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.
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.
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.
Our main result is what follows.
2. Proof of Theorem 1.1
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
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 .
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.
and (i) follows.
Proof of Theorem 1.1.
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.
This is impossible.
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,
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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