You are viewing the new article page. Let us know what you think. Return to old version

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 $a∈C$, we use the following notations frequently used in Nevanlinna theory (see ): $m(r,f)$, $N(r,f)$, $m(r,a)=m(r, 1 f − a )$, $N(r,a)=N(r, 1 f − a )$, … . 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 ). 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 $c≠0$ (see ).

In 1956, Shan and Singh  proved the following theorem.

Theorem A 

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  and Weitsman  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

$lim r → + ∞ T ( r , f ′ ) T ( r , f ) =2−δ(∞,f),$

and

$lim r → + ∞ N ( r , 1 f ′ ) T ( r , f ′ ) =0.$

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

$lim r → + ∞ N ( r , a ) T ( r , f ) =1−δ(a,f)$
(1.2)

holds for every $a∈C$.

Let $f(z)$ be a transcendental meromorphic function of order less than one. Bergweiler and Langley  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)

$lim r → + ∞ T ( r , Δ c f ) T ( r , f ) =2−δ(∞,f)$.

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  proved the following theorem.

Theorem C 

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  proved the following theorem.

Theorem D 

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  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 $Δ c f≠0$, $δ(0,f)=1$, $δ(−1,f)=1$, $δ(∞,f)=0$. 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 

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 $0<ε<1−σ$, 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 

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,

$lim sup r → ∞ T ( r , Δ c f ) T ( r , f ) ≤1+ lim sup r → ∞ N ( r , f ) T ( r , f ) =2−δ(∞,f).$
(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

$lim inf r → ∞ T ( r , Δ c f ) T ( r , f ) ≥ ∑ a ∈ C δ(a,f)=2−δ(∞,f).$

Then

$lim r → + ∞ T ( r , Δ c f ) T ( r , f ) =2−δ(∞,f).$
(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

$∑ a ∈ C δ(a)≤δ(0, Δ c f).$

□

## References

1. 1.

Hayman WK Oxford Mathematical Monographs. In Meromorphic Functions. Clarendon Press, Oxford; 1964.

2. 2.

Yang L: Value Distribution Theory. Springer, Berlin; 1993. Translated and revised from the 1982 Chinese original

3. 3.

Zheng JH: Value Distribution of Meromorphic Functions. Tsinghua University Press, Beijing; 2010.

4. 4.

Weitsman A: Meromorphic functions with maximal deficiency sum and a conjecture of F. Nevanlinna. Acta Math. 1969, 123: 115–139. 10.1007/BF02392387

5. 5.

Halburd RG, Korhonen RJ: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn., Math. 2006, 31: 463–478.

6. 6.

Shan SM, Singh SK: On the derivative of a meromorphic function with maximum defect. Math. Z. 1956, 65: 171–174. 10.1007/BF01473876

7. 7.

Edrei A: Sums of deficiencies of meromorphic functions II. J. Anal. Math. 1967, 19: 53–74. 10.1007/BF02788709

8. 8.

Singh SK, Gopalakrishna HS: Exceptional values of entire and meromorphic functions. Math. Ann. 1971, 191: 121–142. 10.1007/BF02330565

9. 9.

Bergweiler W, Langley JK: Zeros of differences of meromorphic functions. Math. Proc. Camb. Philos. Soc. 2007, 142: 133–147. 10.1017/S0305004106009777

10. 10.

Singh SK, Kulkarni VN: Characteristic function of a meromorphic function and its derivative. Ann. Pol. Math. 1973, 28: 123–133.

11. 11.

Fang ML: A note on a result of Singh and Kulkarni. Int. J. Math. Math. Sci. 2000, 23: 285–288. 10.1155/S016117120000082X

12. 12.

Chiang YM, Feng SJ:On the Nevanlinna characteristic of $f(z+η)$ and difference equations in the complex plane. Ramanujan J. 2008, 16: 105–129. 10.1007/s11139-007-9101-1

13. 13.

Chen ZX, Shon KH: Value distribution of meromorphic solutions of certain difference Painleve equations. J. Math. Anal. Appl. 2010, 364: 556–566. 10.1016/j.jmaa.2009.10.021

14. 14.

Xu HY, Cao TB, Liu BX: The growth of solutions of systems of complex q-shift difference equations. Adv. Differ. Equ. 2012, 2012(216):1–22.

## 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).

## Author information

Correspondence to Zhaojun Wu.

### Competing interests

The author declares that he has no competing interests.

## Rights and permissions

Reprints and Permissions 