Advertisement

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

Open Access
Research Article

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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

A transcendental meromorphic function is meromorphic in the complex plane Open image in new window and not rational. We assume that the readers are familiar with the Nevanlinna theory of meromorphic functions and the standard notations such as Nevanlinna deficiency Open image in new window of Open image in new window with respect to Open image in new window and Nevanlinna characteristic Open image in new window of Open image in new window . And the lower order Open image in new window and the order Open image in new window are in turn defined as follows:

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.

we define

Zheng in [4] proved the following theorem.

Theorem A

Let Open image in new window and Open image in new window be both transcendental meromorphic functions, and let Open image in new window be of finite order Open image in new window and such that for some Open image in new window and an integer Open image in new window . For Open image in new window pair of real numbers Open image in new window satisfying (1.2) and

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.

Let Open image in new window and Open image in new window be both transcendental meromorphic functions, and let Open image in new window be of finite order Open image in new window and such that for some Open image in new window and an integer Open image in new window . For Open image in new window pair of real numbers Open image in new window satisfying (1.2) and
where Open image in new window , assume that Open image in new window are Open image in new window distinct complex numbers, and let Open image in new window be positive integers or Open image in new window satisfying

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 ,

(2) Open image in new window

(3) Open image in new window .

A sequence Open image in new window satisfying (1), (2), and (3) in Lemma 2.1 is called Polya peak of order Open image in new window outside Open image in new window in this article. For Open image in new window and Open image in new window define

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

Lemma 2.2.

Let Open image in new window be transcendental and meromorphic in Open image in new window with the finite lower order Open image in new window and the order Open image in new window and for some Open image in new window . Then for arbitrary Polya peak Open image in new window of order Open image in new window , we have

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 .

Nevanlinna theory on angular domain will play a key role in the proof of theorems. Let Open image in new window be a meromorphic function on the angular domain Open image in new window , where Open image in new window . Nevanlinna defined the following notations (see [8]):
where Open image in new window and Open image in new window are the poles of Open image in new window on Open image in new window appearing according to their multiplicities. Open image in new window is called the angular counting function of the poles of Open image in new window on Open image in new window and Nevanlinna's angular characteristic is defined as follows:
Throughout, we denote by Open image in new window a quantity satisfying

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.

Let Open image in new window be meromorphic on Open image in new window . Then for arbitrary complex number Open image in new window , we have

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.

According to our notations, we have
By Lemma 2.4,

and (i) follows.

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

Proof of Theorem 1.1.

Suppose Open image in new window . For convenience, below we omit the subscript of all the notations, such as Open image in new window and Open image in new window . By applying Lemma 2.5 to Open image in new window and (1.6), we have
This implies that Open image in new window . We have also (2.14) for alternation of Open image in new window and Open image in new window , then
By (1.8), we have

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

Applying Lemma 2.1 to Open image in new window gives the existence of the Polya peak Open image in new window of order Open image in new window of Open image in new window such that Open image in new window , and then from Lemma 2.2 for sufficiently large Open image in new window we have
since Open image in new window . We can assume for all the Open image in new window , (13) holds. Set
Then from (2.18) and (2.20) it follows that
It is easy to see that there exists a Open image in new window such that for infinitely many Open image in new window , we have
We can assume for all the Open image in new window , (2.23) holds. Set Open image in new window . Thus from the definition (2.1) of Open image in new window it follows that

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

Combining (2.24) with (2.25) gives
Thus from (1.5) in Lemma 2.1 for Open image in new window , 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.

Let Open image in new window and Open image in new window be both transcendental meromorphic functions and let Open image in new window be of finite lower order Open image in new window and such that for some Open image in new window and an integer Open image in new window . For Open image in new window pair of real numbers Open image in new window satisfying (1.2) and

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.

Let Open image in new window and Open image in new window be both transcendental meromorphic functions and let Open image in new window be of finite lower order Open image in new window and such that for some Open image in new window and an integer Open image in new window . For Open image in new window pair of real numbers Open image in new window satisfying (1.2) and

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

  1. 1.
    Hayman WK: Meromorphic Functions. Clarendon Press, Oxford, UK; 1964.MATHGoogle Scholar
  2. 2.
    Nevanlinna R: Le théorème de Picard-Borel et la Théorie des Fonctions Méromorphes. Gauthier-Villars, Paris, France; 1929.MATHGoogle Scholar
  3. 3.
    Yi HX, Yang C-C: Uniqueness Theorey of Meromorphic Functions. Kluwer Academic Publishers, Boston, Mass, USA; 2003.Google Scholar
  4. 4.
    Zheng JH: On uniqueness of meromorphic functions with shared values in some angular domains. Canadian Mathematical Bulletin 2004, 47: 152–160. 10.4153/CMB-2004-016-1MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Yang L: Borel directions of meromorphic functions in an angular domain. Sci.Sinica 1979, 149–163.Google Scholar
  6. 6.
    Edrei A: Meromorphic functions with three radially distributed values. Transactions of the American Mathematical Society 1955, 78: 276–293. 10.1090/S0002-9947-1955-0067982-9MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Baernstein A: Proof of Edrei's spread conjecture. Proceedings of the London Mathematical Society 1973, 26: 418–434. 10.1112/plms/s3-26.3.418MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Goldberg AA, Ostrovskii IV: The Distribution of Values of Meromorphic Functions. Nauka, Moscow, Russia; 1970.Google Scholar
  9. 9.
    Yang L, Yang C-C: Angular distribution of value of . Science in China 1994, 37: 284–294.MathSciNetMATHGoogle Scholar
  10. 10.
    Zheng JH: On transcendental meromorphic functions with radially distributed values. Science in China 2004, 47: 401–416. 10.1360/02ys0210CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Zu-Xing Xuan. 2009

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.

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityHaidian District, BeijingChina
  2. 2.Department of Basic CoursesBeijing Union UniversityChaoyang District, BeijingChina

Personalised recommendations