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Value distribution for difference operator of meromorphic functions with maximal deficiency sum
Journal of Inequalities and Applications volume 2013, Article number: 530 (2013)
Abstract
The main purpose of this paper is to investigate the relationship between the characteristic function of a meromorphic function with maximal deficiency sum and that of the exact difference . As an application, the author also establishes an inequality on the zeros and poles for and gives an example to show that the upper bound of the inequality is accurate.
MSC:30D30, 39A05.
1 Introduction
If is a meromorphic function in the complex plane ℂ and , we use the following notations frequently used in Nevanlinna theory (see [1–3]): , , , , … . Denote by any quantity such that , without restriction if 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
If , then one should replace in the above formula by . The classical second fundamental theorem of Nevanlinna theory asserts that the total deficiency of any meromorphic function satisfies the inequality
If the above equality holds, then we say that f has maximal deficiency sum. The Valiron-Mo’honko identity states that if the function is rational in f and has small meromorphic coefficients, then
Certain relationship between the characteristic function of a meromorphic function with maximal deficiency sum and that of derivative 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 with maximal deficiency sum and that of the exact difference , where (see [5]).
In 1956, Shan and Singh [6] proved the following theorem.
Theorem A [6]
Suppose that is a transcendental meromorphic function of finite order and . Then
After that, Edrei [7] and Weitsman [4] proved the following theorem, respectively.
Suppose that is a transcendental meromorphic function of finite order with maximal deficiency sum. Then
and
Under the condition of Theorem B, Singh and Gopalakrishna [8] proved that
holds for every .
Let be a transcendental meromorphic function of order less than one. Bergweiler and Langley [9] proved that outside some exceptional set. Motivated by this result, we extend Theorem B to the exact difference and prove the following theorem.
Theorem 1.1 (main)
Suppose that is a transcendental meromorphic function of order less than one with maximal deficiency sum. Then we have
-
(1)
.
-
(2)
.
Consequently, we have that the deficiency of with respect to 0 is 1, i.e.,
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 is a transcendental meromorphic function of finite order with maximal deficiency sum. Then
where
In 2000, Fang [11] proved the following theorem.
Theorem D [11]
Suppose that is a transcendental meromorphic function of finite order with maximal deficiency sum. Then
In fact, Fang [11] proved that Theorem D is valid for higher order derivatives of . In this paper, we shall extend Theorem D to the exact difference and prove the following theorem.
Theorem 1.2 (main)
Suppose that is a transcendental meromorphic function of order less than one with maximal deficiency sum. Then
where
The following example shows that the upper bound of the inequality in Theorem 1.2 is accurate.
Example 1.3 Let , then . Then , , , . Thus is a meromorphic function with maximal deficiency sum. It is obvious that , and . It follows from (1.2) that
and from Valiron-Mo’honko identity (1.1) that
Therefore, .
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 by ?
Corollary 1.5 Let be a transcendental meromorphic function of order less than one with maximal deficiency sum, and assume . Then
Consequently, we have that the deficiency of with respect to ∞ is 1, i.e.,
As the end of this paper, we shall prove the following theorem.
Theorem 1.6 (main)
Let be a transcendental meromorphic function of order less than one, and assume . Then
2 Some lemmas
Lemma 2.1 [12]
Let be a meromorphic function of finite order σ, and let c be a non-zero complex number. Then, for each , we have
Lemma 2.2 Let be a transcendental meromorphic function of order σ (<1), and let c be a non-zero complex number. Then
Proof Since the order of is less than one, then, for any , it follows from Lemma 2.1 that
Therefore, we have
 □
Lemma 2.3 [12]
Let be a meromorphic function with the exponent of convergence of poles , and let c be a non-zero complex number. Then, for each , we have
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 be a transcendental meromorphic function of order less than one, and let c be a non-zero complex number. Then
3 Proof of Theorem 1.1
Proof By combining the first main theorem of Nevanlinna theory and Lemmas 2.2, 2.4, we have
Hence,
Let be a sequence of distinct complex numbers in â„‚ containing all the finite deficient values of . For any positive q, define
Since and , we deduce from Lemma 2.2 that
This relation yields
By combining the first main theorem of Nevanlinna theory, (3.2) and Valiron-Mo’honko identity (1.1), we have
Hence,
Thus
Since q is arbitrary, we have
Then
On the other hand, by combining the first main theorem of Nevanlinna theory and (3.2), we have
Thus
We derive from (3.3) that
Thus
It follows from (3.3) that
Since q is arbitrary, we have
Then
Therefore,
 □
4 Proof of Theorem 1.2
Proof It follows from Lemma 2.4 that
The above inequality implies that
By Theorem 1.1(1), we have
Therefore,
This relation and Theorem 1.1(2) together yield
 □
5 Proof of Theorem 1.6
Proof If , Theorem 1.6 is valid in this case. In the following, we assume that . Let be a sequence of distinct complex numbers in â„‚ containing all the finite deficient values of . For any positive integer q, as we did in the proof of Theorem 1.1(2), we can get that
holds for any q finite complex numbers in . Therefore, we have
Hence, from (3.1) we can get
Since q is arbitrary and , we have
 □
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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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Wu, Z. Value distribution for difference operator of meromorphic functions with maximal deficiency sum. J Inequal Appl 2013, 530 (2013). https://doi.org/10.1186/1029-242X-2013-530
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DOI: https://doi.org/10.1186/1029-242X-2013-530