# Some Properties of Meromorphic Solutions of Systems of Complexq-Shift Difference Equations

Authors: Hong-Yan Xu; Bing-Xiang Liu; Ke-Zong Tang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

In view of Nevanlinna theory, we study the properties of meromorphic solutions of systems of a class of complex difference equations. Some results obtained improve and extend the previous theorems given by Gao.


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## Body

## 1. Introduction and Main Results

The purpose of this paper is to study some properties of meromorphic solutions of complexq-shift difference equations. The fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used (see [1–3]). Besides, for meromorphic function f, a meromorphic function a(z) is called small function with respect to f if T(r,a(z))=o(T(r,f))=S(r,f) for all r outside a possible exceptional set E of finite logarithmic measure limr→∞∫[1,r)∩E‍(dt/t)<∞.In recent years, it has been a heated topic to study difference equations, difference product, andq-difference in the complex plane ℂ. There were articles focusing on the growth of solutions of difference equations, value distribution and uniqueness of differences analogues of Nevanlinna’s theory (see [4–9]). Chiang and Feng [10] and Halburd and Korhonen [11] established a difference analogue of the logarithmic derivative lemma independently, and Barnett et al. [5] also established an analogue of the logarithmic derivative lemma on q-difference operators. By applying these theorems, a number of results on meromorphic solutions of complex difference and q-difference equations were obtained (see [12–19]).In 2011, Korhonen [20] investigated the properties of finite-order meromorphic solution of the equation
(1)H(z,ω)P(z,ω)=Q(z,ω),
where P(z,ω)=P(z,ω(z),ω(z+c1),…,ω(z+cn)),c1,…,cn∈ℂ and obtained the following result.Theorem 1 (see [20]).
Letω(z) be a finite-order meromorphic solution of (1), where P(z,ω) is a homogeneous difference polynomial with meromorphic coefficients and H(z,ω) and Q(z,ω) are polynomials in ω(z) with meromorphic coefficients having no common factors. If max{degω(H),degω(Q)-degω(P)}>min{degω(P),ord0(Q)}-ord0(P), then N(r,ω)≠S(r,ω), where ord0(P) denotes the order of zero of P(z,x0,x1,…,xn) at x0=0 with respect to the variable x0.Letcj∈ℂ for j=1,…,n, and let I be a finite set of multi-indexes λ=(λ0,…,λn). Then a difference polynomial of a meromorphic function ω(z) is defined as
(2)P(z,ω)=P(z,ω(z),ω(z+c1),…,ω(z+cn))=∑λ∈I‍cλ(z)w(z)λ0w(z+c1)λ1⋯ω(z+cn)λn,
where the coefficients cλ(z) are small with respect to ω(z) in the sense that T(r,cλ)=o(T(r,ω)) as r tends to infinity outside of an exceptional set E of finite logarithmic measure.At the same year, Zheng and Chen [21] consider the value distribution of meromorphic solutions of zero order of a kind of q-difference equations and obtained the following result which is an extension of Theorem 1.Theorem 2 (see [21, Theorem 1]).
Suppose thatf is a nonconstant meromorphic solution of zero order of a q-difference equation of the form
(3)∑λ∈I‍cλ(z)f(qz)iλ,1f(q2z)iλ,2⋯f(qnz)iλ,n=P(z,f(z))Q(z,f(z))=(ak(z)(f(z))k+ak+1(z)(f(z))k+1+⋯+as(z)(f(z))sak(z)(f(z))k+ak+1(z)(f(z))k+1)×(b0(z)+b1(z)f(z)+⋯+bt(z)(f(z))t)-1,
where I={(iλ1,iλ2,…,iλn)} is a finite index set and iλ1+iλ2+⋯+iλn=σ>0 for all λ∈I and q(≠0,1)∈ℂ. Moreover, suppose that 0≤k≤s,ak(z)as(z)bt(z)≢0, the P(z,f) and Q(z,f) have no common factors, and that all meromorphic coefficients in (3) are of growth of o(T(r,f)) on a set of logarithmic density 1. If
(4)max{t,s-σ}>min{σ,k},
then
(5)N(r,f)≠o(T(r,f))
on any set of logarithmic density 1.Remark 3.
The logarithmic density of a setF is defined by
(6)limsupr→∞1logr∫[1,r]∩F‍1tdt.Recently, Gao [22–24] and others [25, 26] also investigated the growth and existence of meromorphic solutions of some systems of complex difference equations; one system of complex difference equation is based on (1) and obtained some interesting results.Inspired by the idea of [21–24, 27], we will investigate the properties of meromorphic solutions of systems of a class of complex q-shift difference equations of the form
(7)Ω1(z,w1,w2)=R1(z,w1),Ω2(z,w1,w2)=R2(z,w2),
where q(≠0,1),cj(j=1,…,n)∈ℂ,I,J are two finite sets of multi-indexes (i1,…,in), (j1,…,jn), and Ω1(z,w1,w2),Ω2(z,w1,w2) are two homogeneous difference polynomials to be defined as
(8)Ω1(z,w1,w2)=Ω1(z,w1(qz+c1),w2(qz+c1),…,w1(qnz+cn),w2(qnz+cn))=∑(i)‍a(i)(z)∏k=12‍(wk(qz+c1))ik1⋯(wk(qnz+cn))ikn,Ω2(z,w1,w2)=Ω2(z,w1(qz+c1),w2(qz+c1),…,w1(qnz+cn),w2(qnz+cn))=∑(j)‍b(j)(z)∏k=12‍(wk(qz+c1))jk1⋯(wk(qnz+cn))jkn.
The coefficients {a(i)},{b(j)} are small with respect to w1, w2 in the sense that T(r,a(i))=o(T(r,wl)), T(r,b(j))=o(T(r,wl)), l=1,2, as r tends to infinity outside of an exceptional set E of finite logarithmic measure. The weights of Ω1(z,w1,w2), Ω2(z,w1,w2) are defined by
(9)σ11=max(i){∑l=1n‍i1l},σ12=max(i){∑l=1n‍i2l},σ21=max(j){∑l=1n‍j1l},σ22=max(j){∑l=1n‍j2l},R1(z,w1)=P1(z,w1)Q1(z,w1)=(ck11(z)(w1(z))k1+ck1+11(z)(w1(z))k1+1+⋯+cs11(z)(w1(z))s1(w1(z))k1+1)×((z)(w1(z))t1d01(z)+d11(z)w1(z)+⋯,+dt11(z)(w1(z))t1)-1,R2(z,w2)=P2(z,w2)Q2(z,w2)=(ck22(z)(w2(z))k2+ck2+12(z)(w2(z))k2+1+⋯+cs2(z)(w2(z))s2(w1(z))k1+1)×((w2(z))t2d02(z)+d12(z)w2(z)+⋯+dt22(z)(w2(z))t2)-1.
The coefficients {ckii(z)}, {dtii(z)} are meromorphic functions and small functions,
(10)S(r)=∑‍T(r,a(i))+∑‍T(r,b(j))+∑‍T(r,ckii)+∑‍T(r,dtii).Now, we will show our main results as follows.Theorem 4.
Let(w1,w2) be meromorphic solution of systems (7) satisfying ρ=ρ(w1,w2)=0. Moreover, suppose that 0≤ki≤si,ck1i(z)cs1i(z)dtii(z)≢0,i=1,2, the Pi(z,wi) and Qi(z,wi) are polynomials in wi(z) with meromorphic coefficients having no common factors, and that all meromorphic coefficients in (7) are of growth of o(T(r,f)) for all r on a set of logarithmic density 1 or outside of an exceptional set of logarithmic density 0. If
(11)max{t1,s1-σ11}>min{σ11,k1}+σ11+σ12,max{t2,s2-σ22}>min{σ22,k2}+σ22+σ21,
then N(r,w1)=o(T(r,w1)) and N(r,w2)=o(T(r,w2)) cannot hold both at the same time, for all r possibly outside of an exceptional set of logarithmic density 0, where the order of meromorphic solution (w1,w2) of systems (7) is defined by
(12)ρ=ρ(w1,w2)=max{ρ(w1),ρ(w2)},ρ(wi)=limsupr→∞logT(r,wi)logr,i=1,2.Theorem 5.
Let(w1,w2) be meromorphic solution of systems (7) satisfying ρ=ρ(w1, w2)=0. Moreover, suppose that 0≤ki≤si,cki(z)csi(z)dti(z)≢0,i=1,2, the Pi(z,wi) and Qi(z,wi) are polynomials in wi(z) with meromorphic coefficients having no common factors, and that all meromorphic coefficients in (7) are of growth of o(T(r,f)) for all r on a set of logarithmic density 1 or outside of an exceptional set of logarithmic density 0, and
(13)A=2σ11-max{s1,t1+σ11}+min{σ11,k1},B=2σ22-max{s2,t2+σ22}+min{σ22,k2}.
If
(14)A<0,B<0,AB>9σ21σ12,
then m(r,wk)=o(T(r,wk)), k=1,2 hold for r that runs to infinity possibly outside of an exceptional set of logarithmic density 0.


## 2. Some Lemmas

Lemma 6 (Valiron-Mohon’ko) ([28]).
Letf(z) be a meromorphic function. Then for all irreducible rational functions in f,
(15)R(z,f(z))=∑i=0mai(z)f(z)i∑j=0nbj(z)f(z)j,
with meromorphic coefficients ai(z), bj(z), the characteristic function of R(z,f(z)) satisfies that
(16)T(r,R(z,f(z)))=dT(r,f)+O(Ψ(r)),
where d=max{m,n} and Ψ(r)=maxi,j{T(r,ai),T(r,bj)}.Lemma 7 (see [27]).
Letf(z) be a nonconstant zero-order meromorphic function and q∈ℂ∖{0}. Then
(17)m(r,f(qz+η)f(z))=o(T(r,f))=S(r,f),
on a set of logarithmic density 1 or outside of an exceptional set of logarithmic density 0.Lemma 8 (see [29]).
Letf(z) be a transcendental meromorphic function of zero order, and let q, η be two nonzero complex constants. Then
(18)T(r,f(qz+η))=T(r,f(z))+S(r,f),N(r,f(qz+η))≤N(r,f)+S(r,f),
on a set of logarithmic density 1 or outside of a possibly exceptional set of logarithmic density 0.


## 3. The Proof of Theorem4

From the definitions ofΩi(z,w1,w2), by Lemma 7, it follows that
(19)m(r,Ω1(z,w1,w2)w1σ11)≤σ12m(r,w2)+o(T(r,w1)),r∉E1′,(20)m(r,Ω2(z,w1,w2)w2σ22)≤σ21m(r,w1)+o(T(r,w2)),r∉E2′,
where E1′, E2′ are two sets of logarithmic density 0. By Lemma 6, we have
(21)T(r,Ω1(z,w1,w2)w1σ11)=T(r,P1(z,w1)Q1(z,w1)w1σ11)=(max{t1+σ11,s1}-min{σ11,k1})×T(r,w1)+o(T(r,w1)),r∉E3′,(22)T(r,Ω2(z,w1,w2)w2σ22)=T(r,P2(z,w2)Q2(z,w2)w2σ22)=(max{t2+σ22,s2}-min{σ22,k2})×T(r,w2)+o(T(r,w2)),r∉E4′,
where E3′, E4′ are two sets of logarithmic density 0. Thus, from the assumptions of Theorem 4, combining (19) and (21), (20) and (22), respectively, we have
(23)N(r,Ω1(z,w1,w2)w1σ11)≥(1+σ12+σ11)T(r,w1)-σ12m(r,w2)+o(T(r,w1)),r∉E1=E1′∪E3′,N(r,Ω2(z,w1,w2)w2σ22)≥(1+σ21+σ22)T(r,w1)-σ21m(r,w1)+o(T(r,w2)),r∉E2=E2′∪E4′.Sinceρ=ρ(w1,w2)=0, from Lemma 8, we have
(24)N(r,Ω1(z,w1,w2)w1σ11)≤N(r,Ω1(z,w1,w2))+σ11N(r,1w1)≤σ11N(r,w1)+σ12N(r,w2)+σ11N(r,1w1)+o(T(r,w1))+o(T(r,w2)),r∉E5′,N(r,Ω2(z,w1,w2)w2σ22)≤N(r,Ω2(z,w1,w2))+σ22N(r,1w2)≤σ22N(r,w2)+σ21N(r,w1)+σ22N(r,1w2)+o(T(r,w1))+o(T(r,w2)),r∉E6′,
where E5′, E6′ are the sets of logarithmic density 0.From (23) and (24), it follows that
(25)(1+σ12+σ11)T(r,w1)≤σ11N(r,w1)+σ12N(r,w2)+σ11N(r,1w1)+σ12m(r,w2)+o(T(r,w1))+o(T(r,w2))≤σ11N(r,w1)+σ12T(r,w2)+σ11T(r,w1)+o(T(r,w1))+o(T(r,w2)),r∉E3=E1∪E5′,(1+σ21+σ22)T(r,w1)≤σ22N(r,w2)+σ21N(r,w1)+σ22N(r,1w2)+σ21m(r,w1)+o(T(r,w1))+o(T(r,w2))≤σ22N(r,w2)+σ21T(r,w1)+σ22T(r,w2)+o(T(r,w1))+o(T(r,w2)),r∉E4=E2∪E6′.Suppose now on the contrary to the assertion of Theorem4 that N(r,w1)=o(T(r,w1)) and N(r,w2)=o(T(r,w2)), from (25); it follows that
(26)(1+σ12)T(r,w1)≤σ12T(r,w2)+o(T(r,w1))+o(T(r,w2)),(1+σ21)T(r,w2)≤σ21T(r,w1)+o(T(r,w1))+o(T(r,w2)),
that is,
(27)(1+σ12+o(1))T(r,w1)≤(σ12+o(1))T(r,w2),(1+σ21+o(1))T(r,w2)≤(σ21+o(1))T(r,w1).
From (27), we can get that
(28)(1+σ12)(1+σ21)≤σ12σ21.
From the previous inequality, we can get a contradiction.Therefore, this completes the proof of Theorem4.


## 4. The Proof of Theorem5

Sinceρ=ρ(w1,w2)=0, from the assumptions concerning the coefficients of systems (7), by Lemma 7, and from the definitions of logarithmic measure and logarithmic density, we have
(29)N(r,Ω1(z,w1,w2)w1σ11)≤σ11[N(r,w1)+N(r,1w1)]+σ12[N(r,w2)+N(r,1w2)]+σ12N(r,w2)+o(T(r,w1))+o(T(r,w2)),r∉E5,
where E5 is a set of logarithmic density 0.From (29), we have
(30)N(r,Ω1(z,w1,w2)w1σ11)≤σ11[N(r,w1)+N(r,1w1)]+σ12[2N(r,w2)+N(r,1w2)]+o(T(r,w1))+o(T(r,w2))≤σ11[2T(r,w1)-m(r,w1)]+σ12[3T(r,w2)-2m(r,w2)]+o(T(r,w1))+o(T(r,w2)),r∉E5.From (19) and (29), we have
(31)N(r,Ω1(z,w1,w2)w1σ11)+σ12m(r,w2)≥N(r,Ω1(z,w1,w2)w1σ11)+m(r,Ω1(z,w1,w2)w1σ11)=T(r,P1(z,w1)Q1(z,w1)w1σ11)=(max{t1+σ11,s1}-min{σ11,k1})×T(r,w1)+o(T(r,w1)),r∉F1=E1′∪E5.
From the previous inequality and (30), we have for r∉F1(32)(max{t1+σ11,s1}-min{σ11,k1})T(r,w1)-σ12m(r,w2)≤σ11[2T(r,w1)-m(r,w1)]+σ12[3T(r,w2)-2m(r,w2)]+o(T(r,w1))+o(T(r,w2)).By using the same argument as in the previously mentioned, there exists a setF2 of logarithmic density 0, for r∉F2, and we have
(33)(max{t2+σ22,s2}-min{σ22,k2})T(r,w2)-σ21m(r,w1)≤σ22[2T(r,w2)-m(r,w2)]+σ21[3T(r,w1)-2m(r,w1)]+o(T(r,w1))+o(T(r,w2)).
From (32) and (33), we have
(34)σ11m(r,w1)≤[2σ11-(max{t1+σ11,s1}-min{σ11,k1})+o(1)]T(r,w1)+(3σ12+o(1))T(r,w2),r∉F1,[(max{t2+σ22,s2}-min{σ22,k2})-2σ22+o(1)]T(r,w2)≤(3σ21+o(1))T(r,w1)-σ21m(r,w1),r∉F2.
From (34), we have
(35)σ11m(r,w1)≤[2σ11-(max{t1+σ11,s1}-min{σ11,k1})+o(1)σ11-(max{t1+σ11,s1}-min{σ11,k1})]T(r,w1)+((3σ12+o(1))×[(3σ21+o(1))T(r,w1)-σ21m(r,w1)](3σ12+o(1)))×((max{t2+σ22,s2}-min{σ22,k2})-2σ22)-1,r∉F=F1∪F2,
that is,
(36)(σ11-3σ12σ21B)m(r,w1)≤[A-9σ12σ21+o(1)B]T(r,w1),r∉F=F1∪F2,
where A=2σ11-max{s1,t1+σ11}+min{σ11,k1} and B=2σ22-max{s2,t2+σ22}+min{σ22,k2}. From (14) and (36), we have
(37)m(r,w1)=o(T(r,w1))
for all r outside of F, a set of logarithmic density 0.Similarly, we can obtain(38)m(r,w2)=o(T(r,w2)
for all r possibly outside of F′, a set of logarithmic density 0.Thus, this completes the proof of Theorem5.



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