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# On growth of meromorphic solutions for linear difference equations with meromorphic coefficients

- Yanxia Liu
^{1}Email author

**2013**:60

https://doi.org/10.1186/1687-1847-2013-60

© Liu; licensee Springer 2013

**Received:**2 December 2012**Accepted:**26 February 2013**Published:**19 March 2013

## Abstract

In this paper, we consider the value distribution of meromorphic solutions for linear difference equations with meromorphic coefficients.

**MSC:**30D35, 39A10.

## Keywords

- difference equation
- meromorphic coefficient
- growth
- zero

## 1 Introduction and preliminaries

Recently, several papers (including [1–7]) have been published regarding value distribution of meromorphic solutions of linear difference equations. We recall the following results. Chiang and Feng proved the following theorem.

**Theorem A** ([2])

*Let*${P}_{0}(z),\dots ,{P}_{n}(z)$

*be polynomials such that there exists an integer*

*l*, $0\le l\le n$,

*such that*

*holds*.

*Suppose*$f(z)$

*is a meromorphic solution of the difference equation*

*Then we have* $\sigma (f)\ge 1$.

In this paper, we use the basic notions of Nevanlinna’s theory (see [8, 9]). In addition, we use the notation $\sigma (f)$ to denote the order of growth of the meromorphic function $f(z)$, and $\lambda (f)$ to denote the exponent of convergence of zeros of $f(z)$.

Chen [1] weakened the condition (1.1) of Theorem A and proved the following results.

**Theorem B** ([1])

*Let*${P}_{n}(z),\dots ,{P}_{0}(z)$

*be polynomials such that*${P}_{n}{P}_{0}\not\equiv 0$

*and*

*Then every finite order meromorphic solution* $f(z)$ (≢0) *of equation* (1.2) *satisfies* $\sigma (f)\ge 1$, *and* $f(z)$ *assumes every nonzero value* $a\in \mathbb{C}$ *infinitely often and* $\lambda (f-a)=\sigma (f)$.

**Theorem C** ([1])

*Let*$F(z)$, ${P}_{n}(z),\dots ,{P}_{0}(z)$

*be polynomials such that*$F{P}_{n}{P}_{0}\not\equiv 0$

*and*(1.3).

*Then every finite order transcendental meromorphic solution*$f(z)$

*of the equation*

*satisfies* $\sigma (f)\ge 1$ *and* $\lambda (f)=\sigma (f)$.

**Theorem D** ([1])

*Let* $F(z)$, ${P}_{n}(z),\dots ,{P}_{0}(z)$ *be polynomials such that* $F{P}_{n}{P}_{0}\not\equiv 0$. *Suppose that* $f(z)$ *is a meromorphic solution with infinitely many poles of* (1.2) (*or* (1.4)). *Then* $\sigma (f)\ge 1$.

Chiang and Feng proved the following result.

**Theorem E** ([2])

*Let*${A}_{0}(z),\dots ,{A}_{n}(z)$

*be entire functions such that there exists an integer*

*l*, $0\le l\le n$,

*such that*

*If* $f(z)$ *is a meromorphic solution of* (1.5), *then we have* $\sigma (f)\ge \sigma ({A}_{l})+1$.

Laine and Yang proved the following theorem.

**Theorem F** ([5])

*Let*${A}_{0},\dots ,{A}_{n}$

*be entire functions of finite order so that among those having the maximal order*$\sigma :=max\{\sigma ({A}_{k}):0\le k\le n\}$,

*exactly one has its type strictly greater than the others*.

*Then for any meromorphic solution of*

*we have* $\sigma (f)\ge \sigma +1$.

**Remark 1.1**If ${A}_{0},\dots ,{A}_{n}$ are meromorphic functions satisfying (1.6), then Theorem E does not hold. For example, the equation

has a solution $y(z)={e}^{iz}-1$, which $\sigma (y)=1<\sigma ({A}_{0})+1$.

This example shows that for the linear difference equation with meromorphic coefficients, the condition (1.6) cannot guarantee that every transcendental meromorphic solution $f(z)$ of (1.7) satisfies $\sigma (f)\ge \sigma ({A}_{l})+1$.

Thus, a natural question to ask is what conditions will guarantee every transcendental meromorphic solution $f(z)$ of (1.7) with meromorphic coefficients satisfies $\sigma (f)\ge \sigma ({A}_{l})+1$.

In this note, we consider this question and prove the following results.

**Theorem 1.1** *Let* ${c}_{1}$, ${c}_{2}$ ($\ne {c}_{1}$), *a* *be nonzero constants*, ${h}_{1}(z)$ *be a nonzero meromorphic function with* $\sigma ({h}_{1})<1$, $B(z)$ *be a nonzero meromorphic function*.

*If*$B(z)$

*satisfies any one of the following three conditions*:

- (i)
$\sigma (B)>1$

*and*$\delta (\mathrm{\infty},B)>0$; - (ii)
$\sigma (B)<1$;

- (iii)
$B(z)={h}_{0}(z){e}^{bz}$

*where**b**is a nonzero constant*, ${h}_{0}(z)$ (≢0)*is a meromorphic function with*$\sigma ({h}_{0})<1$,

*then every meromorphic solution*

*f*(≢0)

*of the difference equation*

*satisfies* $\sigma (f)\ge max\{\sigma (B),1\}+1$.

*Further*,

*if*$\phi (z)$ (≢0)

*is a meromorphic function with*

*then*

**Corollary**

*Under conditions of Theorem*1.1,

*every finite order solution*$f(z)$ (≢0)

*of*(1.8)

*has infinitely many fixed points*,

*satisfies*$\tau (f)=\sigma (f)$,

*and for any nonzero constant*

*c*,

**Example 1.1**

satisfies conditions of Theorem 1.1 and has a solution $f(z)={e}^{{z}^{2}}$ satisfying $\lambda (f)=0$ and $\tau (f)=\sigma (f)=2$. This example shows that under conditions of Theorem 1.1, a meromorphic solution of (1.8) may have no zero.

**Theorem 1.2**

*Let*${h}_{1}(z)$, ${c}_{1}$, ${c}_{2}$,

*a*, $B(z)$

*satisfy conditions of Theorem*1.1,

*and let*$F(z)$ (≢0)

*be a meromorphic function with*$\sigma (F)<max\{\sigma (B),1\}+1$.

*Then all meromorphic solutions with finite order of the equation*

*satisfy*

*with at most one possible exceptional solution with* $\sigma (f)<max\{\sigma (B),1\}+1$.

**Remark 1.2**Under conditions of Theorem 1.1, equation (1.8) has no rational solution. But equation (1.9) in Theorem 1.2 may have a rational solution. For example, the equation

satisfies conditions of Theorem 1.2 and has a solution $f(z)=z$. This shows that in Theorem 1.2, there exists one possible exceptional solution with $\sigma (f)<max\{\sigma (B),1\}+1$.

## 2 Proof of Theorem 1.1

We need the following lemmas to prove Theorem 1.1.

*Given two distinct complex constants*${\eta}_{1}$, ${\eta}_{2}$,

*let*

*f*

*be a meromorphic function of finite order*

*σ*.

*Then*,

*for each*$\epsilon >0$,

*we have*

**Lemma 2.2** (see [11])

*Suppose that*$P(z)=(\alpha +i\beta ){z}^{n}+\cdots $ (

*α*,

*β*

*are real numbers*, $|\alpha |+|\beta |\ne 0$)

*is a polynomial with degree*$n\ge 1$,

*that*$A(z)$ (≢0)

*is an entire function with*$\sigma (A)<n$.

*Set*$g(z)=A(z){e}^{P(z)}$, $z=r{e}^{i\theta}$, $\delta (P,\theta )=\alpha cosn\theta -\beta sinn\theta $.

*Then*,

*for any given*$\epsilon >0$,

*there exists a set*${H}_{1}\subset [0,2\pi )$

*that has the linear measure zero such that for any*$\theta \in [0,2\pi )\mathrm{\setminus}({H}_{1}\cup {H}_{2})$,

*there is*$R>0$

*such that for*$|z|=r>R$,

*we have that*

- (i)
*if*$\delta (P,\theta )>0$,*then*$exp\{(1-\epsilon )\delta (P,\theta ){r}^{n}\}<\left|g\left(r{e}^{i\theta}\right)\right|<exp\{(1+\epsilon )\delta (P,\theta ){r}^{n}\};$(2.1) - (ii)
*if*$\delta (P,\theta )<0$,*then*$exp\{(1+\epsilon )\delta (P,\theta ){r}^{n}\}<\left|g\left(r{e}^{i\theta}\right)\right|<exp\{(1-\epsilon )\delta (P,\theta ){r}^{n}\},$(2.2)

*where* ${H}_{2}=\{\theta \in [0,2\pi );\delta (P,\theta )=0\}$ *is a finite set*.

**Lemma 2.3**

*Let*${c}_{1}$, ${c}_{2}$ ($\ne {c}_{1}$),

*a*

*be nonzero constants*, ${A}_{j}(z)$ ($j=0,1,2$), $F(z)$

*be nonzero meromorphic functions*.

*Suppose that*$f(z)$

*is a finite order meromorphic solution of the equation*

*If* $\sigma (f)>max\{\sigma (F),\sigma ({A}_{j})\phantom{\rule{0.25em}{0ex}}(j=0,1,2)\}$, *then* $\lambda (f)=\sigma (f)$.

*Proof*Suppose that $\sigma (f)=\sigma $, $max\{\sigma (F),\sigma ({A}_{j})\phantom{\rule{0.25em}{0ex}}(j=0,1,2)\}=\alpha $. Then $\sigma >\alpha $. Equation (2.3) can be rewritten as the form

*ε*($0<\epsilon <min\{\frac{1}{4},\frac{\sigma -\alpha}{4}\}$), and for sufficiently large

*r*, we have that

where *M* (>0) is some constant.

*ε*is arbitrary, by (2.10), we obtain

Hence, $\lambda (f)=\sigma (f)=\sigma $. □

*Proof of Theorem 1.1*Suppose that $f(z)$ (≢0) is a meromorphic solution of equation (1.8) with $\sigma (f)<\mathrm{\infty}$.

- (1)Suppose that $B(z)$ satisfies the condition (i): $\sigma (B)>1$ and $\delta (\mathrm{\infty},B)=\delta >0$. Thus, for sufficiently large
*r*,$m(r,B)>\frac{\delta}{2}T(r,B).$(2.11)

*ε*($0<\epsilon <\frac{\sigma (B)-1}{3}$),

*M*(>0) is some constant. Combining (2.17) and $\epsilon <\frac{\sigma (B)-1}{3}$, it follows that

- (2)
Suppose that $B(z)$ satisfies the condition (ii): $\sigma (B)<1$. Using the same method as in (1), we can obtain $\sigma (f)\ge max\{\sigma (B),1\}+1$.

- (3)
Suppose that $B(z)$ satisfies the condition (iii): $B(z)={h}_{0}(z){e}^{bz}$, where

*b*is a nonzero constant, ${h}_{0}(z)$ (≢0) is a meromorphic function with $\sigma ({h}_{0})<1$.

In what follows, we divide this proof into three subcases: (a) $arga\ne argb$; (b) $arga=argb$ and $|a|\ne |b|$; (c) $a=b$.

*r*,

By $\delta (bz,{\theta}_{0})=cos(argb+{\theta}_{0})>0$, $\sigma (f)=\alpha <2$ and ${\epsilon}_{1}<\frac{2-\alpha}{2}$, it is easy to see that (2.22) is a contradiction. Hence, $\sigma (f)\ge 2$.

By ${\epsilon}_{2}<\frac{|b|-|a|}{2(|b|+|a|)}$, we see that (2.27) is a contradiction.

Now suppose that $|b|<|a|$. Using the same method as above, we can also deduce a contradiction.

Hence, $\sigma (f)\ge 2$ in Subcase (b).

Subcase (c). We first affirm that $f(z)$ cannot be a nonzero rational function. In fact, if $f(z)$ is a rational function, then ${e}^{az}[{h}_{1}(z)f(z+{c}_{1})+{h}_{0}(z)f(z)]=-f(z+{c}_{2})$ is a rational function. So that ${h}_{1}(z)f(z+{c}_{1})+{h}_{0}(z)f(z)\equiv 0$, that is, $f(z+{c}_{2})\equiv 0$, a contradiction.

- (4)Suppose that $\phi (z)$ (≢0) is a meromorphic function with $\sigma (\phi )<max\{\sigma (B),1\}+1$. Set $g(z)=f(z)-\phi (z)$. Substituting $f(z)=g(z)+\phi (z)$ into (1.8), we obtain$\begin{array}{c}g(z+{c}_{2})+{h}_{1}(z){e}^{az}g(z+{c}_{1})+B(z)g(z)\hfill \\ \phantom{\rule{1em}{0ex}}=-[\phi (z+{c}_{2})+{h}_{1}(z){e}^{az}\phi (z+{c}_{1})+B(z)\phi (z)].\hfill \end{array}$(2.29)

Thus, Theorem 1.1 is proved. □

## 3 Proof of Theorem 1.2

But ${f}^{\ast}-{f}_{0}$ is a solution of the corresponding homogeneous equation (1.8) of (1.9). By Theorem 1.1, we have $\sigma ({f}^{\ast}-{f}_{0})\ge max\{\sigma (B),1\}+1$, a contradiction. Hence equation (1.9) possesses at most one exceptional solution ${f}_{0}$ with $\sigma ({f}_{0})<max\{\sigma (B),1\}+1$.

*f*is a meromorphic solution of (1.9) with

Thus, Theorem 1.2 is proved.

## Declarations

### Acknowledgements

The author is grateful to the referees for a number of helpful suggestions to improve the paper. This research was partly supported by the National Natural Science Foundation of China (grant no. 11171119).

## Authors’ Affiliations

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