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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 793810, 8 pages
http://dx.doi.org/10.1155/2013/793810
Research Article

Borel Directions and Uniqueness of Meromorphic Functions

1School of Mathematics, Shandong University, Jinan, Shandong 250100, China
2Department of Mathematics, Qilu Normal University, Jinan, Shandong 250013, China
3Department of Informatics and Engineering, Jingdezhen Ceramic Institute, Jingdezhen, Jiangxi 333403, China

Received 15 April 2013; Accepted 23 July 2013

Academic Editor: Paul Eloe

Copyright © 2013 Keyu Zhang et al. 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.

Abstract

We investigate the relationship between Borel directions and uniqueness of meromorphic functions and obtain some results of meromorphic functions sharing four distinct values IM and one set in an angular domain containing a Borel line. Our result is an improvement of a recent theorem given by Long and Wu (2012).

1. Introduction and Main Results

We use to denote the open complex plane, to denote the extended complex plane, and  () to denote an angular domain. The fundamental results and the standard notations of the Nevanlinna value distribution theory of meromorphic functions will be used in [1, 2]. In addition, the order of meromorphic function is defined by

Let be a set of distinct elements in and . Define where if and .

Let and be two nonconstant meromorphic functions in . If , then we say and share the set CM (counting multiplicities) in . If , then we say and share the set IM (ignoring multiplicities) in . In particular, when , , we say and share the value CM in if , and we say and share the value IM in if . When , we give the simple notation as before, , and so on (see [3]).

Nevanlinna (see [4]) proved the following well-known theorems.

Theorem 1 (see [4]). If and are two nonconstant meromorphic functions that share five distinct values IM in , then .

Theorem 2 (see [4]). If and are two distinct nonconstant meromorphic functions that share four distinct values CM in , then is a Möbius transformation of , two of the shared values, say and , are Picard values, and the cross ratio .

After their very work, many investigations studied the uniqueness of meromorphic functions with shared values in the whole complex plane (see [5]). Zheng studied the uniqueness problem under the condition that five values and four values are shared in some angular domain in around 2003 (see [6, 7]). It is an interesting topic to investigate the uniqueness with shared values in the angular domain; see [3, 612]. The basic notations and definitions of meromorphic functions in an angular domain will be introduced as follows (see [1, 6, 7]).

Let be a meromorphic function on the angular domain and . Define where and are the poles of on counted according to their multiplicities. is called the Nevanlinna’s angular characteristic, and is called the angular counting function of the poles of on , and is the reduced function of . Similarly, when , we will use the notations , , , , and so on.

It is well known that angular distribution is an interesting topic of value distribution of meromorphic function in complex analysis, and Borel directions played an important role in the topic of angular distribution (see [1324]). Valiron [16] proved that every meromorphic function of finite order has at least one Borel direction of order . Chuang [25] investigated the existence of Borel direction of meromorphic function of infinite order. To state the Chuang’s results, we will introduce the definition as follows.

Definition 3 (see [25]). Let be a meromorphic function of infinite order, is a real function satisfying the following conditions: (i) is continuous, nondecreasing for and as ; (ii)where ; (iii) Then is called infinite order of meromorphic function . This definition is given by Chuang [25].

Let be infinite order of meromorphic function ; we will denote by the set of meromorphic function satisfying ; that is,

Let , , , and . The definition of Borel direction of meromorphic functions of infinite order is given as follows.

Definition 4 (see [25]). Let be meromorphic functions of infinite order ; if for any , the equality holds for any complex number , at most except two exceptions, where is the counting function of zeros of the function in the angular domain , counting multiplicities, then the ray is called a Borel direction of order of meromorphic function .

Remark 5. Chuang [25] proved that every meromorphic function of infinite order has at least one Borel direction of infinite order .

In 2012, Long and Wu [26] investigated the problem concerning Borel direction and shared value of meromorphic functions and obtained the following theorems.

Theorem 6 (see [26, Theorem 1.1]). Let be a meromorphic function of infinite order , and let , be one Borel direction of order of meromorphic function ; let be five distinct complex numbers. If and share IM in the angular domain for any , then .

Theorem 7 (see [26, Theorem 1.2]). Let be a meromorphic function of infinite order , and let , be one Borel direction of order of meromorphic function ; let be four distinct complex numbers. If and share CM in the angular domain for any , then is a Möbius transformation of .
Thus, a question arises naturally: could the nature of sharing the values be further relaxed in Theorems 6 and 7?

In this paper, we will deal with the above question and obtain the following result which is an improvement of Theorem 6.

Theorem 8. Let be a meromorphic function of infinite order , and let , be one Borel direction of order of meromorphic function ; we assume that and share four distinct values IM in , and , for any , where , and . Then and share all values CM; thus, it follows that either or is a Möbius transformation of . Furthermore, if the number of the values in is odd, then .

Remark 9. The special case of this theorem immediately yields Theorem 6. In fact, when , set . If , share IM, which implies , then by Theorem 8, we can get .

2. Some Lemmas

To prove our result, we require the following lemmas.

Lemma 10 (see [27]). Let be a nonconstant meromorphic function on . Then for arbitrary complex number , we have where as .

Lemma 11 (see [7, 28]). Suppose that is a nonconstant meromorphic function in one angular domain with ; then, for arbitrary distinct , we have where the term will be replaced by when some and

Lemma 12 (see [27, Page 138]). Let be a nonconstant meromorphic function in the whole complex plane . Given one angular domain on . Then, for any , we have where and is a positive constant not depending on and .

Remark 13. Nevanlinna conjectured that when tends to outside an exceptional set of finite linear measure, and he proved that when the function is meromorphic in and has finite order. In 1975, Goldberg [28] constructed a counter example to show that (13) is not valid.

Remark 14. From Lemmas 11 and 12, we can get the following conclusion: where is stated as in (10), is infinite order of meromorphic function , and is a set of finite linear measure.

Remark 15. From the definition of , , , , and Lemmas 1012, we can see that the properties of , and are the same as for the more familiar quantities , , and , respectively.

Lemma 16 (see [29]). Let be meromorphic function of infinite order . Then the ray is one Borel direction of order of meromorphic function if and only if satisfies the equality for any .

By using the same argument as in [8, Lemma 1] and [5], we can get the lemma below easily.

Lemma 17. Suppose that is a nonconstant meromorphic function with infinite order , and the ray is one Borel direction of order of meromorphic function . Let be a polynomial of with degree , where the coefficients are constants, and let be distinct finite complex numbers. Then for any ,

Lemma 18. Suppose that is a nonconstant meromorphic function with infinite order , and the ray is one Borel direction of order of meromorphic function . We assume that for any for any , and share four distinct values   IM in angular domain and . Let Then, .

Proof. Since the ray is one Borel direction of meromorphic function of order, thus, for any , we can assume that and (or ) with multiplicity and (or ) with multiplicity . From (17), we can get Hence, is analytic in . By Lemmas 12 and 17, we have where is a polynomial of degree no more than 2 in and is a polynomial of degree no more than 2 in .
Thus, we complete the proof of this lemma.

3. Proof of Theorem 8

Proof. Since is a meromorphic function of infinite order and is one Borel direction of order of the meromorphic function , by Lemma 16, we can get for any And since , we have Set as where is a set of finite linear measure; then as , .
Since and share four distinct values IM in , suppose that and none of the is . By Lemma 11, we have and by interchanging and , we can get that Thus, it follows and
Let be the function expressed in Lemma 18. Then, . By Lemma 11 and (24), for any , we have that is, Similarly, we have From (26) and (27), it follows for any .
Set By Lemma 17 and (28), we can get that Thus, it follows that , . From (28), we see that “almost all” of poles and -points of and in the angular domain are simple. Since , share the four distinct values , in the angular domain and , we can easily get that . Therefore, we have
Since , we can have
Let be the set of those -points of and in the angular domain in which the multiplicities of and at these points are and , respectively. For any , by simple computation, we have Hence, Similarly, we can see that (34) holds for any .
Now, two cases will be considered below.
Case  1. Suppose that , for all positive integers , .
First, we use to denote the counting function of in with respect to the set , and we also use to denote the corresponding reduced counting function. Thus, we have
From the above two equations and (28)–(32), we can see that . And by (34) each zero of is a zero of , so with the help of Lemma 10 and , we can get for some . Since , it follows From the above inequality and (24), we can get Since is of infinite order and , we can get a contradiction to (38).
Case  2. Suppose that , for some positive integers , . From the definitions of and , we have
Next, we take the following two subcases into consideration.
Subcase  2.1. Suppose that . Without loss of generality, we may assume that . For some two positive integers and , if for some , then (39) implies that . Hence, , and which means that any -points of in are multiple. By Lemma 11 and (24), we can get it follows that
From (41) and (42), we can see that “almost all” of -points of have multiplicity 2, and “almost all” of -points of are simple in . Without loss of generality, we may assume that and attain the values and in . Set
Since is analytic at the poles of and of and also at those common -points of and which have multiplicity 1 with respect to and multiplicity 2 with respect to , by Lemma 17, we have . If , then , which contradicts to (41). Then, . Similarly, we have . Therefore, from the definitions of and , we have
Since , from (44), we have which implies that and share CM in . Since and assume the value , there exist positive integers such that . From the considerations above we get , contradicting the fact that and share CM.
Subcase  2.2. Suppose that .
In this subcase, (39) becomes which implies that and share the four values CM in . Then by applying Theorem 7, is a Möbius transformation of . Furthermore, two of the four values, say , , are Picard exceptional values of and in . Set Similar to the discussion in Subcase 2.1 for , , we can get We define the Möbius transformations , , and by Then, we have Obviously, and are the fixed points of . Therefore, there exist no fixed points of in the set . Let some be given. Then, in vies of , there exists a such that , and from we obtain So is invariant under . Furthermore, we have where denotes the identical transformation. Hence, must contain an even number of values. Thus, the proof of Theorem 8 is completed.

Acknowledgments

This project is supported by the National Natural Science Foundation of China (project nos. 11171184, 61202313) and the Natural Science Foundation of Jiangxi Province in China (nos. 2010GQS0119 and 20132BAB211001).

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