# On Uniqueness of Meromorphic Functions with Multiple Values in Some Angular Domains

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## Abstract

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.

### Keywords

Positive Integer Real Number Complex Number Meromorphic Function Positive Real Number## 1. Introduction

For the references, please see [1]. An Open image in new window is called an IM (ignoring multiplicities) shared value in Open image in new window of two meromorphic functions Open image in new window and Open image in new window if in Open image in new window , Open image in new window if and only if Open image in new window . It is Nevanlinna [2] who proved the first uniqueness theorem, called the Five Value Theorem, which says that two meromorphic functions Open image in new window and Open image in new window are identical if they have five distinct IM shared values in Open image in new window . After his very fundamental work, the uniqueness of meromorphic functions with shared values in the whole complex plane attracted many investigations (see [3]). Recently, Zheng in [4] suggested for the first time the investigation of uniqueness of a function meromorphic in a precise subset of Open image in new window , and this is an interesting topic.

Zheng in [4] proved the following theorem.

Theorem A

where Open image in new window , assume that Open image in new window and Open image in new window have five distinct IM shared values in Open image in new window . If Open image in new window , then Open image in new window

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 Open image in new window to denote the set of zeros of Open image in new window in Open image in new window , with multiplicities no greater than Open image in new window , in which each zero counted only once.

Our main result is what follows.

Theorem 1.1.

where Open image in new window . If Open image in new window then Open image in new window

## 2. Proof of Theorem 1.1

First we introduce several lemmas which are crucial in our proofs. The following result was proved in [5] (also see [6]).

Lemma 2.1 (see [5]).

Let Open image in new window be transcendental and meromorphic in Open image in new window with the lower order Open image in new window and the order Open image in new window . Then for arbitrary positive number Open image in new window satisfying Open image in new window and a set Open image in new window with finite linear measure, there exists a sequence of positive numbers Open image in new window such that

(1) Open image in new window , Open image in new window ,

(3) Open image in new window .

The following result is a special version of the main result of Baernstein [7].

Lemma 2.2.

Although Lemma 2.2 was proved in [7] for the Polya peak of order Open image in new window , the same argument of Baernstein [7] can derive Lemma 2.2 for the Polya peak of order Open image in new window .

where Open image in new window denotes a set of positive real numbers with finite linear measure. It is not necessarily the same for every occurrence in the context [9].

Lemma 2.3.

and Open image in new window .

Lemma 2.4.

where the term Open image in new window will be replaced by Open image in new window when some Open image in new window .

We use Open image in new window to denote the zeros of Open image in new window in Open image in new window whose multiplicities are no greater than Open image in new window and are counted only once. Likewise, we use Open image in new window to denote the zeros of Open image in new window in Open image in new window whose multiplicities are greater than Open image in new window and are counted only once.

Lemma 2.5.

where the term Open image in new window will be replaced by Open image in new window when some Open image in new window .

Proof.

and (i) follows.

Furthermore, Open image in new window , and on combining this with (i), we get (ii).

Proof of Theorem 1.1.

We assume that Open image in new window . By the same argument we can show Theorem 1.1 for the case when Open image in new window . By applying Lemma 2.3 and (2.16), we estimate

The following method comes from [10]. But we quote it in detail here because of its independent significance. Note that Open image in new window . We need to treat two cases.

(I)?? Open image in new window Then Open image in new window . And by the inequality (1.5), we can take a real number Open image in new window such that

On the other hand, by the definition (2.4) of Open image in new window and (2.14), we have

This is impossible.

(II)?? Open image in new window Then Open image in new window By the same argument as in (I) with all the Open image in new window replaced by Open image in new window , we can derive

This is impossible. Theorem 1.1 follows.

Remark 2.6.

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.

Corollary 2.7.

In Theorem 1.1,

(i)if Open image in new window , then Open image in new window ,

(ii)if Open image in new window then Open image in new window ,

(iii)if Open image in new window , then Open image in new window ,

(iv)if Open image in new window , then Open image in new window ,

(v)if Open image in new window , then Open image in new window ,

(vi)if Open image in new window , then Open image in new window ,

Corollary 2.8.

where Open image in new window , assume that Open image in new window are Open image in new window distinct complex numbers satisfying that Open image in new window , where Open image in new window is an integer or Open image in new window . If Open image in new window , then Open image in new window .

Corollary 2.9.

where Open image in new window , assume that Open image in new window are Open image in new window distinct complex numbers satisfying that Open image in new window , Open image in new window , then Open image in new window .

Question 1.

For two meromorphic functions defined in Open image in new window , there are many uniqueness theorems when they share small functions ( Open image in new window is called a small function of Open image in new window if Open image in new window ) (see [3]). So we ask an interesting question: are there similar results when they share small functions in some precise domain Open image in new window ?

## Notes

### Acknowlegment

The work is supported by NSF of China (no. 10871108).

### References

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