- Research Article
- Open Access
On Uniqueness of Meromorphic Functions with Multiple Values in Some Angular Domains
© Zu-Xing Xuan. 2009
- Received: 4 March 2009
- Accepted: 30 June 2009
- Published: 3 August 2009
This article deals with problems of the uniqueness of transcendental meromorphic function with shared values in some angular domains dealing with the multiple values which improve a result of J. Zheng.
- Positive Integer
- Real Number
- Complex Number
- Meromorphic Function
- Positive Real Number
For the references, please see . An is called an IM (ignoring multiplicities) shared value in of two meromorphic functions and if in , if and only if . It is Nevanlinna  who proved the first uniqueness theorem, called the Five Value Theorem, which says that two meromorphic functions and are identical if they have five distinct IM shared values in . After his very fundamental work, the uniqueness of meromorphic functions with shared values in the whole complex plane attracted many investigations (see ). Recently, Zheng in  suggested for the first time the investigation of uniqueness of a function meromorphic in a precise subset of , and this is an interesting topic.
Zheng in  proved the following theorem.
where , assume that and have five distinct IM shared values in . If , then
However, it was not discussed whether there are similar results dealing with multiple values in some angular domains. In this paper we investigate this problem.
We use to denote the set of zeros of in , with multiplicities no greater than , in which each zero counted only once.
Our main result is what follows.
where . If then
Lemma 2.1 (see ).
Let be transcendental and meromorphic in with the lower order and the order . Then for arbitrary positive number satisfying and a set with finite linear measure, there exists a sequence of positive numbers such that
(1) , ,
The following result is a special version of the main result of Baernstein .
where denotes a set of positive real numbers with finite linear measure. It is not necessarily the same for every occurrence in the context .
where the term will be replaced by when some .
We use to denote the zeros of in whose multiplicities are no greater than and are counted only once. Likewise, we use to denote the zeros of in whose multiplicities are greater than and are counted only once.
where the term will be replaced by when some .
and (i) follows.
Furthermore, , and on combining this with (i), we get (ii).
Proof of Theorem 1.1.
We assume that . By the same argument we can show Theorem 1.1 for the case when . By applying Lemma 2.3 and (2.16), we estimate
The following method comes from . But we quote it in detail here because of its independent significance. Note that . We need to treat two cases.
(I)?? Then . And by the inequality (1.5), we can take a real number such that
On the other hand, by the definition (2.4) of and (2.14), we have
This is impossible.
(II)?? Then By the same argument as in (I) with all the replaced by , we can derive
This is impossible. Theorem 1.1 follows.
so Theorem A is a special case of Theorem 1.1. Meanwhile, Zheng in [4, pages 153–154] gave some examples to indicate that the conditions are necessary. So the conditions in theorem are also necessary.
In Theorem 1.1,
(i)if , then ,
(ii)if then ,
(iii)if , then ,
(iv)if , then ,
(v)if , then ,
(vi)if , then ,
where , assume that are distinct complex numbers satisfying that , where is an integer or . If , then .
where , assume that are distinct complex numbers satisfying that , , then .
For two meromorphic functions defined in , there are many uniqueness theorems when they share small functions ( is called a small function of if ) (see ). So we ask an interesting question: are there similar results when they share small functions in some precise domain ?
The work is supported by NSF of China (no. 10871108).
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