Value distribution for difference operator of meromorphic functions with maximal deficiency sum

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R E S E A R C H

Open Access

Value distribution for difference operator of

meromorphic functions with maximal

deﬁciency sum

Zhaojun Wu

*

*_{Correspondence:}

[email protected] School of Mathematics and Statistics, Hubei University of Science and Technology, Xianning, 437100, P.R. China

Abstract

The main purpose of this paper is to investigate the relationship between the characteristic function of a meromorphic functionf(z) with maximal deﬁciency sum and that of the exact diﬀerence

cf=f(z+c) –f(z). As an application, the author also establishes an inequality on the zeros and poles for

cfand gives an example to show that the upper bound of the inequality is accurate.

MSC: 30D30; 39A05

Keywords: diﬀerence operator; maximal deﬁciency sum; meromorphic function

1 Introduction

Iff(z) is a meromorphic function in the complex planeCanda∈C, we use the follow-ing notations frequently used in Nevanlinna theory (see [–]):m(r,f),N(r,f),m(r,a) =

m(r,_f–a ),N(r,a) =N(r,_f_–a ), . . . . Denote byS(r,f) any quantity such thatS(r,f) =o(T(r,f)),

r→ ∞without restriction iff(z) is of finite order and otherwise except possibly for a set of values ofrof finite linear measure. The Nevanlinna deficiency off with respect to a finite complex numberais defined by

δ(a,f) =lim inf r→∞

m(r,a)

T(r,f) =  –lim supr→∞

N(r,a)

T(r,f).

Ifa=∞, then one should replacem(r,a)(N(r,a)) in the above formula bym(r,f)(N(r,f)). The classical second fundamental theorem of Nevanlinna theory asserts that the total de-ﬁciency of any meromorphic functionf(z) satisﬁes the inequality

a∈C

δ(a,f) +δ(∞,f)≤.

If the above equality holds, then we say thatf has maximal deﬁciency sum. The Valiron-Mo’honko identity states that if the functionR(z,f) is rational inf and has small mero-morphic coeﬃcients, then

Tr,R(z,f)=deg_f(R)T(r,f) +S(r,f). (.) Certain relationship between the characteristic function of a meromorphic functionf(z) with maximal deﬁciency sum and that of derivativef(z) plays a key role in the study of a

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conjecture of Nevanlinna (see []). The main contribution of this paper is to study the rela-tionship between the characteristic function of a meromorphic functionf(z) with maximal deﬁciency sum and that of the exact diﬀerencecf =f(z+c) –f(z), wherec=  (see []).

In , Shan and Singh [] proved the following theorem.

Theorem A[] Suppose that f(z)is a transcendental meromorphic function of ﬁnite order and_a∈_Cδ(a,f) = .Then

Tr,f∼T(r,f), r→+∞.

After that, Edrei [] and Weitsman [] proved the following theorem, respectively.

Theorem B[, ] Suppose that f(z)is a transcendental meromorphic function of ﬁnite order with maximal deﬁciency sum.Then

lim r→+∞

T(r,f)

T(r,f) =  –δ(∞,f),

and

lim r→+∞

N(r,_f)

T(r,f) = .

Under the condition of Theorem B, Singh and Gopalakrishna [] proved that

lim r→+∞

N(r,a)

T(r,f) =  –δ(a,f) (.) holds for everya∈C.

Letf(z) be a transcendental meromorphic function of order less than one. Bergweiler and Langley [] proved thatcf(z)∼f(z) outside some exceptional set. Motivated by this

result, we extend Theorem B to the exact diﬀerencecfand prove the following theorem.

Theorem .(main) Suppose that f(z)is a transcendental meromorphic function of order less than one with maximal deﬁciency sum.Then we have

() lim_r→+∞T(r,_T(r,f)cf)=  –δ(∞,f). () lim_r→+∞N(r,



cf) T(r,cf) = .

Consequently,we have that the deﬁciency ofcf with respect tois,i.e.,

δ(,cf) =  –lim sup r→+∞

N(r,_cf)

T(r,cf)

= .

For the zeros and poles involving the derivative of a transcendental meromorphic func-tion of ﬁnite order with maximal deﬁciency sum, Singh and Kulkarni [] proved the fol-lowing theorem.

Theorem C[] Suppose that f(z)is a transcendental meromorphic function of ﬁnite order with maximal deﬁciency sum.Then

 –δ(∞,f)  –δ(∞,f)≤K

f≤( –δ(∞,f))

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where

Kf=lim sup r→+∞

N(r,f) +N(r,_f)

T(r,f) .

In , Fang [] proved the following theorem.

Theorem D[] Suppose that f(z)is a transcendental meromorphic function of ﬁnite order with maximal deﬁciency sum.Then

Kf=( –δ(∞,f))  –δ(∞,f) .

In fact, Fang [] proved that Theorem D is valid for higher order derivatives off(z). In this paper, we shall extend Theorem D to the exact diﬀerencecf and prove the following

theorem.

Theorem .(main) Suppose that f(z)is a transcendental meromorphic function of order less than one with maximal deﬁciency sum.Then

K(cf)≤

( –δ(∞,f))  –δ(∞,f) ,

where

K(cf) =lim sup r→+∞

N(r,cf) +N(r,cf)

T(r,cf)

.

The following example shows that the upper bound of the inequality in Theorem . is accurate.

Example . Letf(z) = 

ez_–, thencf = (–e c_)ez

(ec_ez_–)(ez_–). Thencf = ,δ(,f) = ,δ(–,f) =

,δ(∞,f) = . Thusf(z) is a meromorphic function with maximal deﬁciency sum. It is obvious thatδ(,ez_{) = ,}_δ₍_∞_,_ez_{) =  and}_N₍_r_,

cf) =N(r,ez= ) +N(r,ez=e–c). It follows

from (.) that

Nr,ez_{= }₌_N_r_,_ez₌_e–c_∼_T_r_,_ez_, _r_→₊_∞

and from Valiron-Mo’honko identity (.) that

T(r,cf)∼T

r,ez, r→+∞.

Therefore,K(cf) = (–_–δδ(∞,f(∞,f))) = .

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Question . Under the condition of Theorem ., can we replaceK(cf)≤(–_–δδ(∞,f(∞,f)))by

K(cf) =(–_–δδ(∞,f(∞,f)))?

Corollary . Let f(z)be a transcendental meromorphic function of order less than one with maximal deﬁciency sum,and assumeδ(∞,f) = .Then

lim r→+∞

N(r,cf)

T(r,cf)

= .

Consequently,we have that the deﬁciency ofcf with respect to∞is,i.e.,

δ(∞,cf) =  –lim sup r→+∞

N(r,cf)

T(r,cf)

= .

As the end of this paper, we shall prove the following theorem.

Theorem .(main) Let f(z)be a transcendental meromorphic function of order less than one,and assumeδ(∞,f) = .Then

a∈C

δ(a,f)≤δ(,cf).

2 Some lemmas

Lemma .[] Let f(z)be a meromorphic function of ﬁnite orderσ,and let c be a non-zero complex number.Then,for eachε> ,we have

m

r,f(z+c)

f(z)

=Orσ–+ε.

Lemma . Let f(z)be a transcendental meromorphic function of orderσ (< ),and let c be a non-zero complex number.Then

m

r,f(z+c)

f(z)

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

Proof Since the order off(z) is less than one, then, for any  <ε<  –σ, it follows from Lemma . that

m

r,f(z+c)

f(z)

=Orσ–+ε=O().

Therefore, we have

m

r,f(z+c)

f(z)

=O() =oT(r,f)=S(r,f).

Lemma .[] Let f(z)be a meromorphic function with the exponent of convergence of polesλ(_f) =λ< +∞,and let c be a non-zero complex number.Then,for eachε> ,we have

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From Lemma ., using a similar method as that in the proof of Lemma ., we can prove the following lemma.

Lemma . Let f(z)be a transcendental meromorphic function of order less than one,and let c be a non-zero complex number.Then

Nr,f(z+c)=N(r,f) +S(r,f).

3 Proof of Theorem 1.1

Proof By combining the ﬁrst main theorem of Nevanlinna theory and Lemmas ., ., we have

T(r,cf) =m(r,cf) +N(r,cf)

=m

r,fcf

f

+N(r,cf) ≤m

r,cf

f

+m(r,f) +N(r,f) +Nr,f(z+c)+O() =T(r,f) +N(r,f) +S(r,f).

Hence,

lim sup r→∞

T(r,cf)

T(r,f) ≤ +lim supr→∞

N(r,f)

T(r,f) =  –δ(∞,f). (.) Let{ai}be a sequence of distinct complex numbers inCcontaining all the ﬁnite deﬁcient

values off(z). For any positiveq, deﬁne

F(z) =

q

i=



f –ai

.

Since T(r,f(z) – ai) = T(r,f(z)) +O() and c(f(z) – ai) = cf(z), we deduce from

Lemma . that

mr,F(z)cf(z)

≤ q

i=

m

r,c(f(z) –ai)

f(z) –ai

+logq=S(r,f).

This relation yields

mr,F(z)=m

r,F(z)cf(z)



cf(z)

≤m

r, 

cf

+S(r,f). (.)

By combining the ﬁrst main theorem of Nevanlinna theory, (.) and Valiron-Mo’honko identity (.), we have

qT(r,f) +N

r, 

cf

=Tr,F(z)+N

r, 

cf

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=mr,F(z)+Nr,F(z)+N r, 

cf

+O()

≤mr,F(z)+N

r, 

cf + q i=

N(r,ai) +O()

≤m

r, 

cf

+N

r, 

cf + q i=

N(r,ai) +S(r,f)

=T(r,cf) + q

i=

N(r,ai) +S(r,f).

Hence,

q≤lim inf r→∞

T(r,cf)

T(r,f) +

q

i= lim sup

r→∞

N(r,ai)

T(r,f)

=lim inf r→∞

T(r,cf)

T(r,f) +

q

i=

 –δ(ai,f) .

Thus

lim inf r→∞

T(r,cf)

T(r,f) ≥

q

i=

δ(ai,f).

Sinceqis arbitrary, we have

lim inf r→∞

T(r,cf)

T(r,f) ≥

a∈C

δ(a,f) =  –δ(∞,f).

Then

lim r→+∞

T(r,cf)

T(r,f) =  –δ(∞,f). (.) On the other hand, by combining the ﬁrst main theorem of Nevanlinna theory and (.), we have

q

i=

m(r,ai) +N

r, 

cf

≤mr,F(z)+N

r, 

cf

+O()

≤T(r,cf) +S(r,f).

Thus

q

i=

m(r,ai)

T(r,cf)

+N(r,

 cf)

T(r,cf)

≤ + S(r,f)

T(r,cf)

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We derive from (.) that

q

i= lim inf

r→+∞

m(r,ai)

T(r,cf)

+lim sup r→+∞

N(r,_cf)

T(r,cf) ≤ +lim sup

r→+∞

S(r,f)

T(r,cf) ≤ +lim sup

r→+∞

S(r,f)

T(r,f)

T(r,cf)

= .

Thus

≥lim sup r→+∞

N(r,  cf)

T(r,cf)

+

q

i= lim inf

r→+∞

m(r,ai)

T(r,cf)

≥lim sup

r→+∞

N(r,_cf)

T(r,cf)

+

q

i= lim inf

r→+∞

m(r,ai)

T(r,f) lim infr→+∞

T(r,f)

T(r,cf)

.

It follows from (.) that

N(r,_cf)

T(r,cf)

+

q

i=δ(ai,f)

 –δ(∞,f). Sinceqis arbitrary, we have

N(r,  cf)

T(r,cf)

+ .

Then

lim sup r→+∞

N(r,_cf)

T(r,cf)

= .

Therefore,

lim r→+∞

N(r,_cf)

T(r,cf)

= .

Proof It follows from Lemma . that

N(r,cf)≤N(r,f) +S(r,f).

The above inequality implies that

N(r,cf)

T(r,cf)

T(r,f) ≤

N(r,f)

T(r,f)+

S(r,f)

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By Theorem .(), we have

 –δ(∞,f)lim sup r→+∞

N(r,cf)

T(r,cf)

≤ –δ(∞,f).

Therefore,

lim sup r→+∞

N(r,cf)

T(r,cf)≤

( –δ(∞,f))  –δ(∞,f) . This relation and Theorem .() together yield

K(cf)≤

( –δ(∞,f))

 –δ(∞,f) .

Proof If_a∈_Cδ(a,f) = , Theorem . is valid in this case. In the following, we assume that_a∈_Cδ(a,f) > . Let{aμ}be a sequence of distinct complex numbers inCcontaining all the ﬁnite deﬁcient values off(z). For any positive integerq, as we did in the proof of Theorem .(), we can get that

q

μ=

m(r,aμ) +N

r, 

cf

≤T(r,cf) +S(r,f)

holds for anyqﬁnite complex numbers in{aμ}. Therefore, we have

N(r,_cf)

T(r,cf)

+ T(r,f)

T(r,cf) q

μ=m(r,aμ)

T(r,f) –o()

≤, r→+∞.

Hence, from (.) we can get

_N₍_r_,  cf)

T(r,cf)

+ T(r,f)

T(r,cf) q

μ=m(r,aμ)

T(r,f) –o()

≥lim sup

r→+∞

N(r,_cf)

T(r,cf)

+lim inf r→+∞

T(r,f)

T(r,cf) q

μ=m(r,aμ)

T(r,f) –o()

≥lim sup

r→+∞

N(r,_cf)

T(r,cf)

+lim inf r→+∞

T(r,f)

T(r,cf) lim inf

r→+∞

q

μ=m(r,aμ)

T(r,f)

≥lim sup

r→+∞

N(r,_cf)

T(r,cf)

+

q

j=δ(aμ,f)

 –δ(∞,f) . Sinceqis arbitrary andδ(∞,f) = , we have

a∈C

δ(a)≤δ(,cf).

Competing interests

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Acknowledgements

This research was partly supported by the National Natural Science Foundation of China (Grant No. 11201395) and by the Science Foundation of Educational Commission of Hubei Province (Grant No. Q20132801, D20132804).

Received: 13 August 2013 Accepted: 30 August 2013 Published:11 Nov 2013 References

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