Further Extending Results of Some Classes of Complex Difference and Functional Equations

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doi:10.1155/2010/404582

Research Article

Further Extending Results of Some Classes of

Complex Difference and Functional Equations

Jian-jun Zhang and Liang-wen Liao

Department of Mathematics, Nanjing University, Nanjing 210093, China

Correspondence should be addressed to Liang-wen Liao,[email protected]

Received 29 March 2010; Revised 24 August 2010; Accepted 21 September 2010

Academic Editor: Binggen Zhang

Copyrightq2010 J.-j. Zhang and L.-w. Liao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The main purpose of this paper is to present some properties of the meromorphic solutions of complex diﬀerence equation of the form_λ_∈_Iαλznν1fzcνlλ,ν/μ∈Jβμznν1fz

cνmμ,ν Rz, fz, whereI {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,

cνν 1,2, . . . , nare distinct, nonzero complex numbers,αλz λ ∈ Iandβμz μ ∈ Jare

small functions relative tofz, Rz, fzis a rational function infzwith coeﬃcients which are small functions offz. We also consider related complex functional equations in the paper.

1. Introduction and Main Results

Let fz be a meromorphic function in the complex plane. 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 Tr, f, proximity function mr, f, counting functionNr, f, the first and second main theoremssee, e.g.,1–4 . We also use Nr, f to denote the counting function of the poles of fz whose every pole is counted only once. The notation Sr, f denotes any quantity that satisfies the condition: Sr, f oTr, f as r → ∞ possibly outside an exceptional set of r of finite linear measure. A meromorphic function az is called a small function of fz if and only if Tr, az Sr, f.

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Theorem A. If the second-order diﬀerence equation

fz1 fz−1 Rz, fz a0z a1zf· · ·apzf

p

b0z b1zf· · ·bqzfq, 1.1

with polynomial coeﬃcients ai (i 1,2, . . . , p) and bj (j 1,2, . . . , q), admits a transcendental meromorphic solution of finite order, thendmax{p, q} ≤2.

Theorem B. If the second-order diﬀerence equation

fz1fz−1 Rz, fz a0z a1zf· · ·apzf

p

with polynomial coeﬃcients ai (i 1,2, . . . , p) and bj (j 1,2, . . . , q), admits a transcendental meromorphic solution of finite order, thendmax{p, q} ≤2.

One year later, Heittokangas et al.7 extended the above two results to the case of higher-order diﬀerence equations of more general type. They got the following.

Theorem C. Letc1, c2, . . . , cn∈C\ {0}. If the diﬀerence equation

n

i1

fzci R

z, fz a0z a1zf· · ·apzf

p

with the coeﬃcients of rational functions ai (i 1,2, . . . , p) and bj (j 1,2, . . . , q) admits a transcendental meromorphic solution of finite order, thendmax{p, q} ≤n.

Theorem D. Letc1, c2, . . . , cn∈C\ {0}. If the diﬀerence equation

n

i1

fzci R

z, fz a0z a1zf· · ·apzf

p

with the coeﬃcients of rational functions ai (i 1,2, . . . , p) and bj (j 1,2, . . . , q) admits a transcendental meromorphic solution of finite order, thendmax{p, q} ≤n.

Laine et al.9 and Huang and Chen8 , respectively, generalized the above results. They obtained the following theorem.

Theorem E. Let c1, c2, . . . , cn be distinct, nonzero complex numbers, and suppose that fz is a transcendental meromorphic solution of the diﬀerence equation

{J}

αJz

⎛

⎝

j∈J

fzcj

⎞_⎠

Rz, fz a0z a1zfz · · ·apzfz

p

b0z b1zfz · · ·bqzfzq,

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with coeﬃcientsαJz, aiz(i 0,1, . . . , p) andbjz(j 0,1, . . . , q), which are small functions relative tofz,where{J}is a collection of all subsets of{1,2, . . . , n}. If the orderρfis finite, then

dmax{p, q} ≤n.

In the same paper, Laine et al. also obtained Tumura-Clunie theorem about diﬀerence equation.

Theorem F. Suppose thatc1, c2, . . . , cn are distinct, nonzero complex numbers and thatfzis a transcendental meromorphic solution of

n

j1

αjzf

zcj

Rz, fz P

z, fz

Qz, fz, 1.6

where the coeﬃcientsαjzare nonvanishing small functions relative tofzand wherePz, fz andQz, fzare relatively prime polynomials infzover the field of small functions relative to

fz. Moreover, we assume thatqdeg_fQ >0,

nmaxp, qmaxdeg_fP,deg_fQ, 1.7

and that, without restricting generality,Qis a monic polynomial. If there existsα∈0, nsuch that for allrsuﬃciently large,

N

⎛

⎝_r,n

j1

αjzf

zcj

⎞_⎠

≤αNrC, fzSr, f, 1.8

whereC:max{|c1|,|c2|, . . . ,|cn|}, then either the orderρf ∞, or

Qz, fz≡fz hzq, 1.9

wherehzis a small meromorphic function relative tofz.

Remark 1.1. Huang and Chen8 proved that the Theorem F remains true when the left hand

side of1.6is replaced by the left hand side of1.5, meanwhile, the condition1.8would be replaced by a corresponding form.

Moreover, Laine et al.9 also gave the following result.

Theorem G. Suppose thatfis a transcendental meromorphic solution of

{J}

αJz

⎛

⎝

j∈J

fzcj

⎞_⎠

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wherepzis a polynomial of degreek≥2,{J}is a collection of all subsets of{1,2, . . . , n}. Moreover, we assume that the coeﬃcientsαJzare small functions relative tofand thatn≥k.Then

Tr, fOlogrαε, 1.11

whereαlogn/logk.

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

Theorem 1.2. Let c1, c2, . . . , cn be distinct, nonzero complex numbers and suppose that fz is a transcendental meromorphic solution of the diﬀerence equation

λ∈Iαλznν1fzcνlλ,ν

μ∈Jβμznν1fzcνmμ,ν

p

b0z b1zfz · · ·bqzfzq

,

1.12

with coeﬃcientsαλzλ ∈ I, βμzμ ∈J, aiz i 0,1, . . . , p, andbjz j 0,1, . . . , qare small functions relative tofz,whereI {λ lλ,1, lλ,2, . . . , lλ,n|lλ,ν ∈N∪ {0}, ν1,2, . . . , n} andJ{μ mμ,1, mμ,2, . . . , mμ,n|mμ,ν∈N∪{0}, ν1,2, . . . , n}are two finite index sets, denote

σνmax λ,μ

lλ,ν, mμ,ν

ν1,2, . . . , n, σ

n

ν1

σν. 1.13

If the orderρf:ρis finite, thendmax{p, q} ≤σ.

Corollary 1.3. Letc1, c2, . . . , cn be distinct, nonzero complex numbers and suppose thatfzis a transcendental meromorphic solution of the diﬀerence equation

λ∈I

αλz

_n

ν1

fzcνlλ,ν

p

b0z b1zfz · · ·bqzfzq, 1.14

with coeﬃcients αλzλ ∈ I, aiz i 0,1, . . . , pand bjz j 0,1, . . . , q, which are small functions relative tofz,whereI {λ lλ,1, lλ,2, . . . , lλ,n |lλ,ν ∈ N∪ {0}, ν 1,2, . . . , n}is a finite index set, denote

σνmax

λ {lλ,ν} ν1,2, . . . , n, σ n

ν1

σν. 1.15

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Remark 1.4. InCorollary 1.3, if we take

max

λ,ν {lλ,ν}1, λ∈I, ν1,2, . . . , n, 1.16

thenCorollary 1.3becomes Theorem E. Therefore,Theorem 1.2is a generalization of Theorem E.

Example 1.5. Letc1arctan 2, c2−π/4.Then it is easy to check thatfz tanzsolves the

following diﬀerence equation:

fzc12fzc2

fzc1 fzc22

f44f33f2−4f−4

2f4₋₁₉_f3₇_f2₋₅_f₃. 1.17

Example 1.6. Letc1arctan 2 andc2arctan−2.It is easy to check thatfz tanzsatisfies

the diﬀerence equation

fzc12fzc2 fzc1fzc22

10f3₋₄₀_f

16f4₋₈_f2₁. 1.18

In above two examples, we both haved σ 4 andρf 1 < ∞.Therefore, the estimations inTheorem 1.2andCorollary 1.3are sharp.

Theorem 1.7. Suppose thatc1, c2, . . . , cnare distinct, nonzero complex numbers and thatfzis a transcendental meromorphic solution of

Rz, fz P

z, fz

Qz, fz, 1.19

where the coeﬃcients αλzλ ∈ I, βμzμ ∈ J are nonvanishing small functions relative to

fzandPz, fzandQz, fzare relatively prime polynomials infzover the field of small functions relative to fz, 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}, ν1,2, . . . , n}are two finite index sets, denote

σνmax λ,μ

lλ,ν, mμ,ν

ν1,2, . . . , n, σ

n

ν1

σν. 1.20

Moreover, we assume thatqdeg_fQ >0,

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and that, without restricting generality,Qis a monic polynomial. If there existsα∈0, σsuch that for allrsuﬃciently large,

N

If the left hand side of1.19inTheorem 1.7is replaced by the left hand side of1.14 inCorollary 1.3, then1.23implies1.22. Since we have

N

by the fundamental property of counting function. Therefore, we get the following result easily. andQz, fzare relatively prime polynomials infzover the field of small functions relative to

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and that, without restricting generality,Qis a monic polynomial. If there existsα∈0, σsuch that for allrsuﬃciently large,

n

ν1

σνN

r, fzcν

≤αNrC, fzSr, f, 1.29

whereC:max{|c1|,|c2|, . . . ,|cn|}, then either the orderρf ∞, or

Qz, fz≡fz hzq, 1.30

wherehzis a small meromorphic function relative tofz.

Finally, we give a result corresponding to Theorem G.

Theorem 1.9. Let c1, c2, . . . , cn be distinct, nonzero complex numbers and suppose that f is a transcendental meromorphic solution of

fpz, 1.31

where pzis a polynomial of degree k ≥ 2,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. Denote

σνmax λ,μ

lλ,ν, mμ,ν

ν1,2, . . . , n, σ

n

ν1

σν. 1.32

Moreover, we assume that the coeﬃcientsαλzλ∈Iandβμzμ∈Jare small functions relative

tofand thatσ≥k.Then

Tr, fOlogrαε, 1.33

whereαlogσ/logk.

2. Main Lemmas

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

Lemma 2.1 see 10 . Let fz be a meromorphic function. Then for all irreducible rational functions inf,

Rz, f P

z, f Qz, f

p

i0aizfi

q

j0bjzfj

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such that the meromorphic coeﬃcientsaiz, bjzsatisfy

we have the following estimation by the proof ofLemma 2.2

T

be a fixed nonzero complex number, then for eachε >0,one has

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Lemma 2.5see12 . Letfzbe a meromorphic function and letφbe given by

φfnan−1fn−1· · ·a0, 2.8

whereaii0,1, . . . , n−1are small meromorphic functions relative tofz. Then either

φ

fan−1 n

n

, 2.9

or

Tr, f≤N

r, 1 φ

Nr, fSr, f. 2.10

Lemma 2.6see9,13 . Letfzbe a nonconstant meromorphic function and letPz, f,Qz, f

be two polynomials in fz with meromorphic coeﬃcients small relative to fz. If Pz, f and

Qz, fhave no common factors of positive degree infzover the field of small functions relative to

fz, then

N

r, 1 Qz, f

≤N

r, P

z, f Qz, f

Sr, f. 2.11

Lemma 2.7 see 14 . Let f be a transcendental meromorphic function, and pz akzk

ak−1zk−1 · · ·a1za0, ak/0 be a nonconstant polynomial of degree k. Given 0 < δ < |ak|, denoteλ|ak|δandμ|ak| −δ. Then givenε >0anda∈C∪ {∞},one has

knμrk, a, f≤nr, a, fpz≤knλrk, a, f,

Nμrk, a, fOlogr≤Nr, a, fpz≤Nλrk, a, fOlogr,

1−εTμrk_{, f}_≤_T_{r, f}_p_z_≤₁_ε_T_λrk_{, f}

2.12

for allrlarge enough.

Lemma 2.8see15 . Letφ:r0,∞ → 0,∞be positive and bounded in every finite interval, and suppose thatφμrm_≤_Aφ_r _B_{holds for all}_r_{large enough, where}_{μ >}₀_{, m >}₁_{, A >}₁_and_B are real constants. Then

φr Ologrα, 2.13

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3. Proof of Theorems

Proof ofTheorem 1.2. We assume thatfzis a meromorphic solution of finite order of1.12.

It follows from Lemmas2.1,2.2, and2.4that for eachε >0,

This yields the asserted result.

Proof ofTheorem 1.7. Supposefzis a transcendental meromorphic solution of1.19and the

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Now assuming the orderρf<∞, then we haveSr, fzcν Sr, fand

It follows from this that

Tr, f−Nr, f≤ σ1 σ αN

r2C, f−Nr, fSr, f. 3.6

We prove the following inequality by induction:

Tr, f−Nr, f≤ σm σ αN

r2mC, f−mNr, fSr, f. 3.7

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we prove that inequality3.7holds formk1.We have

It follows from3.7that

Nr, fz≤ σm σm αN

r2mC, fSr, f. 3.12

Letmbe large enough such that

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Since

Proof ofTheorem 1.9. We assume fz is a transcendental meromorphic solution of 1.31.

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SinceTrC, f≤Tβr, fholds forrlarge enough forβ >1,we may assumerto be large enough to satisfy

1−εTμrk, f≤σ1εTβr, f 3.19

outside a possible exceptional set of finite linear measure. By the standard idea of removing the exceptional setsee4, page 5 , we know that wheneverγ >1,

1−εTμrk, f≤σ1εTγβr, f 3.20

holds for allrlarge enough. Denotetγβr, thus inequality3.20may be written in the form

T

μ

γβkt

k_{, f}

≤ σ1ε

1−ε T

t, f. 3.21

ByLemma 2.8, we have

Tr, fOlogrs, 3.22

where

s logσ1ε/1−ε logk

logσ

logk o1. 3.23

Denoting nowαlogσ/logk, thus we obtain the required form.Theorem 1.9is proved.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions. The research was supported by NSF of ChinaGrant no. 10871089.

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