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Research | Open | Published:

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

Abstract

The main purpose of this paper is to investigate the relationship between the characteristic function of a meromorphic function f(z) with maximal deficiency sum and that of the exact difference Δ c f=f(z+c)f(z). As an application, the author also establishes an inequality on the zeros and poles for Δ c f and gives an example to show that the upper bound of the inequality is accurate.

MSC:30D30, 39A05.

1 Introduction

If f(z) is a meromorphic function in the complex plane and aC, we use the following notations frequently used in Nevanlinna theory (see [13]): m(r,f), N(r,f), m(r,a)=m(r, 1 f a ), N(r,a)=N(r, 1 f a ), … . Denote by S(r,f) any quantity such that S(r,f)=o(T(r,f)), r without restriction if f(z) is of finite order and otherwise except possibly for a set of values of r of finite linear measure. The Nevanlinna deficiency of f with respect to a finite complex number a is defined by

δ(a,f)= lim inf r m ( r , a ) T ( r , f ) =1 lim sup r N ( r , a ) T ( r , f ) .

If a=, then one should replace m(r,a)(N(r,a)) in the above formula by m(r,f)(N(r,f)). The classical second fundamental theorem of Nevanlinna theory asserts that the total deficiency of any meromorphic function f(z) satisfies the inequality

a C δ(a,f)+δ(,f)2.

If the above equality holds, then we say that f has maximal deficiency sum. The Valiron-Mo’honko identity states that if the function R(z,f) is rational in f and has small meromorphic coefficients, then

T ( r , R ( z , f ) ) = deg f (R)T(r,f)+S(r,f).
(1.1)

Certain relationship between the characteristic function of a meromorphic function f(z) with maximal deficiency sum and that of derivative f (z) plays a key role in the study of a conjecture of Nevanlinna (see [4]). The main contribution of this paper is to study the relationship between the characteristic function of a meromorphic function f(z) with maximal deficiency sum and that of the exact difference Δ c f=f(z+c)f(z), where c0 (see [5]).

In 1956, Shan and Singh [6] proved the following theorem.

Theorem A [6]

Suppose that f(z) is a transcendental meromorphic function of finite order and a C δ(a,f)=2. Then

T ( r , f ) 2T(r,f),r+.

After that, Edrei [7] and Weitsman [4] proved the following theorem, respectively.

Theorem B [4, 7]

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 ) =2δ(,f),

and

lim r + N ( r , 1 f ) T ( r , f ) =0.

Under the condition of Theorem B, Singh and Gopalakrishna [8] proved that

lim r + N ( r , a ) T ( r , f ) =1δ(a,f)
(1.2)

holds for every aC.

Let f(z) be a transcendental meromorphic function of order less than one. Bergweiler and Langley [9] proved that Δ c f(z) f (z) outside some exceptional set. Motivated by this result, we extend Theorem B to the exact difference Δ c f and prove the following theorem.

Theorem 1.1 (main)

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

  1. (1)

    lim r + T ( r , Δ c f ) T ( r , f ) =2δ(,f).

  2. (2)

    lim r + N ( r , 1 Δ c f ) T ( r , Δ c f ) =0.

Consequently, we have that the deficiency of Δ c f with respect to 0 is 1, i.e.,

δ(0, Δ c f)=1 lim sup r + N ( r , 1 Δ c f ) T ( r , Δ c f ) =1.

For the zeros and poles involving the derivative of a transcendental meromorphic function of finite order with maximal deficiency sum, Singh and Kulkarni [10] proved the following theorem.

Theorem C [10]

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

1 δ ( , f ) 2 δ ( , f ) K ( f ) 2 ( 1 δ ( , f ) ) 2 δ ( , f ) ,

where

K ( f ) = lim sup r + N ( r , f ) + N ( r , 1 f ) T ( r , f ) .

In 2000, Fang [11] proved the following theorem.

Theorem D [11]

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

K ( f ) = 2 ( 1 δ ( , f ) ) 2 δ ( , f ) .

In fact, Fang [11] proved that Theorem D is valid for higher order derivatives of f(z). In this paper, we shall extend Theorem D to the exact difference Δ c f and prove the following theorem.

Theorem 1.2 (main)

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

K( Δ c f) 2 ( 1 δ ( , f ) ) 2 δ ( , f ) ,

where

K( Δ c f)= lim sup r + N ( r , Δ c f ) + N ( r , 1 Δ c f ) T ( r , Δ c f ) .

The following example shows that the upper bound of the inequality in Theorem 1.2 is accurate.

Example 1.3 Let f(z)= 1 e z 1 , then Δ c f= ( 1 e c ) e z ( e c e z 1 ) ( e z 1 ) . Then Δ c f0, δ(0,f)=1, δ(1,f)=1, δ(,f)=0. Thus f(z) is a meromorphic function with maximal deficiency sum. It is obvious that δ(0, e z )=1, δ(, e z )=1 and N(r, Δ c f)=N(r, e z =1)+N(r, e z = e c ). It follows from (1.2) that

N ( r , e z = 1 ) =N ( r , e z = e c ) T ( r , e z ) ,r+

and from Valiron-Mo’honko identity (1.1) that

T(r, Δ c f)2T ( r , e z ) ,r+.

Therefore, K( Δ c f)= 2 ( 1 δ ( , f ) ) 2 δ ( , f ) =1.

By Theorem D and Example 1.3, we pose the following question.

Question 1.4 Under the condition of Theorem 1.2, can we replace K( Δ c f) 2 ( 1 δ ( , f ) ) 2 δ ( , f ) by K( Δ c f)= 2 ( 1 δ ( , f ) ) 2 δ ( , f ) ?

Corollary 1.5 Let f(z) be a transcendental meromorphic function of order less than one with maximal deficiency sum, and assume δ(,f)=1. Then

lim r + N ( r , Δ c f ) T ( r , Δ c f ) =0.

Consequently, we have that the deficiency of Δ c f with respect tois 1, i.e.,

δ(, Δ c f)=1 lim sup r + N ( r , Δ c f ) T ( r , Δ c f ) =1.

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

Theorem 1.6 (main)

Let f(z) be a transcendental meromorphic function of order less than one, and assume δ(,f)=1. Then

a C δ(a,f)δ(0, Δ c f).

2 Some lemmas

Lemma 2.1 [12]

Let f(z) be a meromorphic function of finite order σ, and let c be a non-zero complex number. Then, for each ε>0, we have

m ( r , f ( z + c ) f ( z ) ) =O ( r σ 1 + ε ) .

Lemma 2.2 Let f(z) be a transcendental meromorphic function of order σ (<1), and let c be a non-zero complex number. Then

m ( r , f ( z + c ) f ( z ) ) =o ( T ( r , f ) ) =S(r,f).

Proof Since the order of f(z) is less than one, then, for any 0<ε<1σ, it follows from Lemma 2.1 that

m ( r , f ( z + c ) f ( z ) ) =O ( r σ 1 + ε ) =O(1).

Therefore, we have

m ( r , f ( z + c ) f ( z ) ) =O(1)=o ( T ( r , f ) ) =S(r,f).

 □

Lemma 2.3 [12]

Let f(z) be a meromorphic function with the exponent of convergence of poles λ( 1 f )=λ<+, and let c be a non-zero complex number. Then, for each ε>0, we have

N ( r , f ( z + c ) ) =N(r,f)+O ( r λ 1 + ε ) +O(logr).

From Lemma 2.3, using a similar method as that in the proof of Lemma 2.2, we can prove the following lemma.

Lemma 2.4 Let f(z) be a transcendental meromorphic function of order less than one, and let c be a non-zero complex number. Then

N ( r , 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 2.2, 2.4, we have

T ( r , Δ c f ) = m ( r , Δ c f ) + N ( r , Δ c f ) = m ( r , f Δ c f f ) + N ( r , Δ c f ) m ( r , Δ c f f ) + m ( r , f ) + N ( r , f ) + N ( r , f ( z + c ) ) + O ( 1 ) = T ( r , f ) + N ( r , f ) + S ( r , f ) .

Hence,

lim sup r T ( r , Δ c f ) T ( r , f ) 1+ lim sup r N ( r , f ) T ( r , f ) =2δ(,f).
(3.1)

Let { a i } be a sequence of distinct complex numbers in containing all the finite deficient values of f(z). For any positive q, define

F(z)= i = 1 q 1 f a i .

Since T(r,f(z) a i )=T(r,f(z))+O(1) and Δ c (f(z) a i )= Δ c f(z), we deduce from Lemma 2.2 that

m ( r , F ( z ) Δ c f ( z ) ) i = 1 q m ( r , Δ c ( f ( z ) a i ) f ( z ) a i ) +logq=S(r,f).

This relation yields

m ( r , F ( z ) ) =m ( r , F ( z ) Δ c f ( z ) 1 Δ c f ( z ) ) m ( r , 1 Δ c f ) +S(r,f).
(3.2)

By combining the first main theorem of Nevanlinna theory, (3.2) and Valiron-Mo’honko identity (1.1), we have

q T ( r , f ) + N ( r , 1 Δ c f ) = T ( r , F ( z ) ) + N ( r , 1 Δ c f ) + O ( 1 ) = m ( r , F ( z ) ) + N ( r , F ( z ) ) + N ( r , 1 Δ c f ) + O ( 1 ) m ( r , F ( z ) ) + N ( r , 1 Δ c f ) + i = 1 q N ( r , a i ) + O ( 1 ) m ( r , 1 Δ c f ) + N ( r , 1 Δ c f ) + i = 1 q N ( r , a i ) + S ( r , f ) = T ( r , Δ c f ) + i = 1 q N ( r , a i ) + S ( r , f ) .

Hence,

q lim inf r T ( r , Δ c f ) T ( r , f ) + i = 1 q lim sup r N ( r , a i ) T ( r , f ) = lim inf r T ( r , Δ c f ) T ( r , f ) + i = 1 q { 1 δ ( a i , f ) } .

Thus

lim inf r T ( r , Δ c f ) T ( r , f ) i = 1 q δ( a i ,f).

Since q is arbitrary, we have

lim inf r T ( r , Δ c f ) T ( r , f ) a C δ(a,f)=2δ(,f).

Then

lim r + T ( r , Δ c f ) T ( r , f ) =2δ(,f).
(3.3)

On the other hand, by combining the first main theorem of Nevanlinna theory and (3.2), we have

i = 1 q m ( r , a i ) + N ( r , 1 Δ c f ) m ( r , F ( z ) ) + N ( r , 1 Δ c f ) + O ( 1 ) T ( r , Δ c f ) + S ( r , f ) .

Thus

i = 1 q m ( r , a i ) T ( r , Δ c f ) + N ( r , 1 Δ c f ) T ( r , Δ c f ) 1+ S ( r , f ) T ( r , Δ c f ) .

We derive from (3.3) that

i = 1 q lim inf r + m ( r , a i ) T ( r , Δ c f ) + lim sup r + N ( r , 1 Δ c f ) T ( r , Δ c f ) 1 + lim sup r + S ( r , f ) T ( r , Δ c f ) 1 + lim sup r + S ( r , f ) T ( r , f ) T ( r , f ) T ( r , Δ c f ) = 1 .

Thus

1 lim sup r + N ( r , 1 Δ c f ) T ( r , Δ c f ) + i = 1 q lim inf r + m ( r , a i ) T ( r , Δ c f ) lim sup r + N ( r , 1 Δ c f ) T ( r , Δ c f ) + i = 1 q lim inf r + m ( r , a i ) T ( r , f ) lim inf r + T ( r , f ) T ( r , Δ c f ) .

It follows from (3.3) that

1 lim sup r + N ( r , 1 Δ c f ) T ( r , Δ c f ) + i = 1 q δ ( a i , f ) 2 δ ( , f ) .

Since q is arbitrary, we have

1 lim sup r + N ( r , 1 Δ c f ) T ( r , Δ c f ) +1.

Then

lim sup r + N ( r , 1 Δ c f ) T ( r , Δ c f ) =0.

Therefore,

lim r + N ( r , 1 Δ c f ) T ( r , Δ c f ) =0.

 □

4 Proof of Theorem 1.2

Proof It follows from Lemma 2.4 that

N(r, Δ c f)2N(r,f)+S(r,f).

The above inequality implies that

N ( r , Δ c f ) T ( r , Δ c f ) T ( r , Δ c f ) T ( r , f ) 2 N ( r , f ) T ( r , f ) + S ( r , f ) T ( r , f ) .

By Theorem 1.1(1), we have

( 2 δ ( , f ) ) lim sup r + N ( r , Δ c f ) T ( r , Δ c f ) 2 ( 1 δ ( , f ) ) .

Therefore,

lim sup r + N ( r , Δ c f ) T ( r , Δ c f ) 2 ( 1 δ ( , f ) ) 2 δ ( , f ) .

This relation and Theorem 1.1(2) together yield

K( Δ c f) 2 ( 1 δ ( , f ) ) 2 δ ( , f ) .

 □

5 Proof of Theorem 1.6

Proof If a C δ(a,f)=0, Theorem 1.6 is valid in this case. In the following, we assume that a C δ(a,f)>0. Let { a μ } be a sequence of distinct complex numbers in containing all the finite deficient values of f(z). For any positive integer q, as we did in the proof of Theorem 1.1(2), we can get that

μ = 1 q m(r, a μ )+N ( r , 1 Δ c f ) T(r, Δ c f)+S(r,f)

holds for any q finite complex numbers in { a μ }. Therefore, we have

N ( r , 1 Δ c f ) T ( r , Δ c f ) + T ( r , f ) T ( r , Δ c f ) ( μ = 1 q m ( r , a μ ) T ( r , f ) o ( 1 ) ) 1,r+.

Hence, from (3.1) we can get

1 lim sup r + [ N ( r , 1 Δ c f ) T ( r , Δ c f ) + T ( r , f ) T ( r , Δ c f ) ( μ = 1 q m ( r , a μ ) T ( r , f ) o ( 1 ) ) ] lim sup r + N ( r , 1 Δ c f ) T ( r , Δ c f ) + lim inf r + T ( r , f ) T ( r , Δ c f ) ( μ = 1 q m ( r , a μ ) T ( r , f ) o ( 1 ) ) lim sup r + N ( r , 1 Δ c f ) T ( r , Δ c f ) + lim inf r + T ( r , f ) T ( r , Δ c f ) lim inf r + μ = 1 q m ( r , a μ ) T ( r , f ) lim sup r + N ( r , 1 Δ c f ) T ( r , Δ c f ) + j = 1 q δ ( a μ , f ) 2 δ ( , f ) .

Since q is arbitrary and δ(,f)=1, we have

a C δ(a)δ(0, Δ c f).

 □

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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).

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Correspondence to Zhaojun Wu.

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The author declares that he has no competing interests.

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Keywords

  • difference operator
  • maximal deficiency sum
  • meromorphic function