On Malmquist type theorem of systems of complex difference equations

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R E S E A R C H

Open Access

On Malmquist type theorem of systems of

complex difference equations

Jianjun Zhang

*

*_{Correspondence:}

[email protected] Mathematics and Information Technology School, Jiangsu Second Normal University, Nanjing, Jiangsu 210013, P.R. China

Abstract

The main purpose of this paper is to give the Malmquist type result of the

meromorphic solutions of a system of complex diﬀerence equations of the following form:

⎧ ⎪ ⎨ ⎪ ⎩

λ1∈I1,μ1∈J1

α

λ1,μ1(z)(

n

ν=1f(z+cν)

lλ1,νn

ν=1g(z+cν)mμ1,ν) =

p i=0ai(z)g(z)i

q j=0bj(z)g(z)j,

λ2∈I2,μ2∈J2

β

λ2,μ2(z)(

n

ν=1f(z+cν)

lλ2,νn

ν=1g(z+cν)mμ2,ν) =

s

k=0dk(z)f(z)k

t

l=0el(z)f(z)l ,

wherec1,c2,. . .,cnare distinct, nonzero complex numbers, the coeﬃcients

α

λ1,μ1(z)

(λ1∈I1,

μ

1∈J1),

β

λ2,μ2(z) (λ2∈I2,

μ

2∈J2),ai(z) (i= 0, 1,. . .,p),bj(z) (j= 0, 1,. . .,q),

dk(z) (k= 0, 1,. . .,s), andel(z) (l= 0, 1,. . .,t) are small functions relative tof(z) andg(z),

Ii={

λ

i= (lλi,1,lλi,2,. . .,lλi,n)|lλi,ν∈N∪ {0},

ν

= 1, 2,. . .,n}(i= 1, 2) and

Jj={

μ

j= (mμj,1,mμj,2,. . .,mμj,n)|mμj,ν∈N∪ {0},

ν

= 1, 2,. . .,n}(j= 1, 2) are ﬁnite index

sets. The growth of meromorphic solutions of a related system of complex functional equations is also investigated.

Keywords: systems of complex diﬀerence equations; meromorphic functions;

Malmquist type theorem; functional equation

1 Introduction and main results

Letf(z) be a meromorphic function in the complex planeC. We assume that the reader is familiar with the standard notations and results in Nevanlinna’s value distribution theory of meromorphic functions (seee.g.[–]). We useρ(f) to denote the growth order of a meromorphic functionf(z). The notationS(r,f) denotes any quantity that satisﬁes the conditionS(r,f) =o(T(r,f)) asr→ ∞possibly outside an exceptional set of rof ﬁnite logarithmic measure. A meromorphic functiona(z) is called a small function off(z) if and only ifT(r,a(z)) =S(r,f).

In the last ten years, there has been a great deal of interest in studying the properties of complex diﬀerence equations (seee.g.[–]). Especially, a number of papers (seee.g.

[, , , , , –]) focusing on a Malmquist type theorem of the complex diﬀerence equations emerged. In , Ablowitzet al.[] proved some results on the Malmquist the-orem of the complex diﬀerence equations by utilizing Nevanlinna theory. They obtained the following two results.

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Theorem A If the second-order diﬀerence equation

f(z+ ) +f(z– ) = a(z) +a(z)f+· · ·+ap(z)f

p

b(z) +b(z)f+· · ·+bq(z)fq

,

with polynomial coeﬃcients ai(i= , , . . . ,p)and bj(j= , , . . . ,q),admits a transcendental meromorphic solution of ﬁnite order,then d=max{p,q} ≤.

Theorem B If the second-order diﬀerence equation

f(z+ )f(z– ) =a(z) +a(z)f+· · ·+ap(z)f

p

b(z) +b(z)f+· · ·+bq(z)fq

,

with polynomial coeﬃcients ai(i= , , . . . ,p)and bj(j= , , . . . ,q),admits a transcendental meromorphic solution of ﬁnite order,then d=max{p,q} ≤.

Subsequently, Heittokangaset al.[], Laineet al.[] and Huanget al.[], respectively, gave some generalizations of the above two results. In , the ﬁrst author in this paper and Liao [] obtained the following more general result.

Theorem C Let c,c, . . . ,cnbe distinct,nonzero complex numbers,and suppose that f(z) is a transcendental meromorphic solution of the diﬀerence equation

λ∈I

αλ(z)

n

ν=

f(z+cν)lλ,ν

=a(z) +a(z)f(z) +· · ·+ap(z)f(z)

p

b(z) +b(z)f(z) +· · ·+bq(z)f(z)q

, ()

with coeﬃcientsαλ(z) (λ∈I),ai(z) (i= , , . . . ,p),and bj(z) (j= , , . . . ,q),which are small functions relative to f(z),where I={λ= (lλ,,lλ,, . . . ,lλ,n)|lλ,ν∈N∪ {},ν= , , . . . ,n}is a

ﬁnite index set,and denote

σν=max

λ {lλ,ν} (ν= , , . . . ,n), σ=

n

ν=

σν.

If the orderρ(f)is ﬁnite,then d=max{p,q} ≤σ.

If all the coefficients in the complex difference equation () are rational functions, then in [], we have the following Malmquist type result, which is reminiscent of the classical Malmquist theorem in complex differential equations.

Theorem D Let c,c, . . . ,cnbe distinct,nonzero complex numbers and suppose that f(z) is a transcendental meromorphic solution of the equation

P[z,f] := λ∈I

αλ(z)

n

ν=

f(z+cν)lλ,ν

=Rz,f(z)= P(z,f(z))

Q(z,f(z)),

where I={λ= (lλ,,lλ,, . . . ,lλ,n)|lλ,ν∈N∪ {},ν= , , . . . ,n}is a ﬁnite index set,P and Q

are relatively prime polynomials in f over the ﬁeld of rational functions,the coeﬃcientsαλ (λ∈I)are rational functions.Denoting the degree of P[z,f]by

γP:=max

λ∈I

lλ,+lλ,+· · ·+lλ,n|λ= (lλ,,lλ,, . . . ,lλ,n)

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If f(z)is ﬁnite order and has at most ﬁnitely many poles,then R(z,f)reduces to a polynomial in f of degree d≤γP.

More recently, people began to study the properties of meromorphic solutions of sys-tems of complex diﬀerence equations. In [], Gao discussed the proximity function and counting function of meromorphic solutions of some classes of systems of complex dif-ference equations. In , Wanget al.[] investigated the growth of meromorphic so-lutions of systems of complex diﬀerence equations.

Now, we give the Malmquist type result of a system of complex diﬀerence equations as follows.

Theorem  Let c,c, . . . ,cn be distinct, nonzero complex numbers and suppose that

(f(z),g(z))is a transcendental meromorphic solution of a system of complex diﬀerence equa-tions of the form

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Remark  Obviously, if the conditionmax{p,q}>σ,max{s,t}>σin Theorem  is

re-placed by (max{p,q}–σ)(max{s,t}–σ) =  or (max{p,q}–σ)(max{s,t}–σ) < , the

estimation (max{p,q}–σ)·(max{s,t}–σ)≤σσis still correct. Ifσ=  orσ= ,

then the ﬁrst or second equation in () gets the form of (). For some results as regards (), the reader may refer to the paper [].

Remark  If the condition max{p,q}>σ,max{s,t}>σin Theorem  is replaced by

max{p,q}<σ,max{s,t}<σ, then the estimation (max{p,q}–σ)·(max{s,t}–σ)≤

σσis not true generally. For example, (f(z),g(z)) = (tanz,cotz) satisﬁes the following

system of complex diﬀerence equations:

⎧ ⎨ ⎩

f(z+π_)g(z–π_)g(z+π_)+ g(z+π_)=_gg–_+g_g+_+,

f(z+π_)_f₍_z_–π

)g(z+

π

) +zf(z+

π

) =

–(z+)f–f+z+

f_–_f_+ ,

()

where max{p,q} = , max{s,t}= , σ = , σ = , σ = , σ= , max{p,q} <σ,

max{s,t}<σ. However, (max{p,q}–σ)·(max{s,t}–σ) = (–)×(–) =  >  =σσ.

Ifσ=σ= , then we have the following simpler result.

Corollary  Let c,c, . . . ,cn be distinct, nonzero complex numbers, and suppose that

(f(z),g(z))is a transcendental meromorphic solution of a system of complex diﬀerence equa-tions of the form

⎧ ⎪ ⎨ ⎪ ⎩

λ∈Iαλ(z)(

n

ν=f(z+cν)lλ,ν) =

p i=ai(z)g(z)i

q

j=bj(z)g(z)j,

μ∈Jβμ(z)(

n

ν=g(z+cν)mμ,ν) =

s

k=dk(z)f(z)k

t

l=el(z)f(z)l ,

()

with coeﬃcients αλ(z) (λ∈I), βμ(z) (μ∈J), ai(z) (i= , , . . . ,p), bj(z) (j= , , . . . ,q), dk(z) (k= , , . . . ,s),and el(z) (l= , , . . . ,t)are small functions relative to f(z)and g(z), ap(z),bq(z),ds(z),et(z)≡,where I={λ= (lλ,,lλ,, . . . ,lλ,n)|lλ,ν∈N∪{},ν= , , . . . ,n}and

J={μ= (mμ,,mμ,, . . . ,mμ,n)|mμ,ν∈N∪ {},ν= , , . . . ,n}are two ﬁnite index sets,and

denote

ξν=max

λ {lλ,ν} (ν= , , . . . ,n), σ=

n

ν=

ξν

and

ην=max

μ {mμ,ν} (ν= , , . . . ,n), σ=

n

ν=

ην.

Ifρ(f) < +∞orρ(g) < +∞,thenρ(f) =ρ(g)andmax{p,q} ·max{s,t} ≤σσ.

Example  Letc=arctan,c=arctan(–). It is easy to check that (f(z),g(z)) = (tanz,

cotz) satisﬁes the following system of diﬀerence equations:

⎧ ⎨ ⎩

f(z+c)f(z+c) +f(z+c)f(z+c)=–g

_+_g g_–_g_+, g(z+c)g(z+c) +g(z+c)= –f

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In Example , we have max{p,q}= ,max{s,t}= ,σ = ,σ = ,ρ(f) =ρ(g) =  <

+∞, andmax{p,q} ·max{s,t}=σ·σ = . Therefore, the estimation in Corollary  is

sharp.

In [], Laineet al.also considered the growth of meromorphic solutions of some classes of complex diﬀerence functional equations and obtained the following result.

Theorem E Suppose that f is a transcendental meromorphic solution of the equation

{J} αJ(z)

j∈J

f(z+cj)

=fp(z),

where p(z)is a polynomial of degree k≥,{J}is the collection of all subsets of{, , . . . ,n}. Moreover,we assume that the coeﬃcientsαJ(z)are small functions relative to f and that n≥k.Then

T(r,f) =O(logr)α+ε_,

whereα=log_logn_k.

In , Zhanget al.[] got a more generalized result than Theorem E. Next we give the growth of meromorphic solutions of a system of complex functional equations as fol-lows.

Theorem  Let c,c, . . . ,cn be distinct, nonzero complex numbers and suppose that

(f(z),g(z))is a transcendental meromorphic solution of a system of complex functional equations of the form

λ∈I,μ∈Jαλ,μ(z)( n

ν=f(z+cν)lλ,ν

n

ν=g(z+cν)mμ,ν) =f(p(z)),

λ∈I,μ∈Jβλ,μ(z)( n

ν=f(z+cν)lλ,ν

n

ν=g(z+cν)mμ,ν) =g(p(z)),

()

where p(z)is a polynomial of degree k≥,Ii={λi= (lλi,,lλi,, . . . ,lλi,n)|lλi,ν∈N∪ {},ν=

, , . . . ,n}(i= , )and Jj={μj= (mμj,,mμj,, . . . ,mμj,n)|mμj,ν∈N∪ {},ν= , , . . . ,n}(j= , )are ﬁnite index sets,and denote

ξ,ν=max λ∈I{

lλ,ν}, η,ν=max μ∈J{

mμ,ν},

ξ,ν=max λ∈I{

lλ,ν}, η,ν=max

μ∈J{ mμ,ν}

(ν= , , . . . ,n),

σ=

n

ν=

ξ,ν, σ=

n

ν=

η,ν, σ=

n

ν=

ξ,ν, σ=

n

ν=

η,ν

and

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Moreover,we assume that the coeﬃcientsαλ,μ(z) (λ∈I,μ∈J),βλ,μ(z) (λ∈I,μ∈ J)are small functions relative to f(z)and g(z),and thatσ≥k.Then

T(r,f) =O(logr)α+ε, T(r,g) =O(logr)α+ε,

whereα=log_log_kσ.

Ifσ=σ= , then we can obtain the following result easily.

Corollary  Let c,c, . . . ,cn be distinct, nonzero complex numbers and suppose that

(f(z),g(z))is a transcendental meromorphic solution of a system of complex functional equations of the form

λ∈Iαλ(z)(

n

ν=f(z+cν)lλ,ν) =g(p(z)),

μ∈Jβμ(z)(

n

ν=g(z+cν)mμ,ν) =f(p(z)),

()

where p(z)is a polynomial of degree k≥, I={λ= (lλ,,lλ,, . . . ,lλ,n)|lλ,ν∈N∪ {},ν= , , . . . ,n}and J={μ= (mμ,,mμ,, . . . ,mμ,n)|mμ,ν∈N∪ {},ν= , , . . . ,n}are two ﬁnite

index sets,and denote

ξν=max

λ {lλ,ν} (ν= , , . . . ,n), σ=

n

ν=

ξν,

ην=max

μ {mμ,ν} (ν= , , . . . ,n), σ=

n

ν=

ην

and

σ=max{σ,σ}.

Moreover,we assume that the coeﬃcientsαλ(z) (λ∈I),βμ(z) (μ∈J)are small functions

relative to f(z)and g(z),and thatσ≥k.Then

T(r,f) =O(logr)α+ε_, _T₍_r_,_g_{) =}_O_(log_r₎α+ε_,

whereα=log_log_kσ.

2 Some lemmas

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

Lemma (see []) Let f(z)be a meromorphic function.Then for all irreducible rational functions in f,

R(z,f) = P(z,f)

Q(z,f)=

p i=ai(z)fi

q j=bj(z)fj

,

such that the meromorphic coeﬃcients ai(z),bj(z)satisfy

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we have

Tr,R(z,f)=max{p,q} ·T(r,f) +S(r,f).

In [], AZ Mokhon’ko and VD Mokhon’ko gave an estimation of Nevanlinna’s charac-teristic function of are two ﬁnite index sets. However, the method of the proof was too complex. ForF(z) of the form λ∈If

mation ofT(r,F) was not sharp. For completeness, we give the proof of the following lemma.

n are generated by the poles of the

functionsfj(j= , , . . . ,n), and every pole of multiplicitykoffj(j= , , . . . ,n) has order at

mostkσj. This implies that

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forj= , , . . . ,n. Thus we have

By the deﬁnition ofm(r,f), we immediately conclude that

m

By () and (), the assertion follows.

Remark  If we suppose thatαλ(z) =o(T(r,fj)) (λ∈I) hold for all j∈ {, , . . . ,n}, and

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3 Proofs of theorems

Proof of Theorem We assume that (f(z),g(z)) is a transcendental meromorphic solution of the system of complex diﬀerence equations (). By the ﬁrst equation in (), Lemma , Lemma , and Lemma , we have, for eachε> ,

By the above inequality, we get, for eachε> ,

Similarly, by the second equation in (), we obtain, for eachε> ,

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+Orρ(g)–+ε+O(logr) +S(r,f) +S(r,g)

=

n

ν=

ξ,ν

Tr,f(z)+

n

ν=

η,ν

Tr,g(z)

+O(logr) +S(r,f) +S(r,g)

=σT

r,f(z)+σT

r,g(z)+Orρ(f)–+ε+Orρ(g)–+ε

+O(logr) +S(r,f) +S(r,g). ()

By () andmax{s,t}>σ, we have, for eachε> ,

max{s,t}–σ

T(r,f)

≤σT

r,g(z)+Orρ(f)–+ε+Orρ(g)–+ε

+O(logr) +S(r,f) +S(r,g) ()

and

T(r,f)≤ σ

max{s,t}–σ

Tr,g(z)+Orρ(f)–+ε+Orρ(g)–+ε

+O(logr) +S(r,f) +S(r,g). ()

Using (), we can obtainρ(g)≤ρ(f). Similarly, we can getρ(f)≤ρ(g) from (). There-fore, we haveρ(f) =ρ(g).

It follows from () and () that

max{p,q}–σ

max{s,t}–σ

T(r,f)T(r,g)

≤σσT

r,f(z)Tr,g(z)+oT(r,f)T(r,g). ()

From (), we conclude that

max{p,q}–σ

·max{s,t}–σ

≤σσ.

This yields the asserted result.

Proof of Theorem We assume that (f(z),g(z)) is a transcendental meromorphic solution of a system of complex functional equations (). LetC=max{|c|,|c|, . . . ,|cn|}. According

to the ﬁrst equation in (), Lemma , Lemma , and the last assertion of Lemma , we get

( –ε)Tμrk,f

≤Tr,fp(z)

=T r,

λ∈I,μ∈J

αλ,μ(z) n

ν=

f(z+cν)lλ,ν

n

ν=

g(z+cν)mμ,ν

≤

n

ν=

ξ,νT

r,f(z+cν)

+

n

ν=

η,νT

r,g(z+cν)

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≤ may assumerto be large enough to satisfy

( –ε)Tμrk,f≤σ( +ε)T(βr,f) +σ( +ε)T(βr,g)

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Lettings=logt+

Similarly, from (), we can get

(ks,g)≤M(s,f) +M(s,g). ()

The inequalities () and () hold for allslarge enough. Letting nowα=log_logM_k , namely, M=kα_{. Write}₍_s_,_f_{) =} (s,f)

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Similarly, we have

sup

s∈[k_s ,ks]

(s,f) = sup

s∈[ks,ks]

(ks,f)≤  M+

 M,

sup

s∈[k_s ,ks]

(s,g) = sup

s∈[ks,ks]

(ks,g)≤ M+

 M,

. . . .

Thus, we deduce that

sup

s≥ks

(s,f)≤  M+



M< +∞,

sup

s≥ks

(s,g)≤  M+



M< +∞.

Therefore,(s,f) and(s,g) are bounded for alls≥s. There exist some constantsK,

K,K,K, such that, for anyε> ,

T(t,f) =(s,f) =(s,f)sα≤Ksα=K

logt+ log μ

(γ β)k k– 

α

≤K(logt)α+ε

and

T(t,g) =(s,g) =(s,g)sα_≤_K

sα=K

logt+ log μ

(γ β)k k– 

α

≤K(logt)α+ε.

Therefore, we have

T(r,f) =O(logr)α+ε

and

T(r,g) =O(logr)α+ε,

where

α=logM logk =

logσ

logk +o().

Letting nowα=log_log_kσ, we obtain the required form. Theorem  is proved.

Competing interests

The author declares that there is no conﬂict of interests regarding the publication of the article.

Acknowledgements

The author would like to thank Professor Liangwen Liao and the anonymous referees for their valuable comments and suggestions. The research was supported by NSF of China (11426118), Natural Science Foundation of Jiangsu Province (BK20140767), Natural Science Foundation of the Jiangsu Higher Education Institutions (14KJB110004), Qing Lan Project of Jiangsu Province and Colonel-level topics (JSNU-ZY-01).

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