Normality Criteria of Lahiri's Type and Their Applications
© Xiao-Bin Zhang et al. 2011
Received: 22 September 2010
Accepted: 9 February 2011
Published: 7 March 2011
We prove two normality criteria for families of some functions concerning Lahiri's type, the results generalize those given by Charak and Rieppo, Xu and Cao. As applications, we study a problem related to R. Brück's Conjecture and obtain a result that generalizes those given by Yang and Zhang, Lü, Xu and Chen.
1. Introduction and Main Results
Let denote the complex plane, and let be a nonconstant meromorphic function in . It is assumed that the reader is familiar with the standard notion used in the Nevanlinna value distribution theory such as the characteristic function , the proximity function , the counting function (see, e.g., [1–4]), and denotes any quantity that satisfies the condition as outside of a possible exceptional set of finite linear measure. A meromorphic function is called a small function with respect to , provided that .
Let and be two nonconstant meromorphic functions. Let and be small functions of and . means and have the same zeros (counting multiplicity) and means that and have the same poles (counting multiplicity). If whenever , we write . If and , we write . If , then we say that and share .
Let be a family of meromorphic functions defined in a domain . is said to be normal in , in the sense of Montel, if for any sequence , there exists a subsequence such that converges spherically locally uniformly in , to a meromorphic function or .
According to Bloch's principle, every condition which reduces a meromorphic function in to a constant makes a family of meromorphic functions in a domain normal. Although the principle is false in general, many authors proved normality criteria for families of meromorphic functions starting from Picard type theorems, for instance.
Theorem A (see ).
In 2005, Lahiri  got a normality criterion as follows.
In 2009, Charak and Rieppo  generalized Theorem C and obtained two normality criteria of Lahiri's type.
Very recently, Xu and Cao  further extended Theorems D and E by replacing with ; they got
To prove Theorems D–G, the authors used a key lemma (Lemma 2.4 in this paper) besides Zalcman-Pang's Lemma. It's natural to ask whether such normality criteria of Lahiri's type still hold for the general differential monomial . We study this problem and obtain the following theorem
As an application of Theorem 1.1, we obtain the following theorem.
Let be a family of holomorphic functions in a domain , for every , all zeros of have multiplicity at least . Let be two finite values and be nonnegative integers with , , . For every , all zeros of have multiplicity at least , if , then is normal in .
Let and . If , let , where is the root of . For each function , , in . However, it can be easily verified that is not normal in . Example 1.5 shows that the multiplicity restriction on zeros of in Theorem 1.3 is sharp (at least for ).
2. Preliminary Lemmas
Lemma 2.1 (see ).
such that locally uniformly with respect to the spherical metric, where is a nonconstant meromorphic function on , all of whose zeros have multiplicity at least , such that . Here, as usual, is the spherical derivative.
Lemma 2.2 (see [1, page 158]).
Lemma 2.3 (see ).
Lemma 2.4 (see ).
Take nonnegative integers with , and define . Let be a transcendental meromorphic function with the deficiency . Then for any nonzero value , the function has infinitely many zeros. Moreover, if , the deficient condition can be omitted.
The following two lemmas can be seen as supplements of Lemma 2.4.
Take nonnegative integers with , and define . Let be a transcendental meromorphic function whose zeros have multiplicity at least . Then for any nonzero value , the function has infinitely many zeros, provided that and when . Specially, if is transcendental entire, the function has infinitely many zeros.
Since the case has been proved by Charak and Rieppo , we only need to consider .
This proves Lemma 2.6.
Lemma 2.6 is a generalization of Lemma 2.2 in . The proof of Lemma 2.6 is quite different from its proof. Actually, the proof of Lemma 2.2 in  is incorrect. The main problem appears at (2.2) in . Since has only zero with multiplicity at least , then each zero of is of multiplicity at least , but this does not mean that each zero of is of multiplicity at least because the zeros of may not be the zeros of , and thus their multiplicity may less than . Therefore, the inequality of (2.2) in  is not valid, which implies that the proof of Lemma 2.2 in  is not correct.
This proves Lemma 2.8.
Lemma 2.9 (see [2, page 51]).
3. Proof of Theorem 1.1
This completes the proof of Theorem 1.1.
4. Proof of Theorem 1.2
We can arrive at a contradiction by using the same argument as in the latter part of proof of Theorem 1.1.
This completes the proof of Theorem 1.2.
Proof of Theorem 1.3.
We shall divide our argument into two cases.
Suppose that is not normal in , by Lemma 2.1, there exist , , , and such that locally uniformly with respect to the spherical metric, where is a nonconstant entire function, all of whose zeros have multiplicity at least . By Hurwitz's theorem, we have
but by Lemmas 2.5 and 2.6 we get a contradiction again.
This proves Theorem 5.1.
Further more, using Theorem 1.3, we obtain a uniqueness theorem related to R. Brück's Conjecture. Firstly, we recall this conjecture.
R. Brück's Conjecture
It's interesting to ask what happens if is replaced by in Brück's Conjecture. Recently, Yang and Zhang  considered this problem and got the following theorem.
Lü et al.  improves Theorem H and obtained the following theorem.
We obtain a more general result as follows.
Proof of Theorem 5.1.
This completes the proof of Theorem 5.1.
The authors thank the referees for reading the manuscript very carefully and making a number of valuable suggestions to improve the readability of the paper. The authors were supported by NSF of China (no. 10771121), NSF of Shandong Province (no. Z2008A01) and NSF of Guangdong Province (no. 9452902001003278).
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