Value distribution for difference operator of meromorphic functions with maximal deficiency sum
© Wu; licensee Springer. 2013
Received: 13 August 2013
Accepted: 30 August 2013
Published: 11 November 2013
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.
Keywordsdifference operator maximal deficiency sum meromorphic function
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 ). 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 ).
In 1956, Shan and Singh  proved the following theorem.
Theorem A 
holds for every .
Let be a transcendental meromorphic function of order less than one. Bergweiler and Langley  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)
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 
In 2000, Fang  proved the following theorem.
Theorem D 
In fact, Fang  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)
The following example shows that the upper bound of the inequality in Theorem 1.2 is accurate.
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 ?
As the end of this paper, we shall prove the following theorem.
Theorem 1.6 (main)
2 Some lemmas
Lemma 2.1 
Lemma 2.3 
From Lemma 2.3, using a similar method as that in the proof of Lemma 2.2, we can prove the following lemma.
3 Proof of Theorem 1.1
4 Proof of Theorem 1.2
5 Proof of Theorem 1.6
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).
- Hayman WK Oxford Mathematical Monographs. In Meromorphic Functions. Clarendon Press, Oxford; 1964.Google Scholar
- Yang L: Value Distribution Theory. Springer, Berlin; 1993. Translated and revised from the 1982 Chinese originalMATHGoogle Scholar
- Zheng JH: Value Distribution of Meromorphic Functions. Tsinghua University Press, Beijing; 2010.MATHGoogle Scholar
- Weitsman A: Meromorphic functions with maximal deficiency sum and a conjecture of F. Nevanlinna. Acta Math. 1969, 123: 115–139. 10.1007/BF02392387MathSciNetView ArticleMATHGoogle Scholar
- Halburd RG, Korhonen RJ: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn., Math. 2006, 31: 463–478.MathSciNetMATHGoogle Scholar
- Shan SM, Singh SK: On the derivative of a meromorphic function with maximum defect. Math. Z. 1956, 65: 171–174. 10.1007/BF01473876MathSciNetView ArticleMATHGoogle Scholar
- Edrei A: Sums of deficiencies of meromorphic functions II. J. Anal. Math. 1967, 19: 53–74. 10.1007/BF02788709MathSciNetView ArticleMATHGoogle Scholar
- Singh SK, Gopalakrishna HS: Exceptional values of entire and meromorphic functions. Math. Ann. 1971, 191: 121–142. 10.1007/BF02330565MathSciNetView ArticleMATHGoogle Scholar
- Bergweiler W, Langley JK: Zeros of differences of meromorphic functions. Math. Proc. Camb. Philos. Soc. 2007, 142: 133–147. 10.1017/S0305004106009777MathSciNetView ArticleMATHGoogle Scholar
- Singh SK, Kulkarni VN: Characteristic function of a meromorphic function and its derivative. Ann. Pol. Math. 1973, 28: 123–133.MathSciNetMATHGoogle Scholar
- Fang ML: A note on a result of Singh and Kulkarni. Int. J. Math. Math. Sci. 2000, 23: 285–288. 10.1155/S016117120000082XMathSciNetView ArticleMATHGoogle Scholar
- Chiang YM, Feng SJ:On the Nevanlinna characteristic of and difference equations in the complex plane. Ramanujan J. 2008, 16: 105–129. 10.1007/s11139-007-9101-1MathSciNetView ArticleMATHGoogle Scholar
- 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.2009.10.021MathSciNetView ArticleMATHGoogle Scholar
- 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.MathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.