Journal of Applied MathematicsVolume 2011 (2011), Article ID 790942, 9 pagesdoi:10.1155/2011/790942

Research Article

Normal Criteria of Function Families Concerning Shared Values

Wenjun Yuan,¹ Bing Zhu,² and Jianming Lin³

¹School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China²College of Computer Engineering Technology, Guangdong Institute of Science and Technology, Zhuhai 519090, China³School of Economic and Management, Guangzhou University of Chinese Medicine, Guangzhou 510006, China

Received 10 July 2011; Accepted 7 September 2011

Academic Editor: Hui-Shen Shen

Copyright © 2011 Wenjun Yuan 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 concerning shared values. We consider whether a family of meromorphic functions ℱ is normal in , if, for every pair of functions and in ℱ, and share the value , where and are two finite complex numbers such that , is a positive integer. Some examples show that the conditions in our results are best possible.

1. Introduction and Main Results

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 ([2–4] or [1]).

Bloch's principle [5] states that every condition which reduces a meromorphic function in the plane to be a constant forces a family of meromorphic functions in a domain to be 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 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, in [9–11] and so on. In recent years, this subject has attracted the attention of many researchers worldwide.

We now first introduce a normality criterion related to a Hayman normal conjecture [12].

Theorem 1.1. Let be a family of holomorphic (meromorphic) functions defined in a domain , . If for each function and , then is normal in .

The results for the holomorphic case are due to Drasin [7] for , Pang [13] for , Chen and Fang [14] for , Ye [15] for , and Chen and Gu [16] for the generalized result with and replaced by meromorphic functions. The results for the meromorphic case are due to Li [17], Li [18] and Langley [19] for , Pang [13] for , Chen and Fang [14] for , and Zalcman [20] for , obtained independently.

When and is meromorphic, Theorem 1.1 is not valid in general. Fang and Yuan [21] gave an example to this, and moreover a result added other conditions below.

Example 1.2. The family of meromorphic functions is not normal in . This is deduced by , as and Marty's criterion [2], although, for any .

Here denotes the spherical derivative

Theorem 1.3. Let be a family of meromorphic functions in a domain , and . If and the poles of are of multiplicity ≥3 for each , then is normal in .

In 2008, by the ideas of shared values, Zhang [11] proved the following.

Theorem 1.4. Let be a family of meromorphic (holomorphic) functions in , a positive integer, and , two finite complex numbers such that . If and, for every pair of functions and in , and share the value , then is normal in .

Example 1.5 (see [11]). The family of meromorphic functions is not normal in . Obviously . So for each pair and share the value 0 in , but is not normal at the point , since , as .

Remark 1.6. Example 1.5 shows that Theorem 1.4 is not valid when , and the condition is best possible for meromorphic case.

In this paper, we will consider the similar relations and prove the following results.

Theorem 1.7. Let be a family of meromorphic functions in , a positive integer, and , two finite complex numbers such that . If and, for every pair of functions and in , and share the value , then is normal in .

Example 1.8. The family of holomorphic functions is not normal in . This is deduced by , as and Marty's criterion [2], although, for any .

Remark 1.9. Example 1.8 shows that the condition that added in Theorem 1.7 is best possible. In Theorem 1.7 taking we get Corollary 1.10 obtained by Zhang [22].

Corollary 1.10. Let be a family of meromorphic functions in , , and let be a nonzero finite complex number. If, for every pair of functions and in , and share the value , then is normal in .

A natural problem is what conditions are added such that Theorem 1.7 holds when . Next we give an answer.

Theorem 1.11. Let be a family of meromorphic functions in , and let and be two finite complex numbers such that . Suppose that all of zeros are multiple for each . If, for every pair of functions and in , and share the value , then is normal in .

Remark 1.12. Example 1.8 shows that the condition that all of zeros are multiple for each added in Theorem 1.7 is best possible. In Theorem 1.11 taking we get Corollary 1.13.

Corollary 1.13. Let be a family of meromorphic functions in , and let be a nonzero finite complex number. Suppose that all of zeros are multiple for each . If, for every pair of functions and in , and share the value , then is normal in .

From the proof of Theorem 1.7 we know that the following corollary holds.

Corollary 1.14. Let be a family of meromorphic functions in , be a positive integer and , be two finite complex numbers such that . If for each function in , , then is normal in .

2. Preliminary Lemmas

In order to prove our result, we need the following lemmas. The first one extends a famous result by Zalcman [23] concerning normal families.

Lemma 2.1 (see [24]). Let be a family of meromorphic functions on the unit disc satisfying all zeros of functions in that have multiplicity and all poles of functions in that have multiplicity . Let be a real number satisfying . Then is not normal at 0 if and only if there exist (a); (b); (c); (d)such that converges spherically uniformly on each compact subset of to a nonconstant meromorphic function , whose all zeros have multiplicity and all poles 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 Nevanlinna's characteristic function :

Lemma 2.3 (see [25] or [26]). Let be a meromorphic function and . If has neither simple zero nor simple pole, and , then is constant.

Lemma 2.4 (see [27]). Let be a transcendental meromorphic function of finite order in and have no simple zero, then assumes every nonzero finite value infinitely often.

3. Proof of the Results

Proof of Theorem 1.7. 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 .From (3.1) we know in locally uniformly with respect to the spherical metric, where is the set of all poles of .If , then , where is a constant. This contradicts with being a meromorphic function. So .If , by Lemma 2.3, then is also a constant which is a contradiction with being a nonconstant. Hence, is a nonconstant meromorphic function and has at least one zero.Next we prove that has just a unique zero. On the contrary, let and be two distinct zeros of , and choose small enough such that , where and . From (3.2), by theorem, there exist points , such that for sufficiently large By the hypothesis that, for each pair of functions and in , and share in , we know that for any positive integerFix , take , and note , , then . Since the zeros of have no accumulation point, so Hence, . This contradicts with , and . So has just a unique zero, which can be denoted by . By Lemma 2.4, is not any transcendental function.If is a nonconstant polynomial, then , where A is a nonzero constant, is a positive integer, because . Set , then and . Note that the zeros of are of multiplicity . But has only one zero , so has only the same zero too. Hence, which contradicts with . Therefore, and are rational functions which are not polynomials, and has just a unique zero .Next we prove that there exists no rational function such as . Noting that , we can set where is a nonzero constant, . For stating briefly, denote From (3.6), where are polynomials. Since has only a unique zero , set where is a nonzero constant, so where is a polynomial. From (3.8) we also have where is also a polynomial.Let denote the degree of a polynomial .From (3.8) and (3.9), Similarly from (3.11), (3.12) and noting (3.13), Note that , it follows from (3.8) and (3.10) that and . Thus, , and then is a factor of . Hence, we get that . Combining (3.11) and (3.12) we also have . By (3.14) we obtain Since , we know by (3.16) that If , by (3.15), then Noting (3.17), we obtain ; a contradiction.If , from (3.8) and (3.10), then . Noting that , , and , hence . By (3.16), . From (3.17), we obtain ; a contradiction.The proof of Theorem 1.7 is complete.

Proof of Theorem 1.11. The proof of this theorem is the same as the proof of Theorem 1.7, some different places are stated as follows. The zeros of are multiple; The zeros of are of multiplicity ≥4: Noting , by (3.16) we have If , by (3.15), then Noting (3.17), we obtain ; a contradiction.If , from (3.8) and (3.10), then . Noting that , , and , hence . By (3.16), . From (), we obtain ; a contradiction.The proof of Theorem 1.11 is complete.

Acknowledgment

The authors would like to express their hearty thanks to Professor Qingcai Zhang for supplying them his helpful reprint. They wish to thank the managing editor and referees for their very helpful comments and useful suggestions. This work was completed of the support with the NSF of China (10771220) and Doctorial Point Fund of National Education Ministry of China (200810780002).

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