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

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.

1. Introduction

A transcendental meromorphic function is meromorphic in the complex plane 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 of with respect to and Nevanlinna characteristic of . And the lower order and the order are in turn defined as follows:

(1.1)

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.

Given pair of real numbers satisfying

(1.2)

we define

(1.3)

Zheng in [4] proved the following theorem.

Theorem A

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

(1.4)

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.

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

(1.5)

where , assume that are distinct complex numbers, and let be positive integers or satisfying

(1.6)

(1.7)

(1.8)

where . If then

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

(1), ,

(2)

(3).

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

(2.1)

(2.2)

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

Lemma 2.2.

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

(2.3)

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 .

Nevanlinna theory on angular domain will play a key role in the proof of theorems. Let be a meromorphic function on the angular domain , where . Nevanlinna defined the following notations (see [8]):

(2.4)

where and are the poles of on appearing according to their multiplicities. is called the angular counting function of the poles of on and Nevanlinna's angular characteristic is defined as follows:

(2.5)

Throughout, we denote by a quantity satisfying

(2.6)

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.

Let be meromorphic on . Then for arbitrary complex number , we have

(2.7)

and for an integer ,

(2.8)

and .

Lemma 2.4.

Let be meromorphic on . Then for arbitrary distinct , we have

(2.9)

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.

Let be meromorphic on and let be positive integers. Then for arbitrary distinct , we have

(2.10)

where the term will be replaced by when some .

Proof.

According to our notations, we have

(2.11)

By Lemma 2.4,

(2.12)

and (i) follows.

Furthermore, , and on combining this with (i), we get (ii).

Proof of Theorem 1.1.

Suppose . For convenience, below we omit the subscript of all the notations, such as and . By applying Lemma 2.5 to and (1.6), we have

(2.13)

so that

(2.14)

This implies that . We have also (2.14) for alternation of and , then

(2.15)

By (1.8), we have

(2.16)

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

(2.17)

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

(2.18)

where , and

(2.19)

Applying Lemma 2.1 to gives the existence of the Polya peak of order of such that , and then from Lemma 2.2 for sufficiently large we have

(2.20)

since . We can assume for all the , (13) holds. Set

(2.21)

Then from (2.18) and (2.20) it follows that

(2.22)

It is easy to see that there exists a such that for infinitely many , we have

(2.23)

We can assume for all the , (2.23) holds. Set . Thus from the definition (2.1) of it follows that

(2.24)

On the other hand, by the definition (2.4) of and (2.14), we have

(2.25)

Combining (2.24) with (2.25) gives

(2.26)

Thus from (1.5) in Lemma 2.1 for , we have

(2.27)

This is impossible.

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

(2.28)

This is impossible. Theorem 1.1 follows.

Remark 2.6.

In Theorem A, then

(2.29)

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 , then ,

(ii)if then ,

(iii)if , then ,

(iv)if , then ,

(v)if , then ,

(vi)if , then ,

Corollary 2.8.

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

(2.30)

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

Corollary 2.9.

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

(2.31)

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 ?