for sufficiently large r, possibly outside a set of r with finite logarithmic measure. By Lemma 3.3, with $$a>1$$ and (13) we have

\begin{aligned} \sigma(1-\varepsilon)T(r,w_{t})\leq\kappa(1+ \varepsilon)T \bigl(a \vert q \vert r,w_{t} \bigr),\quad t=1,2, \end{aligned}
(14)

for all $$r\geq r_{0}$$. It follows from Lemma 3.5 and (14) that

\begin{aligned} \mu(w_{t})\geq\frac{\log[\sigma(1-\varepsilon)]-\log[\kappa(1+\varepsilon )]}{\log(a \vert q \vert )},\quad t=1,2. \end{aligned}

As $$\varepsilon\rightarrow0^{+}$$ and $$a\rightarrow1^{+}$$, we get

\begin{aligned} \mu(w_{t})\geq\frac{\log\sigma-\log\kappa}{\log \vert q \vert },\quad t=1,2. \end{aligned}

Suppose that only one between $$w_{1}$$ and $$w_{2}$$ is meromorphic, without loss of generality, we assume that $$w_{1}$$ is meromorphic and $$w_{2}$$ is entire. Then, similar to (11), we have

\begin{aligned} \sigma_{1}T(r,w_{1})+S(r,w_{1})\leq \tau_{2}T \bigl( \vert q_{2} \vert r,w_{2} \bigr)+S \bigl( \vert q_{2} \vert r,w_{2} \bigr). \end{aligned}
(15)

Thus, it follows from (10) and (15) that

\begin{aligned} \sigma T(r,w_{t})+S(r,w_{t})\leq \kappa_{1}T \bigl( \vert q \vert r,w_{t} \bigr)+S \bigl( \vert q \vert r,w_{t} \bigr),\quad t=1,2. \end{aligned}
(16)

Similar to the above argument, since $$\sigma>\kappa_{1}$$ and for any small $$\varepsilon>0$$, we know that there exists $$a>1$$ such that

\begin{aligned} \sigma(1-\varepsilon)T(r,w_{t})\leq \kappa_{1}(1+ \varepsilon)T \bigl(a \vert q \vert r,w_{t} \bigr),\quad t=1,2, \end{aligned}
(17)

for all $$r\geq r_{0}$$. Applying Lemma 3.5 to (17) yields that

\begin{aligned} \mu(w_{t})\geq\frac{\log[\sigma(1-\varepsilon)]-\log[\kappa _{1}(1+\varepsilon)]}{\log(a \vert q \vert )},\quad t=1,2. \end{aligned}

By letting $$\varepsilon\rightarrow0^{+}$$ and $$a\rightarrow1^{+}$$, we obtain

\begin{aligned} \mu(w_{t})\geq\frac{\log\sigma-\log\kappa_{1}}{\log \vert q \vert },\quad t=1,2. \end{aligned}

Suppose that both $$w_{1}$$ and $$w_{2}$$ are entire. Then, similar to (10), we have

\begin{aligned} \sigma_{2}T(r,w_{2})+S(r,w_{2})\leq \tau_{1}T \bigl( \vert q_{1} \vert r,w_{1} \bigr)+S \bigl( \vert q_{1} \vert r,w_{1} \bigr). \end{aligned}
(18)

Thus, it follows from (15) and (18) that

\begin{aligned} \sigma T(r,w_{t})+S(r,w_{t})\leq\tau T \bigl( \vert q \vert r,w_{t} \bigr)+S \bigl( \vert q \vert r,w_{t} \bigr),\quad t=1,2. \end{aligned}

Now, $$\sigma>\tau$$, we know that for $$\varepsilon>0$$ there exists $$a>1$$ such that

\begin{aligned} \sigma(1-\varepsilon)T(r,w_{t})\leq\tau(1+\varepsilon)T \bigl(a \vert q \vert r,w_{t} \bigr),\quad t=1,2, \end{aligned}
(19)

for all $$r\geq r_{0}$$. Recalling Lemma 3.5 and letting $$\varepsilon\rightarrow0^{+}$$ and $$a\rightarrow1^{+}$$, we conclude that

\begin{aligned} \mu(w_{t})\geq\frac{\log\sigma-\log\tau}{\log \vert q \vert },\quad t=1,2. \end{aligned}

Case (ii): $$|q_{1}|<1$$ and $$|q_{2}|<1$$. Suppose that both $$w_{1}$$ and $$w_{2}$$ are meromorphic. Then, similar to the previous argument, we have that for $$\varepsilon>0$$ there exists $$a>1$$ such that $$a|q|<1$$, (12) and (14) hold for all $$r\geq r_{0}$$. Since $$\sigma\leq\kappa$$, then $$\frac{\kappa(1+\varepsilon)}{\sigma(1-\varepsilon)}>1$$. Hence, applying Lemma 3.4 to (14) yields that

\begin{aligned} \rho(w_{t})\leq\frac{\log[\kappa(1+\varepsilon)]-\log[\sigma(1-\varepsilon )]}{-\log(a \vert q \vert )},\quad t=1,2, \end{aligned}

which implies

\begin{aligned} \rho(w_{t})\leq\frac{\log\sigma-\log\kappa}{\log \vert q \vert },\quad t=1,2, \end{aligned}

as $$\varepsilon\rightarrow0^{+}$$ and $$a\rightarrow1^{+}$$.

Suppose that only one between $$w_{1}$$ and $$w_{2}$$ is meromorphic. Without loss of generality, we assume that $$w_{1}$$ is meromorphic and $$w_{2}$$ is entire. Then we similarly obtain that, for $$\varepsilon>0$$, there exists $$a>1$$ such that $$a|q|<1$$, (16) and (17) hold for all $$r\geq r_{0}$$. Since $$\sigma\leq\kappa_{1}$$, then $$\frac{\kappa_{1}(1+\varepsilon)}{\sigma(1-\varepsilon)}>1$$. Thus, we conclude by Lemma 3.4 and (17) that

\begin{aligned} \rho(w_{t})\leq\frac{\log[\kappa_{1}(1+\varepsilon)]-\log[\sigma (1-\varepsilon)]}{-\log(a \vert q \vert )},\quad t=1,2, \end{aligned}

and let $$\varepsilon\rightarrow0^{+}$$ and $$\alpha\rightarrow1^{+}$$, it yields

\begin{aligned} \rho(w_{t})\leq\frac{\log\sigma-\log\kappa_{1}}{\log \vert q \vert },\quad t=1,2. \end{aligned}

Suppose that both $$w_{1}$$ and $$w_{2}$$ are entire. Similarly, for $$\varepsilon>0$$, there exists $$a>1$$ such that $$a|q|<1$$, (15), (18), and (19) hold for all $$r\geq r_{0}$$. Since $$\sigma\leq\tau$$, then $$\frac{\tau(1+\varepsilon)}{\sigma(1-\varepsilon)}>1$$. Therefore, recalling Lemma 3.4, we have

\begin{aligned} \rho(w_{t})\leq\frac{\log[\tau(1+\varepsilon)]-\log[\sigma(1-\varepsilon )]}{-\log(a \vert q \vert )},\quad t=1,2, \end{aligned}

which deduces

\begin{aligned} \rho(w_{t})\leq\frac{\log\sigma-\log\tau}{\log \vert q \vert },\quad t=1,2, \end{aligned}

as $$\varepsilon\rightarrow0^{+}$$ and $$a\rightarrow1^{+}$$.

Case (iii): $$|q_{1}|=|q_{2}|=1$$. Suppose that both $$w_{1}$$ and $$w_{2}$$ are meromorphic. Then, from Valiron–Mohon’ko theorem [15, Theorem 2.2.5] and Lemma 3.1, we conclude that

\begin{aligned} [b]\sigma_{2}T(r,w_{2})+S(r,w_{2}) & \leq\tau_{1} \bigl[T(r,w_{1})+h_{1} \overline{N}(r,w_{1})+S(r,w_{1}) \bigr]+S \bigl(r,w_{1}^{(h_{1})} \bigr) \\ & \leq\tau_{1}(h_{1}+1)T(r,w_{1})+S(r,w_{1}), \end{aligned}
(20)

and

\begin{aligned}[b] \sigma_{1}T(r,w_{1})+S(r,w_{1}) & \leq\tau_{2} \bigl[T(r,w_{2})+h_{2} \overline{N}(r,w_{2})+S(r,w_{2}) \bigr]+S \bigl(r,w_{2}^{(h_{2})} \bigr) \\ & \leq\tau_{2}(h_{2}+1)T(r,w_{2})+S(r,w_{2}). \end{aligned}
(21)

From (20) and (21), we have $$\sigma\leq\kappa$$. Furthermore, if $$\kappa_{t}<\sigma\leq\kappa$$, $$t=1,2$$, then

\begin{aligned} \frac{\sigma-\kappa_{2}}{h_{1}\kappa_{2}}T(r,w_{1})+S(r,w_{1}) \leq \overline{N}(r,w_{1})+S(r,w_{1}) \leq T(r,w_{1})+S(r,w_{1}) \end{aligned}

and

\begin{aligned} \frac{\sigma-\kappa_{1}}{h_{1}\kappa_{1}}T(r,w_{2})+S(r,w_{2}) \leq \overline{N}(r,w_{2})+S(r,w_{2}) \leq T(r,w_{2})+S(r,w_{2}), \end{aligned}

which imply that $$\overline{\lambda} (\frac{1}{w_{t}} )=\lambda (\frac{1}{w_{t}} )=\rho(w_{t})$$, $$t=1,2$$.

Suppose that only one between $$w_{1}$$ and $$w_{2}$$ is meromorphic. Without loss of generality, we assume that $$w_{1}$$ is meromorphic and $$w_{2}$$ is entire. Then we get (20) and

\begin{aligned} \sigma_{1}T(r,w_{1})+S(r,w_{1})\leq \tau_{2}T(r,w_{2})+S(r,w_{2}). \end{aligned}
(22)

Hence, it follows from (20) and (22) that $$\sigma\leq \kappa_{1}$$. Furthermore, if $$\tau<\sigma\leq\kappa_{1}$$, it yields

\begin{aligned} \frac{\sigma-\tau}{\tau h_{1}}T(r,w_{1})+S(r,w_{1})\leq\overline {N}(r,w_{1})+S(r,w_{1})\leq T(r,w_{1})+S(r,w_{1}), \end{aligned}

which implies $$\overline{\lambda} (\frac{1}{w_{1}} )=\lambda (\frac{1}{w_{1}} )=\rho(w_{1})$$. Similarly, if $$w_{2}$$ is meromorphic and $$w_{1}$$ is entire, we obtain that $$\overline{\lambda} (\frac{1}{w_{2}} )=\lambda (\frac {1}{w_{2}} )=\rho(w_{2})$$ when $$\tau<\sigma\leq\kappa_{2}$$.

Suppose that both $$w_{1}$$ and $$w_{2}$$ are entire. Then, similar to the above argument, we can get (22) and

\begin{aligned} \sigma_{2}T(r,w_{2})+S(r,w_{2})\leq \tau_{1}T(r,w_{1})+S(r,w_{1}). \end{aligned}
(23)

Thus, it follows from (22) and (23) that $$\sigma\leq\tau$$.

From Cases (i)–(iii), the proof of Theorem 1.1 is completed. □

### Proof of Theorem 1.2

Suppose first that $$(w_{1},w_{2})$$ is a pair of transcendental solutions of system (9). In what follows, we consider three cases.

Case (i): Suppose that both $$w_{1}$$ and $$w_{2}$$ are meromorphic. Then, by Valiron–Mohon’ko theorem [15, Theorem 2.2.5], Lemma 3.1, and Lemma 3.2, we get

$$\sigma_{1}T \bigl(r,w_{1} \bigl(g_{1}(z) \bigr) \bigr)+S \bigl(r,w_{1} \bigl(g_{1}(z) \bigr) \bigr)\leq\tau _{2}(h_{2}+1)T \bigl( \vert q_{2} \vert r,w_{2} \bigr)+S \bigl( \vert q_{2} \vert r,w_{2} \bigr)$$
(24)

and

$$\sigma_{2}T \bigl(r,w_{2} \bigl(g_{2}(z) \bigr) \bigr)+S \bigl(r,w_{2} \bigl(g_{2}(z) \bigr) \bigr)\leq\tau _{1}(h_{1}+1)T \bigl( \vert q_{1} \vert r,w_{1} \bigr)+S \bigl( \vert q_{1} \vert r,w_{1} \bigr).$$
(25)

By Lemma 3.6, for given $$0<\delta_{1}<|\alpha_{\gamma_{1}}|$$, $$0<\delta_{2}<|\beta_{\gamma_{2}}|$$, and $$\mu_{1}=|\alpha_{\gamma_{1}}|-\delta_{1}$$, $$\mu_{2}=|\beta_{\gamma _{2}}|-\delta_{2}$$, we know that for any small $$\varepsilon>0$$ there exists two sets $$E_{1}$$, $$E_{2}$$ of finite logarithmic measure such that

$$\sigma_{1}(1-\varepsilon)T \bigl(\mu_{1}r^{\gamma_{1}},w_{1} \bigr)\leq\tau _{2}(h_{2}+1) (1+\varepsilon)T \bigl( \vert q_{2} \vert r,w_{2} \bigr),\quad r\notin E_{1},$$
(26)

and

$$\sigma_{2}(1-\varepsilon)T \bigl(\mu_{2}r^{\gamma_{2}},w_{2} \bigr)\leq\tau _{1}(h_{1}+1) (1+\varepsilon)T \bigl( \vert q_{1} \vert r,w_{1} \bigr),\quad r\notin E_{2}.$$
(27)

Thus, for sufficiently large r and $$r\notin E_{1}\cup E_{2}$$, we can deduce from (26) and (27) that

$$\sigma(1-\varepsilon)^{2}T \biggl(\frac{\mu_{1}\mu_{2}^{\gamma _{1}}}{ \vert q_{2} \vert ^{\gamma_{1}}}r^{\gamma_{1}\gamma_{2}},w_{1} \biggr) \leq\kappa(1+\varepsilon)^{2}T \bigl( \vert q_{1} \vert r,w_{1} \bigr)$$
(28)

and

$$\sigma(1-\varepsilon)^{2}T \biggl(\frac{\mu_{2}\mu_{1}^{\gamma _{2}}}{ \vert q_{1} \vert ^{\gamma_{2}}}r^{\gamma_{1}\gamma_{2}},w_{2} \biggr) \leq\kappa(1+\varepsilon)^{2}T \bigl( \vert q_{2} \vert r,w_{2} \bigr).$$
(29)

By Lemma 3.3, with $$a>1$$ and (28), we have

$$\sigma(1-\varepsilon)^{2}T \biggl(\frac{\mu_{1}\mu_{2}^{\gamma _{1}}}{ \vert q_{2} \vert ^{\gamma_{1}}}r^{\gamma_{1}\gamma_{2}},w_{1} \biggr) \leq\kappa(1+\varepsilon)^{2}T \bigl(a \vert q_{1} \vert r,w_{1} \bigr)$$
(30)

for all $$r\geq r_{0}$$. Set $$R=a|q_{1}|r$$. Then (30) can be rewritten as

$$T \biggl(\frac{\mu_{1}\mu_{2}^{\gamma_{1}}}{ \vert q_{2} \vert ^{\gamma_{1}} \vert aq_{1} \vert ^{\gamma _{1}\gamma_{2}}}R^{\gamma_{1}\gamma_{2}},w_{1} \biggr) \leq\frac{\kappa(1+\varepsilon)^{2}}{\sigma(1-\varepsilon)^{2}}T(R,w_{1}).$$
(31)

If $$\sigma\leq\kappa$$, then $$\frac{\kappa(1+\varepsilon)^{2}}{\sigma(1-\varepsilon)^{2}}\geq1$$. Since $$\frac{\mu_{1}\mu_{2}^{\gamma_{1}}}{|q_{2}|^{\gamma_{1}}|aq_{1}|^{\gamma _{1}\gamma_{2}}}>0$$, $$\gamma_{t}\geq2$$ ($$t=1,2$$), applying Lemma 3.7 to (31) yields that

\begin{aligned} T(r,w_{1})=O \bigl((\log r)^{\alpha_{1}} \bigr), \end{aligned}

where

\begin{aligned} \alpha_{1}=\frac{\log[\kappa(1+\varepsilon)^{2}]-\log[\sigma(1-\varepsilon )^{2}]}{\log(\gamma_{1}\gamma_{2})}, \end{aligned}

which deduces

\begin{aligned} \alpha_{1}=\frac{\log\kappa-\log\sigma}{\log(\gamma_{1}\gamma_{2})}, \end{aligned}

as $$\varepsilon\rightarrow0^{+}$$. Similarly, from (29), we conclude that

\begin{aligned} T(r,w_{2})=O \bigl((\log r)^{\alpha_{1}} \bigr), \end{aligned}

where

\begin{aligned} \alpha_{1}=\frac{\log\kappa-\log\sigma}{\log(\gamma_{1}\gamma_{2})}. \end{aligned}

Case (ii): Suppose that only one between $$w_{1}$$ and $$w_{2}$$ is meromorphic. Without loss of generality, we assume that $$w_{2}$$ is meromorphic and $$w_{1}$$ is entire. By Valiron–Mohon’ko theorem [15, Theorem 2.2.5], Lemma 3.1, and Lemma 3.2, we get (24) and

$$\sigma_{2}T \bigl(r,w_{2} \bigl(g_{2}(z) \bigr) \bigr)+S \bigl(r,w_{2} \bigl(g_{2}(z) \bigr) \bigr)\leq \tau_{1}T \bigl( \vert q_{1} \vert r,w_{1} \bigr)+S \bigl( \vert q_{1} \vert r,w_{1} \bigr).$$
(32)

Thus, by an argument similar to the proof of Case (i) of Theorem 1.2, we can deduce

\begin{aligned} T \biggl(\frac{\mu_{1}\mu_{2}^{\gamma_{1}}}{ \vert q_{2} \vert ^{\gamma_{1}} \vert aq_{1} \vert ^{\gamma _{1}\gamma_{2}}}R^{\gamma_{1}\gamma_{2}},w_{1} \biggr) \leq \frac{\kappa_{2}(1+\varepsilon)^{2}}{\sigma(1-\varepsilon)^{2}}T(R,w_{1}). \end{aligned}

If $$\sigma\leq\kappa_{2}$$, then $$\frac{\kappa_{2}(1+\varepsilon)^{2}}{\sigma(1-\varepsilon)^{2}}\geq1$$. Since $$\frac{\mu_{1}\mu_{2}^{\gamma_{1}}}{|q_{2}|^{\gamma_{1}}|aq_{1}|^{\gamma _{1}\gamma_{2}}}>0$$, $$\gamma_{t}\geq2$$ ($$t=1,2$$), it follows from Lemma 3.7 that

\begin{aligned} T(r,w_{1})=O \bigl((\log r)^{\alpha_{2}} \bigr), \end{aligned}

where

\begin{aligned} \alpha_{2}=\frac{\log\kappa_{2}-\log\sigma}{\log(\gamma_{1}\gamma_{2})}. \end{aligned}

Similarly, we have

\begin{aligned} T(r,w_{2})=O \bigl((\log r)^{\alpha_{2}} \bigr), \end{aligned}

where

\begin{aligned} \alpha_{2}=\frac{\log\kappa_{1}-\log\sigma}{\log(\gamma_{1}\gamma_{2})}. \end{aligned}

Case (iii): Suppose that both $$w_{1}$$ and $$w_{2}$$ are entire. Then, similar to the above argument, we can get (32) and

\begin{aligned} \sigma_{1}T \bigl(r,w_{1} \bigl(g_{1}(z) \bigr) \bigr)+S \bigl(r,w_{1} \bigl(g_{1}(z) \bigr) \bigr)\leq \tau_{2}T \bigl( \vert q_{2} \vert r,w_{2} \bigr)+S \bigl( \vert q_{2} \vert r,w_{2} \bigr). \end{aligned}

Hence, by an argument similar to the proof of Case (ii) of Theorem 1.2, if $$\sigma\leq\tau$$, then we can obtain

\begin{aligned} T(r,w_{t})=O \bigl((\log r)^{\alpha_{3}} \bigr),\quad t=1,2, \end{aligned}

where

\begin{aligned} \alpha_{3}=\frac{\log\tau-\log\sigma}{\log(\gamma_{1}\gamma_{2})}. \end{aligned}

From Cases (i)–(iii), the proof of Theorem 1.2 is completed. □

## References

1. 1.

Ablowitz, M.J., Halburd, R., Herbst, B.: On the extension of the Painlevé property to difference equations. Nonlinearity 13, 889–905 (2000)

2. 2.

Bank, S.: A general theorem concerning the growth of solutions of first-order algebraic differential equations. Compos. Math. 25, 61–70 (1972)

3. 3.

Bergweiler, W., Ishizaki, K., Yanagihara, N.: Meromorphic solutions of some functional equations. Methods Appl. Anal. 5, 248–258 (1998). Correction: Methods Appl. Anal. 6, 617–618 (1999)

4. 4.

Bergweiler, W., Ishizaki, K., Yanagihara, N.: Growth of meromorphic solutions of some functional equations. I. Aequ. Math. 63, 140–151 (2002)

5. 5.

Chen, M.F., Gao, Z.S., Du, Y.F.: Existence of entire solutions of some non-linear differential-difference equations. J. Inequal. Appl. 2017, Article ID 90 (2017)

6. 6.

Chen, M.F., Jiang, Y.Y., Gao, Z.S.: Growth of meromorphic solutions of certain types of q-difference differential equations. Adv. Differ. Equ. 2017, Article ID 37 (2017)

7. 7.

Gao, L.Y.: On meromorphic solutions of a type of system of composite functional equations. Acta Math. Sci. Ser. B Engl. Ed. 32, 800–806 (2012)

8. 8.

Gao, L.Y.: Systems of complex difference equations of Malmquist type. Acta Math. Sinica (Chin. Ser.) 55, 293–300 (2012)

9. 9.

Goldstein, R.: Some results on factorization of meromorphic functions. J. Lond. Math. Soc. 4, 357–364 (1971)

10. 10.

Goldstein, R.: On meromorphic solutions of certain functional equations. Aequ. Math. 18, 112–157 (1978)

11. 11.

Gundersen, G.G., Heittokangas, J., Laine, I., Rieppo, J., Yang, D.Q.: Meromorphic solutions of generalized Schröder equations. Aequ. Math. 63, 110–135 (2002)

12. 12.

Hayman, W.K.: Meromorphic Functions. Clarendon Press, Oxford (1964)

13. 13.

Heittokangas, J., Korhonen, R., Laine, I., Rieppo, J., Tohge, K.: Complex difference equations of Malmquist type. Comput. Methods Funct. Theory 1, 27–39 (2001)

14. 14.

Heittokangas, J., Laine, I., Rieppo, J., Yang, D.G.: Meromorphic solutions of some linear functional equations. Aequ. Math. 60, 148–166 (2000)

15. 15.

Laine, I.: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin (1993)

16. 16.

Malmquist, J.: Sur les fonctions a un nombre fini de branches définies par les équations différentielles du premier order. Acta Math. 36, 297–343 (1913)

17. 17.

Rieppo, J.: On a class of complex functional equations. Ann. Acad. Sci. Fenn., Math. 32, 151–170 (2007)

18. 18.

Silvennoinen, H.: Meromorphic Solutions of Some Composite Functional Equations. Ann. Acad. Sci. Fenn. Math. Diss., vol. 133 (2003)

19. 19.

Xu, H.Y., Liu, S.Y., Li, Q.P.: The existence and growth of solutions for several systems of complex nonlinear difference equations. Mediterr. J. Math. 16, Article ID 8 (2019)

20. 20.

Xu, H.Y., Liu, B.X., Tang, K.Z.: Some properties of meromorphic solutions of systems of complex q-shift difference equations. Abstr. Appl. Anal. 2013, Article ID 680956 (2013)

21. 21.

Yang, C.C., Yi, H.Y.: Uniqueness Theory of Meromorphic Functions. Kluwer Academic, New York (2003)

### Acknowledgements

The authors are very grateful to the editor and anonymous referees for their valuable comments and suggestions, which improved the presentation of this manuscript.

### Availability of data and materials

Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.

## Funding

This work was supported by the Innovation Research for the Postgraduates of Guangzhou University under Grant No. 2018GDJC-D04.

## Author information

Authors

### Contributions

Both authors drafted the manuscript, read and approved the final manuscript.

### Corresponding author

Correspondence to Hongqiang Tu.

## Ethics declarations

### Competing interests

The authors declare that they have no competing interests. 