The main purpose of this paper is to investigate the growth order of the meromorphic solutions of complex functional difference equation of the form (∑λ∈Iαλ(z)(∏ν=1nf(z+cν)lλ,ν))/(∑μ∈Jβμ(z)(∏ν=1nf(z+cν)mμ,ν))=Q(z,f(p(z))), where I={λ=(lλ,1,lλ,2,…,lλ,n)∣lλ,ν∈ℕ⋃{0}, ν=1,2,…,n} and J={μ=(mμ,1,mμ,2,…,mμ,n)∣mμ,ν∈ℕ⋃{0}, ν=1,2,…,n} are two finite index sets, cν(ν=1,2,…,n) are distinct complex numbers, αλ(z) (λ∈I) and βμ(z) (μ∈J) are small functions relative to f(z), and
Q(z,u) is a rational function in u with coefficients which are small functions of f(z), p(z)=pkzk+pk-1zk-1+⋯+p0∈ℂ[z] of degree k≥1. We also give some examples to show that our results are sharp.

1. Introduction and Main Results

Let f(z) be a function meromorphic 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 such as the characteristic function T(r,f), proximity function m(r,f), counting function N(r,f), and the first and second main theorems (see, e.g., [1–4]). We also use N¯(r,f) to denote the counting function of the poles of f(z) whose every pole is counted only once. The notations ρ(f) and μ(f) denote the order and the lower order of f(z), respectively. S(r,f) denotes any quantity that satisfies the condition: S(r,f)=o(T(r,f)) as r→∞ possibly outside an exceptional set of r of finite linear measure. A meromorphic function a(z) is called a small function of f(z) or a small function relative to f(z) if and only if T(r,a(z))=S(r,f).

Recently, some papers (see, e.g., [5–7]) focusing on complex difference and functional difference equations emerged. In 2005, Laine et al. [5] firstly considered the growth of meromorphic solutions of the complex functional difference equations by utilizing Nevanlinna theory. They obtained the following result.

Theorem A.

Suppose that f is a transcendental meromorphic solution of the equation
(1)∑{J}αJ(z)(∏j∈Jf(z+cj))=f(p(z)),
where {J} is a collection of all subsets of {1,2,…,n}, cj’s are distinct complex constants, and p(z) is a polynomial of degree k≥2. Moreover, we assume that the coefficients αJ(z) are small functions relative to f and that n≥k. Then
(2)T(r,f)=O((logr)α+ɛ),
where α=logn/logk.

In 2007, Rieppo [6] gave an estimation of growth of meromorphic solutions of complex functional equations as follows.

Theorem B.

Suppose that f is a transcendental meromorphic function. Let Q(z,f), R(z,f) be rational functions in f with small meromorphic coefficients relative to f such that 0<q:=degfQ≤d:=degfR and p(z)=pkzk+pk-1zk-1+⋯+p0∈ℂ[z] of degree k>1. If f is a solution of the functional equation
(3)R(z,f(z))=Q(z,f(p(z))),
then qk≤d, and for any ɛ,0<ɛ<1, there exist positive real constants K1 and K2 such that
(4)K1(logr)α-ɛ≤T(r,f)≤K2(logr)α+ɛ,α=logd-logqlogk,
when r is large enough.

Rieppo [6] also considered the growth order of meromorphic solutions of functional equation (3) when k=1 and got the following.

Theorem C.

Suppose that f is a transcendental meromorphic solution of (3), where p(z)=az+b, a,b∈ℂ, a≠0 and |a|≠1. Then
(5)μ(f)=ρ(f)=logd-logqlog|a|.

Two years later, Zheng et al. [7] extended Theorem A to more general type and obtained a similar result of Theorem C. In fact, they got the following two results.

Theorem D.

Suppose that f is a transcendental meromorphic solution of the equation
(6)∑{J}αJ(z)(∏j∈Jf(z+cj))=Q(z,f(p(z))),
where {J} is a collection of all nonempty subsets of {1,2,…,n}, cj(j=1,…,n) are distinct complex constants, p(z)=pkzk+pk-1zk-1+⋯+p0∈ℂ[z] of degree k>1, and Q(z,u) is a rational function in u of deguQ=q(>0). Also suppose that all the coefficients of (6) are small functions relative to f. Then qk≤n, and
(7)T(r,f)=O((logr)α+ɛ),
where α=(logn-logq)/logk.

Theorem E.

Suppose that f is a transcendental meromorphic solution of (6), where {J} is a collection of all nonempty subsets of {1,2,…,n}, cj(j=1,…,n) are distinct complex constants, p(z)=az+b, a,b∈ℂ, and Q(z,u) is a rational function in u of deguQ=q(>0). Also suppose that all the coefficients of (6) are small functions relative to f.

If 0<|a|<1, then we have
(8)μ(f)≥logq-logn-log|a|.

If |a|>1, then we have q≤n and
(9)ρ(f)≤logn-logqlog|a|.

If |a|=1,q>n, then we have ρ(f)=μ(f)=∞.

In this paper, we will consider a more general class of complex functional difference equations. We prove the following results, which generalize the above related results.

Theorem 1.

Suppose that f(z) is a transcendental meromorphic solution of the functional difference equation
(10)∑λ∈Iαλ(z)(∏ν=1nf(z+cν)lλ,ν)∑μ∈Jβμ(z)(∏ν=1nf(z+cν)mμ,ν)=Q(z,f(p(z))),
where cν(ν=1,…,n) are distinct complex constants, I={λ=(lλ,1,lλ,2,…,lλ,n)∣lλ,ν∈ℕ⋃{0}, ν=1,2,…,n} and J={μ=(mμ,1,mμ,2,…,mμ,n)∣mμ,ν∈ℕ⋃{0}, ν=1,2,…,n} are two finite index sets, p(z)=pkzk+pk-1zk-1+⋯+p0∈ℂ[z] of degree k>1, and Q(z,u) is a rational function in u of deguQ=q(>0). Also suppose that all the coefficients of (10) are small functions relative to f. Denoting
(11)σν=maxλ,μ{lλ,ν,mμ,ν}(ν=1,2,…,n),σ=∑ν=1nσν.
Then qk≤σ, and
(12)T(r,f)=O((logr)α+ɛ),
where α=(logσ-logq)/logk.

Theorem 2.

Suppose that f is a transcendental meromorphic solution of the equation
(13)∑λ∈Iαλ(z)(∏ν=1nf(z+cν)lλ,ν)∑μ∈Jβμ(z)(∏ν=1nf(z+cν)mμ,ν)=Q(z,f(az+b)),
where cν(ν=1,…,n) are distinct complex constants, I={λ=(lλ,1,lλ,2,…,lλ,n)∣lλ,ν∈ℕ⋃{0}, ν=1,2,…,n} and J={μ=(mμ,1,mμ,2,…,mμ,n)∣mμ,ν∈ℕ⋃{0}, ν=1,2,…,n} are two finite index sets, a,b∈ℂ, and Q(z,u) is a rational function in u of deguQ=q(>0). Also suppose that all the coefficients of (10) are small functions relative to f. Denoting
(14)σν=maxλ,μ{lλ,ν,mμ,ν}(ν=1,2,…,n),σ=∑ν=1nσν.

If 0<|a|<1, then we have
(15)μ(f)≥logq-logσ-log|a|.

If |a|>1, then we have q≤σ and
(16)ρ(f)≤logσ-logqlog|a|.

If |a|=1 and q>σ, then we have μ(f)=ρ(f)=∞.

Next we will give some examples to show that our results are best in some extent.

Example 3.

Let c1=arctan2, c2=-π/4. Then it is easy to check that f(z)=tanz solves the following equation:
(17)f(z+c1)2f(z+c2)f(z+c1)+f(z+c2)2=(-4f(z2)8+8f(z2)7+28f(z2)6-56f(z2)5fff-32f(z2)4+56f(z2)3+28f(z2)2-8f(z2)-4)×(3f(z2)8+10f(z2)7+16f(z2)6+122f(z2)5dddd-6f(z2)4-122f(z2)3+16f(z2)2-10f(z2)+3)-1.
Obviously, we have
(18)μ(f)=ρ(f)=1=logq-logσ-log|a|,
where q=8, σ=4 and a=1/2.

Example 3 shows that the estimate in Theorem 2(i) is sharp.

Example 4.

It is easy to check that f(z)=tanz satisfies the equation
(19)f(z+(π/3))2f(z+(π/6))-f(z+(π/6))f(z+(π/3))f(z+(π/6))2-f(z+(π/3))=3f(2z)2+4f(2z)+3-3f(2z)2+4f(2z)-3.
Clearly, we have
(20)μ(f)=ρ(f)=1=logσ-logqlog|a|,
where σ=4, q=2 and a=2.

Example 4 shows that the estimate in Theorem 2(ii) is sharp.

Example 5.

f(z)=tanz satisfies the equation of the form
(21)f(z+(π/4))2f(z+(π/4))+f(z-(π/4))2=-(f(z/2)2-2f(z/2)-1)38f(z/2)(f(z/2)2-1)(f(z/2)2+2f(z/2)-1),
where σ=4, q=6, and a=1/2.ρ(f)=μ(f)=1>log(3/2)/log2=(logq-logσ)/-log|a|.

Example 5 shows that the strict inequality in Theorem 2 may occur. Therefore, we do not have the same estimation as in Theorem C for the growth order of meromorphic solutions of (13).

The following Example shows that the restriction q>σ in case (iii) in Theorem 2 is necessary.

Example 6.

Meromorphic function f(z)=tanz solves the following equation:
(22)f(z+(π/4))2f(z+(π/4))+f(z-(π/4))2=(f(z)+1)34f(z)(1-f(z)),
where a=1 and 4=σ>q=3, but ρ(f)=μ(f)=1.

Next, we give an example to show that case (iii) in Theorem 2 may hold.

Example 7.

Function f(z)=zeez satisfies the following equation:
(23)(z+log6)(z+log2)5[f(z+log4)4+f(z+log4)](z+log4)f(z+log6)=(z+log4)3f(z+log2)6+(z+log2)6f(z+log2),
where a=1 and q=6>5=σ. Obviously, ρ(f)=μ(f)=∞.

2. Main Lemmas

In order to prove our results, we need the following lemmas.

Lemma 1 (see [<xref ref-type="bibr" rid="B7">4</xref>, <xref ref-type="bibr" rid="B9">8</xref>]).

Let f(z) be a meromorphic function. Then for all irreducible rational functions in f,
(24)R(z,f)=P(z,f)Q(z,f)=∑i=0pai(z)fi∑j=0qbj(z)fj,
such that the meromorphic coefficients ai(z), bj(z) satisfy
(25)T(r,ai)=S(r,f),i=0,1,…,p,T(r,bj)=S(r,f),j=0,1,…,q;
then one has
(26)T(r,R(z,f))=max{p,q}·T(r,f)+S(r,f).

From the proof of Theorem 1 in [9], we have the following estimate for the Nevanlinna characteristic.

Lemma 2.

Let f1,f2,…,fn be distinct meromorphic functions and
(27)F(z)=P(z)Q(z)=∑λ∈Iαλ(z)f1lλ,1f2lλ,2…fnlλ,n∑μ∈Jβμ(z)f1mμ,1f2mμ,2…fnmμ,n.
Then
(28)T(r,F(z))≤∑ν=1nσνT(r,fν)+S(r,f),
where I={λ=(lλ,1,lλ,2,…,lλ,n)∣lλ,ν∈ℕ⋃{0}, ν=1,2,…,n} and J={μ = (mμ,1,mμ,2,…,mμ,n)∣mμ,ν∈ℕ⋃{0},ν=1,2,…,n} are two finite index sets, σν=maxλ,μ{lλ,ν,mμ,ν}(ν=1,2,…,n). αλ(z)=o(T(r,fν)(λ∈I)) and βμ(z)=o(T(r,fν)(μ∈J)) hold for all ν∈{1,2,…,n} and satisfy T(r,αλ)=S(r,f)(λ∈I) and T(r,βμ)=S(r,f)(μ∈J).

Lemma 3 (see [<xref ref-type="bibr" rid="B12">7</xref>]).

Let c be a complex constant. Given ɛ>0 and a meromorphic function f, one has
(29)T(r,f(z±c))≤(1+ɛ)T(r+|c|,f),
for all r>r0, where r0 is some positive constant.

Lemma 4 (see [<xref ref-type="bibr" rid="B7">4</xref>]).

Let g:(0,+∞)→ℝ, h:(0,+∞)→ℝ be monotone increasing functions such that g(r)≤h(r) outside of an exceptional set E of finite linear measure. Then, for any α>1, there exists r0>0 such that g(r)≤h(αr) for all r>r0.

Lemma 5 (see [<xref ref-type="bibr" rid="B2">10</xref>]).

Let f be a transcendental meromorphic function, and p(z)=akzk+ak-1zk-1+⋯+a1z+a0,ak≠0, be a nonconstant polynomial of degree k. Given 0<δ<|ak|, denote λ=|ak|+δ and μ=|ak|-δ. Then given ɛ>0 and a∈ℂ⋃{∞}, one has
(30)kn(μrk,a,f)≤n(r,a,f(p(z)))≤kn(λrk,a,f)N(μrk,a,f)+O(logr)≤N(r,a,f(p(z)))≤N(λrk,a,f)+O(logr)(1-ɛ)T(μrk,f)≤T(r,f(p(z)))≤(1+ɛ)T(λrk,f),
for all r large enough.

Lemma 6 (see [<xref ref-type="bibr" rid="B3">11</xref>]).

Let ϕ:[r0,+∞)→(0,+∞) be positive and bounded in every finite interval, and suppose that ϕ(μrm)≤Aϕ(r)+B holds for all r large enough, where μ>0, m>1, A>1 and B are real constants. Then
(31)ϕ(r)=O((logr)α),
where α=logA/logm.

Lemma 7 (see [<xref ref-type="bibr" rid="B11">6</xref>]).

Let ϕ:(r0,∞)→(1,∞), where r0≥1, be a monotone increasing function. If for some real constant α>1, there exists a real number K>1 such that ϕ(αr)≥Kϕ(r), then
(32)lim_r→∞logϕ(r)logr≥logKlogα.

Lemma 8 (see [<xref ref-type="bibr" rid="B4">12</xref>]).

Let ϕ:(1,∞)→(0,∞) be a monotone increasing function and let f be a nonconstant meromorphic function. If, for some real constant α∈(0,1), there exist real constants K1>0 and K2≥1 such that
(33)T(r,f)≤K1ϕ(αr)+K2T(αr,f)+S(αr,f),
then
(34)ρ(f)≤logK2-logα+lim¯r→∞logϕ(r)logr.

3. Proof of TheoremsProof of Theorem <xref ref-type="statement" rid="thm1">1</xref>.

We assume f(z) is a transcendental meromorphic solution of (10). Denoting C=max{|c1|,|c2|,…,|cn|}. According to Lemmas 1, 2, and 3 and the last assertion of Lemma 5, we get that for any ε1>0,
(35)q(1-ε1)T(μrk,f)+S(r,f)≤qT(r,f(p(z)))+S(r,f)=T(r,Q(z,f(p(z))))=T(r,∑λ∈Iαλ(z)(∏ν=1nf(z+cν)lλ,ν)∑μ∈Jβμ(z)(∏ν=1nf(z+cν)mμ,ν))≤∑ν=1nσνT(r,f(z+cν))+S(r,f)≤∑ν=1nσν(1+ε1)T(r+C,f(z))+S(r,f)=(∑ν=1nσν)(1+ε1)T(r+C,f(z))+S(r,f)=σ(1+ε1)T(r+C,f(z))+S(r,f),
where r is large enough and μ=|pk|-δ for some 0<δ<|pk|. Since T(r+C,f)≤T(βr,f) holds for r large enough for β>1, we may assume r to be large enough to satisfy
(36)q(1-ε1)T(μrk,f)≤σ(1+ε1)T(βr,f)
outside a possible exceptional set of finite linear measure. By Lemma 4, we know that whenever γ>1,
(37)q(1-ε1)T(μrk,f)≤σ(1+ε1)T(γβr,f)
holds for all r large enough. Denote t=γβr; thus the inequality (37) may be written in the form
(38)T(μ(γβ)ktk,f)≤σ(1+ε1)q(1-ε1)T(t,f).
By Lemma 6, we have
(39)T(r,f)=O((logr)α1),
where
(40)α1=log(σ(1+ε1)/q(1-ε1))logk=logσ-logqlogk+log((1+ε1)/(1-ε1))logk.
Denoting now α=(logσ-logq)/logk and ɛ=log((1+ε1)/(1-ε1))/logk; thus we obtain the required form.

Finally, we show that qk≤σ. If qk>σ, then we have α<1. For sufficiently small ɛ>0, we have α+ɛ<1, which contradicts with the transcendency of f. Thus Theorem 1 is proved.

Proof of Theorem <xref ref-type="statement" rid="thm2">2</xref>.

Suppose f(z) is a transcendental meromorphic solution of (13). Denoting C=max{|c1|,|c2|,…,|cn|}.

0<|a|<1. We may assume that q>σ, since the case q≤σ is trivial by the fact that μ(f)≥0. By Lemmas 1–3, we have for any ɛ>0 and β>1,
(41)qT(r,f(p(z)))+S(r,f)=T(r,Q(z,f(p(z))))=T(r,∑λ∈Iαλ(z)(∏ν=1nf(z+cν)lλ,ν)∑μ∈Jβμ(z)(∏ν=1nf(z+cν)mμ,ν))≤∑ν=1nσνT(r,f(z+cν))+S(r,f)≤∑ν=1nσν(1+ɛ)T(r+C,f(z))+S(r,f)=(∑ν=1nσν)(1+ɛ)T(r+C,f(z))+S(r,f)=σ(1+ɛ)T(r+C,f(z))+S(r,f)≤σ(1+ɛ)T(βr,f)+S(r,f),

where r is large enough.

By the last assertion of Lemma 5 and (41), we obtain that, for μ=|a|-δ(0<δ<|a|,0<μ<1), the following inequality
(42)q(1-ɛ)T(μr,f)≤σ(1+ɛ)T(βr,f)
holds, where r is large enough outside of a possible set of finite linear measure. By Lemma 4, we get that for any γ>1 and sufficiently large r,
(43)q(1-ɛ)T(μr,f)≤σ(1+ɛ)T(γβr,f).
Therefore,
(44)q(1-ɛ)σ(1+ɛ)T(r,f)≤T(γβμr,f).
Since β>1, γ>1, 0<μ<1 and q>σ, we have βγ/μ>1 and q(1-ɛ)/σ(1+ɛ)>1 when ɛ is small enough. Using Lemma 7, we see that
(45)μ(f)≥logq(1-ɛ)-logσ(1+ɛ)logγβ-logμ.
Letting ɛ→0, δ→0, β→1 and γ→1, we have
(46)μ(f)≥logq-logσ-log|a|.

|a|>1. By the similar reasoning as is (i), we easily obtain that
(47)q(1-ɛ)T(μr,f)≤qT(r,f(p(z)))≤σ(1+ɛ)T(r+C,f(z))+S(r,f)

for all r large enough. We may select sufficiently small numbers δ>0 and ɛ>0, such that μ=|a|-δ>1 and (1/μ)+ɛ<1. Thus we have
(48)T(μr,f)≤σ(1+ɛ)q(1-ɛ)T(r+C,f(z))+S(r,f);
namely,
(49)T(μr,f)≤σ(1+ɛ)q(1-ɛ)T(r+C,f(z)),
where r is large enough possibly outside of a set of finite linear measure. By Lemma 4, we have for any 1<γ<μ,
(50)T(μr,f)≤σ(1+ɛ)q(1-ɛ)T(γr,f(z));
that is,
(51)T(r,f)≤σ(1+ɛ)q(1-ɛ)T(γμr,f(z))
holds for all sufficiently large r. By Lemma 8, we obtain
(52)ρ(f)≤logσ-logq+log(1+ɛ)-log(1-ɛ)-log(γ/μ).
Letting ɛ→0, δ→0 and γ→1, we have
(53)ρ(f)≤logσ-logqlog|a|.

|a|=1 and q>σ. The proof of this case is completely similar as in the case in (i). In fact, we set μ=|a|-δ=1-δ(0<δ<1,0<μ<1). Similarly, we can get
(54)μ(f)≥logq-logσ-log|a|.

Since |a|=1, we have μ(f)=ρ(f)=∞.Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions. The research was supported by Colonel-level topics (JSNU-ZY-01), (Jsie2012zd01), and NSF of China (11271179).

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