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

- Zu-Xing Xuan
^{1, 2}Email author

**2009**:208516

https://doi.org/10.1155/2009/208516

© Zu-Xing Xuan. 2009

**Received: **4 March 2009

**Accepted: **30 June 2009

**Published: **3 August 2009

## 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

## 1. Introduction

For the references, please see [1]. An is called an IM (ignoring multiplicities) shared value in of two meromorphic functions and if in , if and only if . It is Nevanlinna [2] 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 [3]). Recently, Zheng in [4] 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 [4] proved the following theorem.

Theorem A

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.

Theorem 1.1.

## 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 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 [7].

Lemma 2.2.

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

where 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.

Lemma 2.4.

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.

Lemma 2.5.

where the term will be replaced by when some .

Proof.

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 [10]. 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.

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,

Corollary 2.8.

where , assume that are distinct complex numbers satisfying that , where is an integer or . If , then .

Corollary 2.9.

where , assume that are distinct complex numbers satisfying that , , then .

Question 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 [3]). So we ask an interesting question: are there similar results when they share small functions in some precise domain ?

## Declarations

### Acknowlegment

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

## Authors’ Affiliations

## References

- Hayman WK:
*Meromorphic Functions*. Clarendon Press, Oxford, UK; 1964.MATHGoogle Scholar - Nevanlinna R:
*Le théorème de Picard-Borel et la Théorie des Fonctions Méromorphes*. Gauthier-Villars, Paris, France; 1929.MATHGoogle Scholar - Yi HX, Yang C-C:
*Uniqueness Theorey of Meromorphic Functions*. Kluwer Academic Publishers, Boston, Mass, USA; 2003.Google Scholar - Zheng JH:
**On uniqueness of meromorphic functions with shared values in some angular domains.***Canadian Mathematical Bulletin*2004,**47:**152–160. 10.4153/CMB-2004-016-1MathSciNetView ArticleMATHGoogle Scholar - Yang L:
**Borel directions of meromorphic functions in an angular domain.***Sci.Sinica*1979, 149–163.Google Scholar - Edrei A:
**Meromorphic functions with three radially distributed values.***Transactions of the American Mathematical Society*1955,**78:**276–293. 10.1090/S0002-9947-1955-0067982-9MathSciNetView ArticleMATHGoogle Scholar - Baernstein A:
**Proof of Edrei's spread conjecture.***Proceedings of the London Mathematical Society*1973,**26:**418–434. 10.1112/plms/s3-26.3.418MathSciNetView ArticleMATHGoogle Scholar - Goldberg AA, Ostrovskii IV:
*The Distribution of Values of Meromorphic Functions*. Nauka, Moscow, Russia; 1970.Google Scholar - Yang L, Yang C-C:
**Angular distribution of value of .***Science in China*1994,**37:**284–294.MathSciNetMATHGoogle Scholar - Zheng JH:
**On transcendental meromorphic functions with radially distributed values.***Science in China*2004,**47:**401–416. 10.1360/02ys0210View ArticleMathSciNetMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.