On the hyper exponent of convergence of zeros of f(j)−φ of higher order linear differential equations

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R E S E A R C H

Open Access

On the hyper exponent of convergence of

zeros of

f

(j)

–

ϕ

of higher order linear

differential equations

Hong-Yan Xu

1*

_{, Jin Tu}

2

_{and Xiu-Min Zheng}

2

*_{Correspondence:}

[email protected]

1_{Department of Informatics and}

Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China

Full list of author information is available at the end of the article

Abstract

In this paper, we deal with the relationship between the small function and the derivative of solutions of higher order linear diﬀerential equations

f(k)+Ak–1f(k–1)+· · ·+A0f= 0 (k≥2),

whereAj(z) (j= 0, 1,. . .,k– 1) are entire functions or meromorphic functions. The

theorems of this paper improve the previous results given by Chen, Belaïdi, Liu.

MSC: 34M10; 30D35

Keywords: linear diﬀerential equation; hyper order; type; small function

1 Introduction and main results

Complex oscillation theory of solutions of linear diﬀerential equations in the complex planeCwas started by Bank and Laine [, ]. After their well-known work, many impor-tant results have been obtained on the complex oscillation theory of solutions of linear diﬀerential equations inC, refer to [, ].

To state those correlated results, we require to give some explanation as follows. We shall assume that the reader is familiar with the fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions (see [, , ]). In addition, we will use the notationσ(f) to denote the order of meromorphic functionf(z), λ(f) to denote the exponent of convergence of the zero-sequence off(z) andλ(f) to denote exponent of convergence of distinct zero-sequence of meromorphic functionf(z), andτ(f) to denote the type of an entire functionf(z) with  <σ(f) =σ< +∞, which is deﬁned to be (see [])

τ(f) =lim sup

r→∞

logM(r,f)

rσ .

We useσ(f) to denote the hyper-order off(z),σ(f) is deﬁned to be (see [])

σ(f) =lim sup

r→∞

log logT(r,f)

logr .

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We useλ(f) (λ(f)) to denote the hyper-exponent of convergence of the zero-sequence

(distinct zero-sequence) of meromorphic functionf(z),λ(f),λ(f) are deﬁned to be (see

[, ])

λ(f) =lim sup

r→∞

log logN(r,_f)

logr , and λ(f) =lim supr→∞

log logN(r,_f)

logr .

Letϕ(z) be an entire function withσ(ϕ) <σ(f) orσ(ϕ) <σ(f), the hyper-exponent of

convergence of zeros and distinct zeros off(z) –ϕ(z) are deﬁned to be

λ(f –ϕ) =lim sup

r→∞

log logN(r,_f_–_ϕ)

logr , λ(f–ϕ) =lim supr→∞

log logN(r,_f_–_ϕ)

logr ,

especially ifϕ(z) =z, we useλ(f –z) andλ(f –z) to denote the hyper-exponent of

con-vergence of ﬁxed points and distinct ﬁxed points off(z), respectively. We usemE=_Edt to denote the linear measure of a setE⊂(, +∞) and usemlE=

E dt

t to denote the loga-rithmic measure of a setE⊂[, +∞). We denote byS(r,f) any quantity satisfying

S(r,f) =oT(r,f),

asr→+∞, possibly outside of a set with ﬁnite measure. A meromorphic functionψ(z) is called a small function with respect tof ifT(r,ψ) =S(r,f).

For the second order linear diﬀerential equation

f+A(z)f+B(z)f= , ()

whereA(z) andB(z) (≡) are entire functions, it is an interesting problem to investigate the complex oscillation of solutions of Equation (). Many mathematicians obtained a lot of important and signiﬁcant results (see [, , , , ]) by studying the above equation.

In , Shon [] investigated the hyper-order of the solutions of () and obtained the following result.

Theorem A(see []) Let A(z)and B(z)be entire functions such thatσ(A) <σ(B)orσ(B) <

σ(A) <_, then every function f≡of () satisﬁesσ(f)≥max{σ(A),σ(B)}.

In , Chen and Shon [] investigated the zeros of the solution concerning small functions and ﬁxed points of solutions of second order linear diﬀerential equations and obtained some results as follows.

Theorem B(see []) Let Aj(z)≡(j= , ) be entire functions withσ(Aj) < , suppose that

a, b are complex numbers and satisfy ab= andarga=argb or a=cb ( <c< ). Ifϕ(z)≡ is an entire function of ﬁnite order, then every non-trivial solution f of the equation

f+A(z)eazf+A(z)ebzf = 

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Theorem C(see []) Let A(z)≡,ϕ(z)≡, Q(z)be entire functions withσ(A) < , < σ(Q) <∞andσ(ϕ) <∞, then every non-trivial solution f of the equation

f+A(z)eazf+Q(z)f= 

satisﬁesλ(f–ϕ) =λ(f–ϕ) =λ(f–ϕ) =∞, where a= is a complex number.

In the same year, Liu and Zhang [] investigated the ﬁxed points when the coeﬃcients of the equations are meromorphic functions and obtained the following result.

Theorem D(see []) Suppose that k≥and A(z)is a transcendental meromorphic

func-tion satisfyingδ(∞,A) =lim_r_→∞_Tm(₍r_r_,,_AA₎) =δ> ,σ(A) =σ < +∞. Then every meromorphic solution f ≡of the equation

f(k)+A(z)f =  ()

satisfies that f and f,f, . . . ,f(k)_{all have infinitely many fixed points and} _λ_(f(j)_–_{z) =}_σ

(j= , , . . . ,k).

For Equation (), Belaïdi [], Belaïdi and El Farissi [] investigated the ﬁxed points and the relationship between small functions and diﬀerential polynomials of solutions of Equa-tion () and obtained some results which improve Theorems D and C.

There naturally arises an interesting subject on the problems of ﬁxed points of solutions of the diﬀerential equation

f(k)+Ak–f(k–)+· · ·+Af=  (k≥), ()

whereAj(z) (j= , , . . . ,k– ) are entire functions.

In this paper, we will deal with the above equation and investigate the relationship be-tween small functions and derivative of solutions of Equation () and obtain some theo-rems which improve the previous results given by Chen, Kwon andetc.

Theorem . Let Aj(z), j= , , . . . ,k– be entire functions with ﬁnite order and satisfy one of the following conditions:

(i) max{σ(Aj) :j= , , . . . ,k– }<σ(A) <∞;

(ii)  <σ(Ak–) =· · ·=σ(A) =σ(A) <∞andmax{τ(Aj) :j= , , . . . ,k– }=

τ<τ(A) =τ,

then for every solution f ≡of () and for any entire functionϕ(z)≡satisfyingσ(ϕ) < σ(A), we have

λ(f–ϕ) =λ

f–ϕ=λ

f(i)–ϕ=σ(f) =σ(A) (i∈N).

Theorem . Let Aj(z), j= , , . . . ,k– be polynomials, A(z)be a transcendental entire

function, then for every solution f ≡of () and for any entire functionϕ(z)of ﬁnite order, we have

(i) λ(f–ϕ) =λ(f –ϕ) =σ(f) =∞;

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Theorem . Let Aj(z), j= , , . . . ,k– be meromorphic functions satisfyingmax{σ(Aj) : j= , , . . . ,k– }<σ(A)andδ(∞,A) > . Then for every meromorphic solution f ≡of

() and for any meromorphic functionϕ(z)≡satisfyingσ(ϕ) <σ(A), we haveλ(f(i)– ϕ) =λ(f(i)–ϕ)≥σ(A)(i= , , . . .), where f()=f .

Remark . The following example shows that Theorem . is not valid whenAj(z),j=

, , . . . ,k–  do not satisfy the conditionmax{σ(Aj) :j= , , . . . ,k– }<σ(A).

Example . For the equation

f+e

z₊_ez_{– }

 –ez f

₊ –ez

 –ezf = , ()

we can easily get that Equation () has a solutionf(z) =eez₊_ez_{. And the functions} ez+ez– –ez ,

–ez

–ezare meromorphic and satisfyδ(∞,–e

z

–ez) = _. Takeϕ(z) =ez, thenσ(ϕ) <σ(–e

z

–ez). Thus,

we can get thatλ(f–ϕ) =λ(ee

z

ez_{) = }_{=  =}_σ₍–ez

–ez).

From Theorems .-., ifϕ(z) =z, we can get the following corollaries easily.

Corollary . Under the assumptions of Theorem ., ifϕ(z) =z, for every solution f ≡

of (), we have

λ(f –z) =λ

f–z=λ

f(i)–z=σ(f) =σ(A) (i∈N).

Corollary . Under the assumptions of Theorem ., ifϕ(z) =z, for every solution f ≡

of (), we have

(i) λ(f–z) =λ(f–z) =σ(f) =∞;

(ii) λ(f(i)_–_{z) =}_λ_(f(i)_–_{z) =}_σ_(f(i)_–_{z) =}_∞₍_i_≥_,_i_∈_N_).

Corollary . Under the assumptions of Theorem ., ifϕ(z) =z, for every meromorphic

solution f ≡of (), we haveλ(f(i)–z) =λ(f(i)–z)≥σ(A)(i= , , . . .), where f()=f .

Remark . In Theorem B, ifab=  anda=cb( <c< ), it is easy to see thatσ(Aeaz) =

σ(Aebz) =  andτ(Aeaz) =|a|<τ(Aebz) =|b|. By Theorem ., for every solutionf ≡

of () and for any entire functionϕ(z)≡ withσ(ϕ) < , we haveλ(f–ϕ) =λ(f–ϕ) = λ(f–ϕ) = . Therefore, Theorem . is also a partial extension of Theorem B.

Theo-rem . is the improvement of TheoTheo-rem C. TheoTheo-rem . and Corollary . are the im-provements of Theorem D.

2 Some lemmas

To prove our theorems, we require the following lemmas.

Lemma . Assume f ≡is a solution of Equation (), set g=f –ϕ, then g satisﬁes the

equation

g(k)+Ak–g(k–)+· · ·+Ag= –

ϕ(k)+Ak–ϕ(k–)+· · ·+Aϕ

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Proof Sinceg=f–ϕ, we haveg=f–ϕ, . . . ,g(k)₌_f(k)_–_ϕ(k)_{. Substituting them into}

Equa-And the derivation of () is

f(k+)+Ak–f(k)+

Substituting () into (), we have

f(k+)+

Substituting () into (), we have

g_(k)+

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Lemma . Assume f ≡is a solution of Equation (), set g=f–ϕ, then gsatisﬁes the

equation

g_(k)+U_k_–g_(k–)+· · ·+U_g= –

ϕ(k)+U_k_–ϕ(k–)+· · ·+U_ϕ, ()

where U

j =Uj+

+U

j – U_

U_U



j+, j= , , , . . . ,k– and Uk≡.

Proof Sinceg=f–ϕ, we can get

f=g+ϕ, f=g+ϕ, . . . , f(k+)=g (k)

 +ϕ(k). ()

The derivation of () is

f(k+)+U_k_–f(k+)+U_k_–+U_k_–f(k)+· · ·+U_+U_f+U_f= . ()

Substituting () into (), we have

f(k+)+

U_k_––U

 

U_

f(k+)+

U_k_–+U_k_––U

 

U_U



k–

f(k)

+· · ·+

U_+U_–U

 

U_U

 

f= .

()

SetU

j =Uj+

+U

j – U_ U_U



j+,j= , , , . . . ,k–  andUk≡, then from () we have

f(k+)+U_k_–f(k+)+U_k_–f(k)+· · ·+U_f= . ()

Substituting () into (), we can get ().

Thus, this completes the proof of Lemma ..

Lemma . Assume f ≡is a solution of Equation (), set g=f–ϕ, then gsatisﬁes the

equation

g_(k)+U_k_–g_(k–)+· · ·+U_g= –

ϕ(k)+U_k_–ϕ(k–)+· · ·+U_ϕ, ()

where U_j=U_j_++U_j–U

 

UU



j+, j= , , , . . . ,k– and Uk≡.

Proof Sinceg=f–ϕ, we have

f=g+ϕ, f()=g+ϕ, . . . , f(k+)=g (k)

 +ϕ(). ()

The derivation of () is

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Substituting () into (), we have

f(k+)+

U_k_––U

 

U_

f(k+)+

U_k_–+U_k_––U

 

U_U



k–

f(k+)

+· · ·+

U_+U_–U

 

U_U

 

f= .

()

SetU_j=U_j_++U_j–U

UU



j+,j= , , , . . . ,k–  andUk≡, then from () we have

f(k+)+U_k_–f(k+)+U_k_–f(k+)+· · ·+U_f= . ()

Substituting () into (), we can get ().

Thus, we complete the proof of Lemma ..

Lemma . Assume f ≡is a solution of Equation (), set gi=f(i)–ϕ, then gisatisﬁes the

equation

g_i(k)+U_ki_–g_i(k–)+· · ·+U_igi= –

ϕ(k)+U_ki_–ϕ(k–)+· · ·+U_iϕ, ()

where Ui j=Uji+–

+Ui–

j –

Ui–

Ui_–U i–

j+, j= , , , . . . ,k– , Uki–≡and i∈N.

Proof The inductive method will be used to prove it.

At ﬁrst, from Lemmas .-., we get that () holds fori= , , .

Next, suppose thatgi=f(i)–ϕ,i≤n,n∈Nsatisfy (). Thus,gn=f(n)–ϕsatisﬁes the equation

g_n(k)+U_kn_–g_n(k–)+· · ·+U_ngn= –

ϕ(k)+U_kn_–ϕ(k–)+· · ·+U_nϕ, ()

whereU_jn=U_jn_+–+U_jn––Un–

Un–  U

n–

j+,j= , , , . . . ,k–  andUkn–≡. Sincegn=f(n)–ϕ, we have

f(n+)=g_n+ϕ, f(n+)=g_n+ϕ, . . . , f(k+n)=g_n(k)+ϕ(k). ()

From () and (), we have

f(k+n)+U_kn_–f(k+n–)+U_kn_–f(k+n–)+· · ·+U_nf(n)= . ()

Now we will prove thatgn+=f(n+)–ϕsatisﬁes (). Sincegn+=f(n+)–ϕ, we have

f(n+)=g_n_++ϕ, f(n+)=g_n_++ϕ, . . . , f(n+k+)=g_n(k_+) +ϕ(k). ()

The derivation of () is

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Substituting () into (), we have

This completes the proof of Lemma .. Similar to the proof of [, Lemma .], we can get the following lemma.

Lemma . Let f(z)be a transcendental meromorphic function withσ(f) =σ ≥, then

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From ()-(), there exists a setEwith inﬁnite logarithmic measure such that

lim

r→∞

logm(r,Un+  )

logr =rlim→∞

logm(r,Un

)

logr =σ(A) >σ

=lim sup

r→∞

max≤j≤k–{logm(r,Ujn)}

logr ()

=lim sup

r→∞

max≤j≤k–{logm(r,Ujn+)}

logr .

Thus, the proof of Lemma . is completed.

Lemma . Let Hj(z)(j= , , . . . ,k– ) be meromorphic functions of ﬁnite order. If

lim sup

r→∞

max≤j≤k–{logm(r,Hj)}

logr =β

and there exists a set Ewith inﬁnite logarithmic measure such that

lim

r→∞

logm(r,H)

logr =β>β

holds for all r∈E, then every meromorphic solution of

f(k)+Hk–f(k–)+· · ·+Hf+Hf =  ()

satisﬁesσ(f)≥β.

Proof Assume thatf(z) is a meromorphic solution of (). From (), we have

m(r,H)≤m

r,f

(k)

f

+m

r,f

(k–)

f

+· · ·+m

r,f

f

+ k–

j=

m(r,Hj) +O(). ()

By the lemma on logarithmic derivative and (), we have

m(r,H)≤O

logrT(r,f)+ k–

j=

m(r,Hj), r∈/E, ()

whereE⊂[, +∞) is a set with ﬁnite linear measure. From the assumptions of Lemma .,

there exists a setEwith inﬁnite logarithmic measure such that for all|z|=r∈E–Ewe

have

rβ–ε_≤_O_log_rT(r,_f₎_{+ (k}_{– )r}β+ε_, _()

where  < ε<β–β. From (), we haveσ(f)≥β.

Lemma .(see []) Let f(z)be a transcendental meromorphic function withσ(f) =σ<

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ﬁnite logarithmic measure such that for all z satisfying|z|=r∈/[, ]∪Eand(k,j)∈, we

exists a set Ewith inﬁnite logarithmic measure such that

U_ji≤exp(τ+ε)rσ

, U_i≥exp(τ–ε)rσ_, _()

where i∈Nand j= , , . . . ,k– .

Proof The inductive method will be used to prove it. (i) We ﬁrst prove thatUi

setEwith inﬁnite logarithmic measure such that

U_≥exp

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(ii) Next, we show thatU_ji(j= , , , . . . ,k– ) satisfy () wheni= . FromU

there exists a setEwith inﬁnite logarithmic measure such that

_Ui

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Lemma . Let Bj(z), j= , , . . . ,k–  be meromorphic functions withmax{σ(Bj) :j= , , . . . ,k– }=σ<σ(B) =σandδ(∞,B) =limr→∞mT((rr,,BB)) > . Then every meromorphic

solution f of the equation

f(k)+Bk–f(k–)+· · ·+Bf+Bf =  ()

satisﬁesσ(f)≥σ.

Proof Letf be a meromorphic solution of Equation (), by (), we have

m(r,B)≤m

r,f

(k)

f

+m

r,f

(k–)

f

+· · ·+m

r,f

f

+ k–

j=

m(r,Bj)

≤OlogrT(r,f)+ k–

j=

T(r,Bj), r∈/E,

()

whereE⊂[, +∞) is a set with ﬁnite linear measure. By Lemma ., there exists a setE

with inﬁnite logarithmic measure such that for all|z|=r∈E, we have

lim

r→∞

logT(r,B)

logr =σ, r∈/E. ()

Sinceδ(∞,B) > , then for any givenε( < ε<σ–σ) and for allr∈E, by (), we have

m(r,B)≥rσ–ε. ()

From () and (), we have

rσ–ε≤_O_log_rT(r,_f₎_{+ (k}_{– )r}σ+ε_, _r∈_E_–_E

. ()

From (), we can getσ(f)≥σ=σ(B).

Lemma .(see []) Let f(z)be a transcendental meromorphic function andα> be a

given constant, for any givenε> , there exists a set E⊂[,∞)that has ﬁnite logarithmic

measure and a constant M> that depends only onα and(m,n)(m,n∈ {, . . . ,k}with m<n) such that for all z satisfying|z|=r∈/[, ]∪E, we have

_ff((mn))(z)_(z)

≤M

T(αr,f) r

logαrlogT(αr,f)

n–m .

Lemma . Let Bj(z), j= , , . . . ,k– be meromorphic functions of ﬁnite order. If there ex-ist positive constantsσ,β,β( <β<β) and a set Ewith inﬁnite logarithmic measure

such that

maxBj(z):j= , , . . . ,k– 

≤expβrσ

, B(z)≥exp

βrσ

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Proof Suppose thatf is a meromorphic solution of (), by (), we have

B(z)≤ f(_fk)+

k–

j= Bj(z)

f_f(j). ()

By Lemma ., there exists a setEwith ﬁnite logarithmic measure such that for all|z|=

r∈/E, we have

f_f(j)≤MT(r,f)j, j= , , . . . ,k, ()

whereM>  is a constant. By (), () and the assumptions of Lemma ., for all|z|= r∈E–E, we have

expβrσ

≤MT(r,f)kexpβrσ

. ()

Since  <β<β, by (), we haveσ(f)≥σ.

Lemma .(see []) Let A,A, . . . ,Ak–,F≡be meromorphic functions, if f is a

mero-morphic solution of the equation

f(k)+Ak–f(k–)+· · ·+Af =F,

then we have the following statements:

(i) ifmax{σ(F),σ(Aj);j= , , . . . ,k– }<σ(f) =σ≤ ∞, thenσ(f) =λ(f) =λ(f); (ii) ifmax{σ(F),σ(Aj);j= , , . . . ,k– }<σ(f) =σ, thenσ(f) =λ(f) =λ(f).

Lemma .(see [, Theorem ]) Let Aj(z)(j= , , . . . ,k– ) be entire functions satisfying

 <σ(Ak–) =· · ·=σ(A) =σ(A) <∞,max{τ(Aj) :j= , , . . . ,k– }<τ(A) <∞, then

every solution f ≡of () satisﬁesσ(f) =σ(A).

3 Proofs of theorems

3.1 The proof of Theorem 1.1

To prove the conclusions of Theorem ., we will consider two cases as follows. Case . Suppose thatmax{σ(Aj) :j= , , . . . ,k– }<σ(A) <∞.

(i) First, we prove thatλ(f –ϕ) =σ(f). Assume thatf ≡ is a solution of (), from

ref. [], we haveσ(f) =σ(A). Setg=f –ϕ. Sinceσ(ϕ) <σ(A), thenσ(g) =σ(f) = σ(A) andλ(g) =λ(f –ϕ). By Lemma ., we get thatg satisﬁes Equation (). SetF= ϕ(k)₊_A

k–ϕ(k–)+· · ·+Aϕ. IfF≡, from ref. [], we haveσ(ϕ) =σ(A), we can get a

contradiction. ThenF≡, by Lemma ., we haveλ(g) =λ(g) =σ(g) =σ(A). Thus,

we haveλ(f –ϕ) =λ(f–ϕ) =σ(f) =σ(A).

(ii) Second, we prove thatλ(f–ϕ) =σ(f). Setg=f–ϕ, thenσ(g) =σ(f) =σ(A).

From Lemma ., we get thatgsatisﬁes Equation (). SetF=ϕ(k)+Uk–ϕ(k–)+· · ·+Uϕ,

whereU

j (j= , , . . . ,k– ) are stated as in Lemma .. IfF≡, from Lemma . and

Lemma ., we haveσ(ϕ)≥σ(A), a contradiction withσ(ϕ) <σ(A). HenceF≡. By

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(iii) Now, we prove thatλ(f–ϕ) =σ(f). Setg=f–ϕ, thenσ(g) =σ(f) =σ(A).

From Lemma ., we get thatgsatisﬁes Equation (). SetF=ϕ(k)+Uk–ϕ(k–)+· · ·+Uϕ,

whereU_j(j= , , . . . ,k– ) are stated as in Lemma .. IfF≡, from Lemma . and

Lemma ., we haveσ(ϕ)≥σ(A), a contradiction withσ(ϕ) <σ(A). HenceF≡. By

Lemma ., we getλ(f–ϕ) =λ(f–ϕ) =σ(f).

(iv) Setg=f–ϕ. From Lemmas ., ., . and Lemma ., using the same argument

as in Case (iii), we can getλ(f–ϕ) =λ(f–ϕ) =σ(f) easily.

(v) We will prove thatλ(f(i)–ϕ) =σ(f) (i> ,i∈N). Setgi=f(i)–ϕ, thenσ(gi) =σ(f) = σ(A). From Lemma ., we have gi satisﬁes Equation (). SetFi=ϕ(k)+Uki–ϕ(k–)+ · · ·+U_iϕ, whereU_ji(j= , , . . . ,k– ;i∈N) are stated as in Lemma .. IfFi≡, from Lemma . and Lemma ., we haveσ(ϕ)≥σ(A), a contradiction withσ(ϕ) <σ(A).

HenceFi≡. By Lemma ., we getλ(f(i)–ϕ) =λ(f(i)–ϕ) =σ(f).

Case . Suppose that  < σ(Ak–) =· · · =σ(A) =σ(A) < ∞ and max{τ(Aj) :j= , , . . . ,k– }=τ<τ(A) =τ.

(i) We ﬁrst prove that λ(f –ϕ) =σ(f). Assume that f ≡ is a solution of (), by

Lemma ., we have σ(f) =σ(A) > . Setg =f –ϕ. Sinceϕ≡ is an entire

func-tion satisfyingσ(ϕ) <σ(A), then we haveσ(g) =σ(f) =σ(A) and λ(g) =λ(f –ϕ).

From Lemma ., we get thatg satisﬁes Equation (). We will aﬃrmF≡. IfF≡, by Lemma ., we getσ(ϕ) =σ(A), a contradiction. HenceF≡. From the assumptions

of Theorem ., we get

maxσ(F),σ(Aj) :j= , , . . . ,k– 

<σ(g) =σ(A).

From Lemma .(ii), we haveλ(f–ϕ) =λ(f –ϕ) =σ(f) =σ(A).

(ii) Now we prove thatλ(f–ϕ) =σ(f). Letg=f–ϕ. Sinceσ(ϕ) <σ(A), we have σ(g) =σ(f) =σ(A). By Lemma ., we haveg satisﬁes Equation (). IfF≡, from

Lemma . and Lemma ., we have σ(ϕ)≥σ(A). Then we can get a contradiction

withσ(ϕ) <σ(A). Therefore, we haveF≡. By () and Lemma ., we haveλ(f–ϕ) = λ(f–ϕ) =σ(f) =σ(A).

Similar to the arguments as in Case (iii)-(v) and by using Lemmas .-., . and Lemma ., we can get

λ

f(i)–ϕ=λ

f(i)–ϕ=σ(f) =σ(A) (i∈N).

Thus, the proof of Theorem . is completed.

SinceAj(z) (j= , , . . . ,k– ) are polynomials andA(z) is a transcendental entire function,

then we have thatAj(z) (j= , , , . . . ,k– ) satisfy the condition of Theorem .. By using the same argument as in Theorem . and Lemma .(i), we can get the conclusions of Theorem . easily.

Thus, we omit the process of proving Theorem ..

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Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

HYX completed the main part of this article, HYX, JT and XMZ corrected the main theorems. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi, 333403, China.}2_Institute

of Mathematics and Informatics, Jiangxi Normal University, Nanchang, Jiangxi, 330022, China.

Acknowledgements

The authors thank the referee for his/her valuable suggestions to improve the present article.

This work was supported by the Natural Science Foundation of Jiang-Xi Province in China (Grant No. 2009GQS0013 and No. 2010GQS0119) and the Youth Foundation of Education Bureau of Jiangxi Province in China (Grant No. GJJ11072).

Received: 17 April 2012 Accepted: 4 July 2012 Published: 20 July 2012

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doi:10.1186/1687-1847-2012-114