In this article, we shall consider the linear higher order difference equation

$$ P_{n}{(z)}f(z+n)+P_{n-1}{(z)}f(z+n-1)+\cdots +P_{1}{(z)}f(z+1)+P _{0}{(z)}f(z)=0, $$

(1)

where \(P_{0}(z),\ldots ,P_{n}(z)\) are meromorphic functions. Nevanlinna theory is an important and basic tool in our discussion; for references see [9, 12, 18]. For a meromorphic function \(f(z)\), we denote by \(T(r,f)\), \(N(r,f)\) the characteristic function and the counting function of \(f(z)\), respectively. In particular, we define the order and the lower order of a meromorphic function \(f(z)\) by

$$ \sigma (f)=\limsup_{r\rightarrow \infty }\frac{\log ^{+}T(r,f)}{ \log r} \quad \mbox{and} \quad \mu (f)=\liminf_{r\rightarrow \infty }\frac{\log ^{+}T(r,f)}{\log r}, $$

while

$$ \lambda (f)=\limsup_{r\rightarrow \infty }\frac{\log ^{+}N(r, \frac{1}{f})}{\log r} \quad \mbox{and} \quad \delta (a,f)= \liminf_{r\to \infty }\frac{m(r,\frac{1}{f-a})}{T(r,f)}=1- \limsup _{r\to \infty }\frac{N(r,\frac{1}{f-a})}{T(r,f)} $$

stand for the exponents of convergence of zero sequence of *f* and the deficiency of *f* at the point *a*, respectively. Let \(\alpha (z)\) be a meromorphic function. We say that \(\alpha (z)\) is a small function with respect to \(f(z)\), if \(T(r,\alpha (z))=o(T(r,f))\), possible outside of a set *E* with finite linear measure.

In [5], Chiang and Feng considered the growth of transcendental entire solutions of linear higher order difference equations.

### Theorem A

*Let*
\(P_{j}(z)\) (\(j=0,1,\ldots ,n\)) *be polynomials*. *If there exists an integer*
*l*, \(0\le l\le n\), *such that*

$$ \deg (P_{l})> \max_{0\leq j \leq n,j\neq l}\bigl\{ \deg (P_{j})\bigr\} . $$

(2)

*Then every meromorphic solution*
\(f(z)\)
*of Eq*. (1) *satisfies*
\(\sigma (f)\geq 1\).

The condition (2) shows the degree of \(P_{l}\) is larger than the other coefficients which means the polynomial \(P_{l}\) is dominating coefficient. Chen [3] weakens the condition (2), and obtained.

### Theorem B

*Let*
\(P_{j}(z)\) (\(j=0,\ldots ,n\)) *be polynomials such that*
\(P_{n}P_{0}\neq 0\)
*and*

$$ \deg \{P_{n}+\cdots +P_{0}\}=\max \{\deg P_{j}:j=0,\ldots ,n\}\geq 1. $$

(3)

*If*
\(f(z)\)
*is a meromorphic solution of Eq*. (1), *then*
\(\sigma (f) \geq 1\), *and*
\(f(z)\)
*assumes every finite value*
\(a\in \mathbb{C}\)
*except zero infinitely often and*
\(\lambda (f-a)=\sigma (f)\).

The above theorems deal with the case that the coefficients of Eq. (1) are polynomials, and there exists a dominating coefficient. Then a natural question arises: can we estimate the growth of the Eq. (1) provided the coefficients are entire functions or meromorphic functions. To answer this question, Chiang and Feng [5] obtained the following.

### Theorem C

*Let*
\(P_{0}(z),\ldots ,P_{n}(z)\)
*be entire functions such that there exists an integer*
*l*, \(0\le l\le n\), *such that*

$$ \sigma (P_{l})>\max \bigl\{ \sigma (P_{j}):0 \le j\le n, j\ne l\bigr\} . $$

(4)

*If*
\(f(z)\)
*is a nontrivial meromorphic solution of Eq*. (1), *then*
\(\sigma (f)\ge \sigma (P_{l})+1\).

Clearly, since the order of \(P_{l}\) is larger than that of the others, \(P_{l}\) in Theorem C is a dominating coefficient. But if no such coefficient exists, Laine and Yang [13] also proved a similar conclusion in Theorem D below.

### Definition 1.1

([9])

The type of a meromorphic function \(f(z)\) of order *σ* (\(0<\sigma <\infty \)) is defined by

$$ \tau (f)={ \lim_{r \to \infty }\sup }\frac{T(r,f)}{r ^{\sigma }}. $$

### Theorem D

([13])

*Let*
\(P_{0}(z),\ldots ,P_{n}(z)\)
*be entire functions of finite order so that among those having the maximal order*
\(\sigma :=\max \{\sigma (P_{j}):0\le j\le n \}\), *exactly one has its type strictly greater than the others*. *Then*, *for any meromorphic solution of Eq*. (1), *we have*
\(\sigma (f)\ge \sigma +1\).

The condition about type in Theorem D still means the growth of some coefficient is faster than the others, although there exist several coefficients have the same order. This implies there still exists dominating coefficient. For a linear difference equation with meromorphic coefficients, Chen [4] proved the following result.

### Theorem E

*Let*
\(P_{0}(z),\ldots ,P_{n}(z)\)
*be meromorphic functions such that there exists an integer*
*l*, \(0\le l\le n\), *such that*
\(\sigma (P_{l})>\max \{\sigma (P_{j}):0 \le j\le n, j\ne l\}\)
*and*
\(\delta (\infty , P_{l})>0\). *If*
\(f(z)\)
*is a nontrivial meromorphic solution of Eq*. (1), *then*
\(\sigma (f)\ge \sigma (P_{l})+1\).

Laine and Yang [13] posed the following question, which it is natural to ask.

### Question 1.1

Do all meromorphic solution *f* of Eq. (1) satisfy \(\sigma (f)\ge 1+\max \{\sigma (P_{j}):0\le j\le n\}\), even if there is no dominating coefficient?

In [11], some examples are given to illustrate that the Laine–Yang conjecture is not true in whole. In this paper, we consider this question and shall give estimates of growth of solutions of Eq. (1), and there is no dominating coefficient.

### Theorem 1.1

*Let*
\(P_{j}(z)\) (\(j=0,\ldots ,n\)) *be meromorphic functions*. *If*
\(P_{1}(z)\)
*has a finite deficient value*
*a*, *and*
\(\sigma (P_{j})<\sigma (P_{0})<\frac{1}{2}\), \(j\neq 0,1\), *then every finite order meromorphic solution*
\(f(z)\)
*of Eq*. (1) *satisfies*
\(\sigma (f)\geq \mu (P_{0})-\max \{\sigma (P_{j}),j\neq 0,1\}+1\).

Edrei and Fuchs proved that the number of deficient values cannot be infinite when the zeros and poles of a meromorphic function distributed near some curves. We first give some definitions and results in order to relate their results with Eq. (1).

Let

$$\begin{aligned}& \varOmega (\theta _{1},\theta _{2},r)=\bigl\{ z:\theta _{1}< \arg z< \theta _{2}, \vert z \vert < r\bigr\} , \\& \overline{\varOmega }(\theta _{1},\theta _{2},r)=\bigl\{ z: \theta _{1}\leq \arg z \leq \theta _{2}, \vert z \vert < r \bigr\} . \end{aligned}$$

### Definition 1.2

([6, 7])

Let \(f(z)\) be a meromorphic function with finite order of growth (\(0<\sigma (f)< \infty \)). A ray \(\arg z=\theta \) starting from the origin is called a zero-pole accumulation ray of \(f(z)\), if, for any given real number \(\varepsilon >0\), the following equality holds:

$$ \varlimsup _{r\rightarrow \infty }\frac{\log n \{\overline{\varOmega }(\theta -\varepsilon ,\theta +\varepsilon ,r),f=0 \}+\log n \{\overline{\varOmega }(\theta -\varepsilon ,\theta +\varepsilon ,r),f=\infty \}}{\log r}=\sigma (f). $$

Zhang [18] proved that if \(f(z)\) is a meromorphic function with order \(\sigma =\sigma (f)\), \(0<\sigma <\infty \) and \(f(z)\) has *q* zero-pole accumulation rays and *p* deficient values other than 0 and ∞, then \(p\leq q\). We say \(f(z)\in \mathit{EF}\) if \(p=q\geq 1\), and the set of such \(f(z)\) is called an Edrei–Fuchs set. This means that \(f(z)\) is a positive finite order meromorphic function with *p* zero-pole accumulation rays and *p* nonzero finite deficient values.

Now we are in the position to give a result concerning an Edrei–Fuchs set.

### Theorem 1.2

*Let*
\(P_{j}(z)\) (\(j=0,\ldots ,n\)) *be meromorphic functions*. *If*
\(P_{0}(z)\)
*has a deficient value* ∞, \(P_{1}(z)\in \mathit{EF}\), *and*
\(\sigma (P_{j})<\sigma (P_{0})\)
*for*
\(j\neq 0,1\), *then every meromorphic solution*
\(f(z)\)
*of Eq*. (1) *satisfies*
\(\sigma (f)\geq \sigma (P_{0})-\max \{\sigma (P_{j}),j\neq 0,1\}+1\).

### Remark 1.1

In Theorems 1.1 and 1.2, we require one coefficient in the equation to have one or several finite deficient value, but we do not know which one of \(P_{0}\) and \(P_{1}\) has faster growth, and this means there does not exist a dominating coefficient.

Recently, some results about the connections between complex dynamics with linear differential equations have been obtained; see [10, 14]. In the following, we shall consider Question 1.1 from the dynamical system point of view. Some notations of complex dynamics are needed; see [10].

Let \(f:\mathbb{C}\mapsto \widehat{\mathbb{C}}\) be a transcendental meromorphic function. Denote by \(f^{n}\), \(n\in \mathbb{N}\), the *n*th iterate of *f*, that is, \(f^{1}=f,\ldots ,f^{n}=f\circ (f^{n-1})\). The Fatou set of *f* is denoted by \(F(f)\), and the Julia set of *f* is defined by \(J(f)=\widehat{\mathbb{C}}\setminus F(f)\). Let *U* be a connected component of \(F(f)\), then \(f^{n}(U)\) is contained in a component of \(F(f)\), denoted by \(U_{n}\). We say *U* is a wandering domain, if \(U_{n}\cap U_{m}=\emptyset \). Furthermore, if *U* is wandering, and all \(U_{n}\) are multiply-connected and surround 0 and the Euclidean distance \(\operatorname{dist}(0,U_{n})\rightarrow \infty \) as \(n\rightarrow \infty \), then *U* is called a Baker wandering domain. The reader can refer to [2, 19] for more details.

### Definition 1.3

([17])

Let \(f(z)\) be a meromorphic function of order *σ*, where \(0<\sigma <\infty \). A ray \(\arg z=\theta _{0}\) (\(0\leq \theta _{0}<2\pi \)) is named a Borel direction of order *σ* of \(f(z)\), if for any positive number *ε*, the inequality

$$ \varlimsup _{r\rightarrow \infty }\frac{\log n (r, \theta _{0}-\varepsilon ,\theta _{0}+\varepsilon ,f=a)}{\log r}\geq \mu $$

holds for any finite complex value *a*, with possibly one exceptional value, where \(n(r,\theta _{0}-\varepsilon ,\theta _{0}+\varepsilon ,f=a)\) denotes the number of zeros of \(f(z)-a\) in the region \((|z|\leq r) \cap (\theta _{0}-\varepsilon \leq \arg z\leq \theta _{0}+\varepsilon )\), multiple zeros being counted with their multiplicities.

Suppose that \(f(z)\) is an entire function of finite lower order \(\mu >0\). Let \(q\ (<\infty )\) denote the number of Borel directions and *p* denote the number of finite deficient values of \(f(z)\). Yang in [17] proved that \(p\leq \frac{q}{2}\). We will say that an entire function \(f(z)\) is extremal for Yang’s inequality if \(f(z)\) satisfies the assumptions of Definition 1.3 with \(p=\frac{q}{2}\).

Next, we consider Eq. (1) from the dynamical system point of view, and we can establish the following result.

### Theorem 1.3

*Let*
\(P_{j}(z)\) (\(j=0,\ldots ,n\)) *be entire functions*. *Suppose that*
\(P_{1}(z)\)
*is extremal for Yang’s inequality*, *and*
\(F(P_{0})\)
*has a Baker wandering domain*. *If*
\(\sigma (P_{j})<\sigma (P_{0})\)
*for*
\(j\neq 0,1\), *then every meromorphic solution*
\(f(z)\)
*of Eq*. (1) *satisfies*
\(\sigma (f)\geq \sigma (P_{0})-\max \{\sigma (P_{j}),j\neq 0,1\}+1\).