• No results found

Results on the growth of meromorphic solutions of some linear difference equations with meromorphic coefficients

N/A
N/A
Protected

Academic year: 2020

Share "Results on the growth of meromorphic solutions of some linear difference equations with meromorphic coefficients"

Copied!
13
0
0

Loading.... (view fulltext now)

Full text

(1)

R E S E A R C H

Open Access

Results on the growth of meromorphic

solutions of some linear difference equations

with meromorphic coefficients

Zhang Li Yuan and Qiu Ling

*

*Correspondence:

qiuling1978@bjut.edu.cn College of Applied Science, Beijing University of Technology, Beijing, 100124, China

Abstract

In this paper, we investigate the growth of meromorphic solutions of some linear difference equations. We obtain some new results on the growth of meromorphic solutions when most coefficients in such equations are meromorphic functions, which are supplements of previous results due to Li and Chen (Adv. Differ. Equ. 2012:203, 2012) and Liu and Mao (Adv. Differ. Equ. 2013:133, 2013).

MSC: 30D35; 39A10

Keywords: complex difference equation; meromorphic coefficients; growth

1 Introduction and main results

In this article, a meromorphic function always means meromorphic in the whole complex planeC, andcalways means a nonzero constant. We adopt the standard notations of the Nevanlinna value distribution theory of meromorphic functions such asT(r,f),m(r,f) andN(r,f) as explained in [–]. In addition, we will use notationsρ(f) to denote the order of growth of a meromorphic functionf(z),λ(f) to denote the exponents of convergence of the zero sequence of a meromorphic functionf(z), λ(

f) to denote the exponents of

convergence of the pole sequence of a meromorphic functionf(z), and we define them as follows:

ρ(f) =lim sup

r→∞

logT(r,f)

logr ,

λ(f) =lim inf

r→∞

logN(r,f)

logr ,

λ

f

=lim inf

r→∞

logN(r,f)

logr .

Recently, meromorphic solutions of complex difference equations have become a sub-ject of great interest from the viewpoint of Nevanlinna theory due to the apparent role of the existence of such solutions of finite order for the integrability of discrete difference equations.

About the growth of meromorphic solutions of some linear difference equations, some results can be found in [–]. Laine and Yang [] considered the entire functions coef-ficients case and got the following.

(2)

Theorem A[] Let Aj(z) (≡) (j= , , . . . ,n)be entire functions of finite order such that

among those coefficients having the maximal orderρ:=max≤jnρ(Aj),exactly one has its

type strictly greater than the others.Then,for any meromorphic solution f(z)to

An(z)f(z+n) +· · ·+A(z)f(z+ ) +A(z)f(z) = ,

we haveρ(f)≥ρ+ .

Chiang and Feng [, ] improved Theorem A as follows.

Theorem B[, ] Let Aj(z) (≡) (j= , , . . . ,n)be entire functions such that there exists

an integer l, ≤ln,such that

ρ(Al) > max

≤jn,j=lρ(Aj).

Suppose that f(z)is a meromorphic solution to

An(z)f(z+n) +· · ·+A(z)f(z+ ) +A(z)f(z) = ,

then we haveρ(f)≥ρ(Al) + .

Recently in [], Peng and Chen investigated the order and the hyper-order of solutions of some second-order linear differential equations and proved the following results.

Theorem C[] Suppose that Aj(z) (≡) (j= , )are entire functions andρ(Aj) < .Let

α,αbe two distinct complex numbers such that αα=  ( suppose that|α| ≤ |α|).If

argα=πorα< –,then every solution f(z)≡of the equation

f+ezf+Az+Az

f = 

has infinite order andρ(f) = .

Moreover, Xu and Zhang [] extended the above result from entire coefficients to mero-morphic coefficients.

It is well known that f(z) =f(z+c) –f(z) is regarded as the difference counterpart off. Thus a natural question is: Can we change the above second-order linear differen-tial equation to the linear difference equation? What conditions will guarantee that every meromorphic solution will have infinite order when most coefficients in such equations are meromorphic functions?

Li and Chen [] considered the following difference equation and obtained the follow-ing theorem.

Theorem D[] Let k be a positive integer, p be a nonzero real number and f(z)be a nonconstant meromorphic solution of the difference equation

An(z)f(z+n) +· · ·+A(z)f(z+ ) +

(3)

where Aj(z),B(z),B(z) (≡) (j= , , . . . ,n)are all entire functions andmax{ρ(Aj),ρ(B),

ρ(B)}=σ<k,then we haveρ(f)≥k+ .

The main purpose of this paper is to investigate the growth of meromorphic solutions of certain linear difference equations with meromorphic coefficients. The remainder of the paper studies the properties of meromorphic solutions of a nonhomogeneous linear difference equation. In fact, we prove the following results, in which there are still some coefficients dominating in some angles.

Theorem . Let k be a positive integer.Suppose that Aj(z),B(z) (≡) (j= , , . . . ,n)are

all entire functions andmax{ρ(Aj),ρ(B) : ≤jn}=α<k.Letα,βbe two distinct

complex numbers such thatαβ=  (suppose that|α| ≤ |β|).Letαbe a strictly negative

real constant.Ifargα=π orα<α, then every meromorphic solution f(z)≡of the

equation

An(z)f(z+n) +· · ·+A(z)f(z+ ) +A(z)z k

f(z+ )

+A(z)zk+B(z)eβzkf(z) =  (.)

satisfiesρ(f)≥k+ .

Theorem . Suppose that Aj(z),B(z) (≡) (j= , , . . . ,n)are all meromorphic functions

andmax{ρ(Aj),ρ(B) : ≤jn}=α< .Letα,βbe two distinct complex numbers such

thatαβ=  (suppose that|α| ≤ |β|).Letαbe a strictly negative real constant.Ifargα= πorα<α,then every meromorphic solution f(z)≡of the equation

An(z)f(z+n) +· · ·+A(z)f(z+ ) +A(z)zf(z+ )

+A(z)z+B(z)z

f(z) =  (.)

satisfiesρ(f)≥.

Theorem . Under the assumption for the coefficients of(.)in Theorem.,if f(z)is a finite order meromorphic solution to(.),thenλ(fz) =ρ(f).What is more,either k+ ≤ ρ(f)≤max{λ(f),λ(f)}+ orρ(f) =k+  >max{λ(f),λ(f)}+ .

Theorem . Under the assumption for the coefficients of(.)in Theorem.,if f(z)is a finite order meromorphic solution to(.),thenλ(fz) =ρ(f).What is more,either≤ ρ(f)≤max{λ(f),λ(f)}+ orρ(f) =  >max{λ(f),λ(f)}+ .

Liu and Mao [] considered the meromorphic solutions of the difference equation

ak(z)f(z+k) +· · ·+a(z)f(z+ ) +a(z)f(z) = , (.)

one of their results can be stated as follows.

Theorem E [] Let aj(z) =Aj(z)ePj(z) (j= , , . . . ,k),where Pj(z) =αjnzn+· · ·+αjare

(4)

, , . . . ,k)are distinct complex numbers,then every meromorphic solution f (≡)of Eq. (.)satisfiesρ(f)≥max≤jk{ρ(aj)}+ .

In this paper, we extend and improve the above result from entire coefficients to mero-morphic coefficients in the case where the polynomialsPj(z) are of degree .

Theorem . Let aj(z) =Aj(z)eαjz(j= , , . . . ,n),αjare distinct complex constants,suppose

that Aj(z) (≡)are meromorphic functions andρ(Aj) =α< ,then every meromorphic

solution f (≡)of the equation

an(z)f(z+n) +· · ·+a(z)f(z+ ) +a(z)f(z) =  (.)

satisfiesρ(f)≥.

Theorem . Let aj(z) =Aj(z)eαjz+Dj(z) (j= , , . . . ,n),αjare distinct complex constants,

suppose that Aj(z) (≡),Dj(z) (≡)are meromorphic functions andρ(Aj) =α< ,ρ(Dj) =

β< ,then every meromorphic solution f (≡)of Eq. (.)satisfiesρ(f)≥.

Next we consider the properties of meromorphic solutions of the nonhomogeneous lin-ear difference equation corresponding to (.)

an(z)f(z+n) +· · ·+a(z)f(z+ ) +a(z)f(z) =F(z), (.) whereF(z) (≡) is a meromorphic function.

Theorem . Let aj(z) (j= , , . . . ,n)satisfy the hypothesis of Theorem.or Theorem.,

and let F(z)be a meromorphic function ofρ(F) < ,then at most one meromorphic solution fof Eq. (.)satisfies≤ρ(f)≤andmax{λ(f),λ(f)}=ρ(f),the other solutions f satisfy ρ(f)≥.

2 Some lemmas

In this section, we present some lemmas which will be needed in the sequel.

Lemma .[] Let f(z)be a meromorphic function of finite orderρ, be a positive con-stant,ηandηbe two distinct nonzero complex constants.Then

m

r,f(z+η)

f(z+η)

=Orρ–+ ,

and there exists a subset E⊂(,∞)of finite logarithmic measure such that,for all z

satis-fying|z|=r∈/[, ]∪E,and as r→ ∞,

exp––+ f(z+η)

f(z+η)

≤exp–+.

Lemma .[] Let f(z)be a meromorphic function of finite orderρ,then,for any given

> ,there exists a set E⊂(,∞)of finite logarithmic measure such that,for all z satisfying

|z|=r∈/[, ]∪E,and as r→ ∞,

(5)

Lemma .[] Suppose that P(z) = (α+iβ)zn+· · · (α,βare real numbers,|α|+|β| = )

is a polynomial with degree n≥,A(z) (≡)is an entire function withρ(A) <n.Set g(z) =

A(z)eP(z),z=re,δ(P,θ) =αcosnθβsinnθ.Then,for any given > ,there exists a set

E⊂[, π)that has linear measure zero such that for anyθ∈[, π)\(E∪E),there is

R> such that for|z|=r>R,we have

(i) ifδ(P,θ) > ,then

exp( – )δ(P,θ)rn<greiθ<exp( + )δ(P,θ)rn;

(ii) ifδ(P,θ) < ,then

exp( + )δ(P,θ)rn<greiθ<exp( – )δ(P,θ)rn,

where E={θ∈[, π) :δ(P,θ) = }is a finite set.

Lemma . applies in Theorem . whereA(z) (≡) is an entire function.

Lemma .[] Suppose that n≥is a positive integer.Let Pj(z) =ajnzn+· · · (j= , )

be nonconstant polynomials, where ajq (q= , . . . ,n)are distinct complex numbers and

anan= .Set z=reiθ,ajn=|ajn|eiθj,θj∈[–π,π),δ(Pj,θ) =|ajn|cos(θj+),then there is

a set E⊂[–πn,πn)that has linear measure zero.Ifθ=θ,then there exists a rayargz=θ, θ∈[–πn,πn)\(E∪E)such that

δ(P,θ) > , δ(P,θ) < 

or

δ(P,θ) < , δ(P,θ) > ,

where E={θ∈[–πn,πn) :δ(Pj,θ) = }is a finite set,which has linear measure zero.

In Lemma ., ifθ ∈[–πn,

π

n)\(E∪E) is replaced byθ ∈[

πn,

π

n)\(E∪E), then we

have the same result.

Lemma .[] Consider g(z) =A(z)eaz,where A(z) ()is a meromorphic function with

ρ(A) =α< ,a is a complex constant,a=|a|eiϕ[, π)).Set E

={θ∈[, π) :cos(ϕ+ θ) = },then Eis a finite set.Then,for any given ( < <  –α),there exists a set E⊂ [, π)that has linear measure zero,if z=reiθ,θ[, π)\(E

∪E),then we have(when r

is sufficiently large): (i) ifδ(az,θ) > ,then

exp( – )δ(az,θ)rgreiθ≤exp( + )δ(az,θ)r; (ii) ifδ(az,θ) < ,then

exp( + )δ(az,θ)rgreiθ≤exp( – )δ(az,θ)r,

(6)

Lemma . applies in Theorem . whereA(z) (≡) is a meromorphic function.

for allroutside of a possible exceptional setEwith finite logarithmic measure. Applying Lemma ., we have

for allroutside of a possible exceptional setEwith finite linear measure. Therefore from (.) we can get that

(7)
(8)

Since θ ∈ [–πk,πk)\(E ∪E ∪ E ∪E), we know that cos > , then |eαz k

| =

e–|α|rkcos< . Therefore, by (.) we obtain

A(z)zkexprα+ . (.)

Substituting (.), (.), (.), (.) and (.) into (.), we obtain

Mexp

( + )δαzk,θ

rknexp+ +rρ(f)–+ . (.)

Byδ(αzk,θ) > ,α+ <kandρ(f) –  + <k, we know that (.) is a contradiction. Case .α<α, which isθ=π.

Subcase .. Assume thatθ=ϕ, thenϕ=π. By Lemma ., for the above , there is a rayargz=θ such thatθ∈[–πk,πk)\(E∪E∪E∪E) (whereEandEare defined as in Lemma .,E∪E∪E∪Eis of linear measure zero) satisfyingδ(βzk,θ) > . Since

cos> , we haveδ(αzk,θ) =|α|cos(θ+kθ) = –|α|cos< . For a sufficiently larger, by Lemma ., we get

A(z)z k

<exp( – )δαzk,θ

rk< , (.)

B(z)zk>exp( – βzk,θ

rk. (.)

By (.) and (.), we get

A(z)zk+B(z)eβzkB(z)eβzkA(z)eαzk

>exp( – )δβzk,θ

rk– . (.)

Using the same reasoning as in Subcase ., we can get a contradiction.

Subcase .. Assume thatθ=ϕ, thenθ=ϕ=π. By Lemma ., for the above , there is a rayargz=θ such thatθ∈[π

k,

π

k)\(E∪E∪E∪E) (whereEandEare defined

as in Lemma .,E∪E∪E∪Eis of linear measure zero), thencos< ,δ(αzk,θ) =

|α|cos(θ+kθ) = –|α|cos> ,δ(βzk,θ) =|β|cos(ϕ+) = –|β|cos> . Since|α| ≤ |β|,α=β, andθ=ϕ, then|α|<|β|, thusδ(βzk,θ) >δ(αzk,θ) > .

For a sufficiently larger, we get (.), (.) and (.) hold.

Using the same reasoning as in Subcase ., we can get a contradiction. Thus we have ρ(f)≥k+ .

3.2 The proof of Theorem 1.2

SinceAj(z),B(z) (≡) (j= , , . . . ,n) are meromorphic functions andmax{ρ(Aj),ρ(B) :

≤jn}=α < , then when δ(αz,θ) > ,β(az,θ) <  for a sufficiently large r, by Lemma ., we have

A(z)zexp( – )δ(αz,θ)r

, (.)

B(z)z≤exp

( – )δ(βz,θ)r

< . (.)

(9)

3.3 The proof of Theorems 1.3 and 1.4

Now, for any given > , applying Lemma . and Theorem ., we can deduce that

m torization of a meromorphic function, we writef(z) as follows:

(10)

where

tion to our assumption. This completes our proof.

The proof of Theorem . is similar to that of Theorem ..

3.4 The proof of Theorem 1.5

(11)

By (.), (.) and (.), we get forz=reiθand sufficiently larger/[, ]E,

any given > , there exists a subsetE⊂(,∞) of finite logarithmic measure such that, for allzsatisfying|z|=r∈/[, ]∪E, and asr→ ∞, we have

Then, using a similar argument to that of Theorem . and Theorem . and only replacing (.) (or (.)) by (.) (or (.)), we can prove Theorem ..

3.6 The proof of Theorem 1.7

Letf(z)≡ be a meromorphic solution of (.). Suppose thatρ(f) < , then by Lemma ., we obtain ρ(F) =ρ( kj=aj(z)f(z+j)) = . This contradictsρ(F) < , therefore we have

ρ(f)≥.

(12)

linear difference equation corresponding to (.) and ρ(f–f) < . By Theorem . or Theorem ., we get a contradiction. So Eq. (.) has at most one meromorphic solution

fsatisfying ≤ρ(f)≤. Next we prove max{λ(f),λ(f

)}=ρ(f) in the caseρ(f) = . Suppose that max{λ(f), λ(f

)}<ρ(f), then by the Weierstrass factorization, we obtain

f(z) =

P(z)

P(z)e

Q(z), (.)

whereP(z),P(z) are entire functions such thatρ(P) =λ(P) =λ(f),ρ(P) =λ(P) =λ(f ) andQ(z) is a polynomial of degree .

In the caseaj(z) =Aj(z)eαjz. Substituting (.) into (.), we get n

j=

Aj(z)

P(z+j)

P(z+j)e

αjz+Q(z+j)=F(z). (.)

Sinceαjare distinct complex numbers, by Lemma ., we obtain that the order of the

left-hand side of (.) is . This contradictsρ(F) < .

Foraj(z) =Aj(z)eαjz+Dj(z), by using an argument similar to the above, we also obtain a

contradiction.

It is obvious thatmax{λ(f),λ(f

)}=ρ(f) provided that  <ρ(f) < . Therefore we have

max{λ(f),λ(f

)}=ρ(f).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

ZLY performed and drafted manuscript. All authors read and approved the final manuscript.

Acknowledgements

The author thanks the referee for his/her valuable suggestions to improve the present article. The research was supported by the Beijing Natural Science Foundation (No. 1132013) and the Foundation of Beijing University of Technology (No. 006000514313002).

Received: 9 July 2014 Accepted: 14 November 2014 Published:03 Dec 2014

References

1. Hayman, WK: Meromorphic Functions. Claredon Press, Oxford (1964)

2. Laine, l: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin (1993)

3. Yang, CC, Yi, HX: Uniqueness Theory of Meromorphic Functions. Kluwer Academic, Dordrecht (2003)

4. Zhang, J: Some results on zeros and the uniqueness of one certain type of high difference polynomials. Adv. Differ. Equ.2012, Article ID 160 (2012)

5. Chen, ZX: Growth and zeros of meromorphic solution of some linear difference equations. J. Math. Anal. Appl.373, 235-241 (2011)

6. Ishizaki, K, Yanagihara, N: Wiman-Valiron method for difference equations. Nagoya Math. J.175, 75-102 (2004) 7. Laine, I: Nevanlinna Theory and Complex Differential Equations. de Gruyter, Berlin (1993)

8. Li, S, Gao, ZS: Finite order meromorphic solutions of linear difference equations. Proc. Jpn. Acad., Ser. A, Math. Sci.

87(5), 73-76 (2011)

9. Zheng, XM, Tu, J: Growth of meromorphic solutions of linear difference equations. J. Math. Anal. Appl.384, 349-356 (2011)

10. Chen, ZX: The growth of solutions of a class of second-order differential equations with entire coefficients. Chin. Ann. Math., Ser. B20(1), 7-14 (1999) (in Chinese)

11. Liu, Y: On growth of meromorphic solutions for linear difference equations with meromorphic coefficients. Adv. Differ. Equ.2013, Article ID 60 (2013)

12. Laine, I, Yang, CC: Clunie theorems for difference andq-difference polynomials. J. Lond. Math. Soc.76(2), 556-566 (2007)

(13)

14. Chiang, YM, Feng, SJ: On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans. Am. Math. Soc.361(7), 3767-3791 (2009)

15. Peng, F, Chen, ZX: On the growth of solutions of some second-order linear differential equations. J. Inequal. Appl.

2011, Article ID 635604 (2011)

16. Xu, J, Zhang, X: Some results of meromorphic solutions of second-order linear differential equations. J. Inequal. Appl.

2013, Article ID 304 (2013)

17. Li, S, Chen, B: Results on meromorphic solutions of linear difference equations. Adv. Differ. Equ.2012, Article ID 203 (2012)

18. Liu, H, Mao, Z: On the meromorphic solutions of some linear difference equations. Adv. Differ. Equ.2013, Article ID 133 (2013)

19. Chen, ZX, Shon, KH: On the growth and fixed points of solutions of second order differential equation with meromorphic coefficients. Acta Math. Sin. Engl. Ser.21(4), 753-764 (2005)

20. Halburd, RG, Korhonen, R: Finite-order meromorphic solutions and the discrete Painlevé equations. Proc. Lond. Math. Soc.94, 443-474 (2007)

10.1186/1687-1847-2014-306

Cite this article as:Yuan and Ling:Results on the growth of meromorphic solutions of some linear difference

References

Related documents

We focus on studies that reported changes in brain activity level from before to after ECT as evidenced by increases (which we term activation) and decreases (deactivation) in the

Table 2-1 Definitions of Sustainability 75 Table 2-2 Dominant Ideologies in Environmentalism 79 Table 2-3 Taxonomy of Economic Valuation Techniques 90 Table 4-1 Two Dimensions

Natural language is often used to support the specification of requirements, if only as an intermediate solution before for- mal modelling. As natural language is inherently

Table IV shows how each group coursework (G1 to G17) was evaluated against the list of Tropos goals and UML use cases, gathered around the main actors expressing their

Whether we think of it as an academic discipline, a looser tradition of ideas and texts, a particular methodology, a political project or movement, or a vague name for almost any

To investigate the soil- hydrodynamic phenomena under fully controlled turbulent flow conditions a Soil Protrusion Apparatus (SPA) was developed by re-fabricating

Similarly, as the intensity of competition becomes high, the only industries for which it will be optimal to have a closed economy are those which have a low marginal cost

Thus the 2010 control for England’s Saturday match was taken as the average rate per thousand population aged 16 or over (based on ONS 2010 mid-year estimates) for the four