Journal of Applied Mathematics

Volume 2012 (2012), Article ID 312324, 7 pages

doi:10.1155/2012/312324

Research Article

Normal Criterion Concerning Shared Values

Wei Chen,¹ Yingying Zhang,¹ Jiwen Zeng,² and Honggen Tian¹

¹School of Mathematical Sciences, Xinjiang Normal University, Xinjiang, Urumqi 830054, China

²School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361105, China

Received 12 June 2012; Accepted 21 August 2012

Academic Editor: Yansheng Liu

Copyright © 2012 Wei Chen 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 normal criterion of meromorphic functions shared values, we obtain the following. Let be a family of meromorphic functions in a domain , such that function has zeros of multiplicity at least 2, there exists nonzero complex numbers depending on satisfying is a constant;   for some ;   or , then is normal. These results improve some earlier previous results.

1. Introduction and Main Results

We use to denote the open complex plane, to denote the extended complex plane and to denote a domain in . A family of meromorphic functions defined in is said to be normal, if for any sequence contains a subsequence which converges spherically, and locally, uniformly in to a meromorphic function or . Clearly is said to be normal in if and only if it is normal at every point of see [1].

Let be a domain in . For meromorphic on and , set

Two meromorphic functions and on are said to share the value if . Let and be complex numbers. If whenever , we write

If and , we write

According to Bloch’s principle [2], every condition which reduces a meromorphic function in the plane to constant forces a family of meromorphic functions in domain normal. Although the principle is false in general (see [3]), many authors proved normality criterion for families of meromorphic functions by starting from Liouville-Picard type theorem (see [4]). It is also more interesting to find normality criteria from the point of view of shared values. In this area, Schwick [5] 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 have emerged [6–9]. In recent years, this subject has attracted the attention of many researchers worldwide.

In this paper, we use to denote the spherical distance between and and the definition of the spherical distance can be found in [10].

In , Fang and Zalcman [11] proved the following results.

Theorem 1.1 (see [11]). Let be a transcendental function. Let and be complex numbers, and let be positive integers, then assumes every value infinitely often.

Theorem 1.2 (see [11]). Let be a transcendental function. Let and be complex numbers, and let be positive integers. If for every has multiple zeros, and , then is normal in .

In 2009, Xu et al. [12] proved the following results.

Theorem 1.3 (see [12]). Let be a transcendental function. Let and let be complex numbers, and be positive integers, which satisfy , then assumes each value infinitely often.

Theorem 1.4 (see [12]). Let be a transcendental function. Let and be complex numbers, and let be positive integers, which satisfy . If for every has only zeros of multiplicity at least , and satisfies , then is normal in .

In Theorems 1.2 and 1.4, the constants are the same for each . Now we will prove the condition for the constants be the same can be relaxed to some extent.

Theorem A. Let be a family of meromorphic functions in the unit disc , and be a positive integer and . For every , such that all zeros of have multiplicity at least , there exist finite nonzero complex numbers depending on satisfying that

(i) is a constant; (ii) for some ; (iii).

Then is normal in .

Theorem B. Let be a family of meromorphic functions in the unit disc , and be a positive integer. For every , such that all zeros of have multiplicity at least , there exist finite nonzero complex numbers depending on satisfying that

(i)is a constant; (ii) for some ; (iii).

Then is normal in .

2. Some Lemmas

In order to prove our theorems, we require the following results.

Lemma 2.1 (see [7]). Let be a family of meromorphic functions in a domain , and be a positive integer, such that each function has only zeros of multiplicity at least , and suppose that there exists such that whenever . If is not normal at , then for each , there exist a sequence of points , a sequence of positive numbers , and a subsequence of functions such that

locally uniformly with respect to the spherical metric in , where is a nonconstant meromorphic function, all of whose zeros have multiplicity at least , such that . Morever,   has order at most .

Here as usual, is the spherical derivative.

Lemma 2.2 (see [10]). Let be any positive number. Then, Möbius transformation satisfies , for some constants , and also satisfy the uniform Lipschitz condition

where is a constant depending on .

3. Proof of Theorems

Proof of Theorem A. Let . We can find nonzero constants and satisfying . For each , define a Möbius map by , thus .

Next we will show is normal in . Suppose to the contrary, is not normal in . Then by Lemma 2.1. We can find , and , such that converges locally uniformly with respect to the spherical metric to a nonconstant meromorphic function whose zeros of multiplicity at least and spherical derivative is limited and has order at most .

We now consider three cases.

Case 1. If , then is a polynomial with degree at most , a contradiction.

Case 2. If there exists such that . Noting that . By Hurwitz’s theorem, there exist a sequence of points such that (for large enough )

Hence . This contradicts with the suppose of Theorem A.

Case 3. If . Let be the solution of the equation , then . When is a rational function, then is also a rational function. By Picard Theorem we can deduce that is a constant . Hence is a polynomial with degree at most . This contradicts with has zeros of multiplicity at least . When is a transcendental function, combining with the second main theorem, we have

Hence, , a contradiction.

Hence is normal and equicontinuous in . There given , where is the constant of Lemma 2.2, there exists such that for the spherical distance ,

for each . Hence by Lemma 2.2.

Therefore, the family is equicontinuous in . This completes the proof of Theorem A.

Proof of Theorem B. Let . We can find nonzero constants and satisfying . For each , define a Möbius map by , thus .

Next we will show is normal in . Suppose to the contrary, is not normal in . Then by Lemma 2.1. We can find , and , such that converges locally uniformly with respect to the spherical metric to a nonconstant meromorphic function whose spherical derivate is limited and has order at most .

We will also consider three cases.

Case 1. If , then is a polynomial with degree at most 1, a contradiction.

Case 2. If there exists such that . Noting that . By Hurwitz’s theorem, there exist a sequence of points such that (for large enough )

Hence , then we have by the condition (iii).

Thus

This is a contradiction.

Case 3. If . Let be the solution of the equation , then . When is a rational function, then is also a rational function. By Picard theorem we can deduce that is a constant . Hence is a polynomial with degree at most . This contradicts with has zeros of multiplicity at least . When is a transcendental function, combining with the second main theorem, we have

Hence, , a contradiction.

Hence is normal and equicontinuous in . There given , where is the constant of Lemma 2.2, there exists such that for the spherical distance ,

for each . Hence by Lemma 2.2.

Therefore, the family is equicontinuous in . This completes the proof of Theorem B.

Remark 3.1. Using the similar argument, if the condition (iii) when is replaced by (iii) when , then is normal too.

Authors’ Contribution

W. Chen performed the proof and drafted the paper. All authors read and approved the final paper.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgment

This paper is supported by Nature Science Foundation of Fujian Province (2012J01022). The authors wish to thank the referee for some valuable corrections.

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