Discrete Dynamics in Nature and SocietyVolume 2011 (2011), Article ID 826164, 6 pagesdoi:10.1155/2011/826164

Research Article

Normal Criteria of Function Families Related to a Result of Drasin

Zhu Bing,¹ Lin Jianming,² and Yuan Wenjun³

¹College of Computer Engineering Technology, Guangdong Institute of Science and Technology, Zhuhai 519090, China²School of Economic and Management, Guangzhou University of Chinese Medicine, Guangzhou 512009, China³School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

Received 7 April 2011; Accepted 20 May 2011

Academic Editor: M. De la Sen

Copyright © 2011 Zhu Bing et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the normality of families of meromorphic functions related to a result of Drasin. We consider whether a family meromorphic functions whose each function does not take zero is normal in , if for every pair of functions and in , and share ∞ or and share 0, where . Some examples show that the conditions in our results are best possible.

1. Introduction and Main Result

Let and be two nonconstant meromorphic functions in a domain , and let be a finite complex value. We say that and share CM (or IM) in provided that and have the same zeros counting (or ignoring) multiplicity in . When , the zeros of mean the poles of (see [1]). It is assumed that the reader is familiar with the standard notations and the basic results of Nevanlinna’s value-distribution theory (see [2–4] or [1]).

Influenced from Bloch’s principle [5], every condition which reduces a meromorphic function in the plane to a constant makes a family of meromorphic functions in a domain normal. Although the principle is false in general (see [6]), many authors proved normality criterion for families of meromorphic functions corresponding to Liouville-Picard type theorem (see [7] or [4]).

It is also more interesting to find normality criteria from the point of view of shared values. In this area, Schwick [8] first proved an interesting result that a family of meromorphic functions in a domain is normal if in which every function shares three distinct finite complex numbers with its first derivative. And later, more results about normality criteria concerning shared values can be found, for instance, (see [9–13]) and so on. In recent years, this subject has attracted the attention of many researchers worldwide.

The following result is due to Schiff [14].

Theorem 1.1. Let be a family holomorphic functions in , and a positive integer. If and the zeros of are of multiplicity for each , where and is a positive integer, then is normal in .

In 2001, Fang and Yuan [15] extended Theorem 1.1 as follows.

Theorem 1.2. Let be a family holomorphic functions in , and a positive integer. If and the zeros of are of multiplicity for each , where , then is normal in .

It is natural to ask whether or not Theorems 1.1 and 1.2 hold for meromorphic case or holomorphic case by the idea of shared values. In this paper, we answer above question and prove the following results.

Theorem 1.3. Let be a family meromorphic functions in , and a positive integer. If (i) and the zeros of are of multiplicity for each , (ii) and share for every pair of functions and in , where , then is normal in .

Theorem 1.4. Let be a family holomorphic functions in , and a positive integer. If (i) for each , (ii) and share the value 0 for every pair of functions and in , where , then is normal in .

Example 1.5. The family of holomorphic functions is not normal in . Obviously . On the other hand, , as . This implies that the family fails to be equicontinuous at 0, and thus is not normal at 0.

Example 1.6. The family of holomorphic functions is normal in . Obviously, and . So for each pair and share the value 1 in . Theorem 1.4 implies that the family is normal in .

Example 1.7. The family of meromorphic functions is normal in . The reason is the conditions of Theorems 1.3 and 1.4 hold, that is, and for each in .

Example 1.8. The family of meromorphic functions is normal in . The reason is the conditions of Theorem 1.3 hold, that is, and has only one pole 0 for each ; for each in .

Remark 1.9. Example 1.5 shows that is not replaced by in Theorems 1.3 and 1.4. is not valid when . All of Examples 1.6, 1.7 and 1.8 show that Theorems 1.3 and 1.4 occur.

2. Preliminary Lemmas

In order to prove our result, we need the following lemmas. The first one extends a famous result by Zalcman [16] concerning normal families.

Lemma 2.1 (see [17]). Let be a family of meromorphic functions on the unit disc satisfying all zeros of functions in having multiplicity and all poles of functions in have multiplicity . Let be a real number satisfying . Then, is not normal at 0 if and only if there exist (a)a number ,(b)points with ,(c)functions ,(d)positive numbers such that converges spherically uniformly on each compact subset of to a nonconstant meromorphic function , whose all zeros of functions in have multiplicity and all poles of functions in have multiplicity and order is at most 2.

Remark 2.2. If is a family of holomorphic functions on the unit disc in Lemma 2.1, then is a nonconstant entire function whose order is at most 1.The order of is defined by using the Nevanlinna’s characteristic function Here, denotes the spherical derivative

Lemma 2.3 (see [3]). Let be a meromorphic function in , then

Remark 2.4. Both (2.3) and (2.4) are called Hayman inequality and Milloux inequality, respectively.

2.1. Proof of the Results

Proof of Theorem 1.3. Suppose that is not normal in . Then, there exists at least one point such that is not normal at the point . Without loss of generality, we assume that . By Lemma 2.1, there exist points , positive numbers and functions such that locally uniformly with respect to the spherical metric, where is a nonconstant meromorphic function in . Moreover, the order of is less than 2.Since , by Hurwitz’s theorem, we see .From (2.5), we know also locally uniformly with respect to the spherical metric.If , then is a polynomial with degree . This contradicts .If , then by Hayman inequality (2.3) of Lemma 2.3, we know that is a constant, which is impossible.Hence, is a nonconstant meromorphic function and has at least one zero. By (2.7) and Hurwitz’s theorem, we see that the zeros of are multiple.Next, we prove that has at most one distinct pole. By contraries, let and be two distinct poles of , and choose small enough such that , where and . From (2.5), by Hurwitz’s theorem, there exist points , such that for sufficiently large By the hypothesis that for each pair of functions and in , and share 0 in , we know that for any positive integer Fix , take , and note , , then . Since the zeros of has no accumulation point, so. Hence This contradicts with and . So, has at most one distinct pole.For applying Milloux inequality (2.4) of Lemma 2.3, we deduce From (2.11), we know that is a rational function of degree at most 2. Noting that , we can write where and are two finite complex numbers. Simple calculating shows that has at least distinct simple zeros. This is impossible.The proof of Theorem 1.3 is complete.

Proof of Theorem 1.4. Similarly with the proof of Theorem 1.3, we have that (2.5) and (2.7) also hold. Moreover, is an entire function.If , then is a polynomial with degree . This contradicts .If , then by Hayman inequality (2.3) of Lemma 2.3 we know that is a constant, which is impossible.Hence, is a nonconstant entire function and has at least one zero.Next, we prove that has one distinct zero. By contraries, let and be two distinct zeros of , and choose small enough such that , where and . From (2.7), by Hurwitz’s theorem, there exist points , such that for sufficiently large By the hypothesis that for each pair of functions and in , and share 0 in , we know that for any positive integer Fix , take , and note , , then . Since the zeros of has no accumulation point, so Hence This contradicts with and . So, has one distinct zero.For applying Milloux inequality (2.4) of Lemma 2.3, we deduce From (2.15) and , we know that is a polynomial of degree 1. We can write , where and are two finite complex numbers. Hence, has no only one distinct zero. This is impossible.The proof of Theorem 1.4 is complete.

Acknowlegment

This work was completed with the support of the NSF of China (no. 10771220) and Doctorial Point Fund of National Education Ministry of China (no. 200810780002).

References

C.-C. Yang and H.-X. Yi, Uniqueness Theory of Meromorphic Functions, vol. 557 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.

Y. X. Gu, X. C. Pang, and M. L. Fang, On Rosenbloom's Fix-Points Theorem and Related Results, Science Press, Beijing, China, 2007.

W. K. Hayman, Meromorphic Functions, Oxford Mathematical Monographs, Clarendon Press, Oxford, UK, 1964.

L. Yang, Value Distribution Theory, Springer, Berlin, Germany, 1993.

W. Bergweiler, “Bloch's principle,” Computational Methods and Function Theory, vol. 6, no. 1, pp. 77–108, 2006.

L. A. Rubel, “Four counterexamples to Bloch's principle,” Proceedings of the American Mathematical Society, vol. 98, no. 2, pp. 257–260, 1986.

D. Drasin, “Normal families and the Nevanlinna theory,” Acta Mathematica, vol. 122, pp. 231–263, 1969.

W. Schwick, “Normality criteria for families of meromorphic functions,” Journal d'Analyse Mathématique, vol. 52, pp. 241–289, 1989.

X. Pang and L. Zalcman, “Normal families and shared values,” The Bulletin of the London Mathematical Society, vol. 32, no. 3, pp. 325–331, 2000.

X. Pang and L. Zalcman, “Normality and shared values,” Arkiv för Matematik, vol. 38, no. 1, pp. 171–182, 2000.

W. J. Yuan, Z. R. Li, and B. Xiao, “Normal families of composite holomorphic functions and sharing a meromorphic function,” Scientia Sinica, Series A, vol. 40, no. 5, pp. 429–436, 2010 (Chinese).

Q. Zhang, “Normal families of meromorphic functions concerning shared values,” Journal of Mathematical Analysis and Applications, vol. 338, no. 1, pp. 545–551, 2008.

Q. Zhang, “Some normality criteria of meromorphic functions,” Complex Variables and Elliptic Equations, vol. 53, no. 8, pp. 791–795, 2008.

J. L. Schiff, Normal Families, Universitext, Springer, New York, NY, USA, 1993.

M. Fang and W. Yuan, “On the normality for families of meromorphic functions,” Indian Journal of Mathematics, vol. 43, no. 3, pp. 341–351, 2001.

L. Zalcman, “A heuristic principle in complex function theory,” The American Mathematical Monthly, vol. 82, no. 8, pp. 813–817, 1975.

L. Zalcman, “Normal families: new perspectives,” Bulletin of the American Mathematical Society, vol. 35, no. 3, pp. 215–230, 1998.