The Unique Range Set of Meromorphic Functions in an Angular Domain

Xu, Hong-Yan; Xuan, Zu-Xing; Wang, Hua

doi:https://doi.org/10.1155/2014/564640

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Research Article | Open Access

Volume 2014 | Article ID 564640 | https://doi.org/10.1155/2014/564640

The Unique Range Set of Meromorphic Functions in an Angular Domain

Hong-Yan Xu,¹Zu-Xing Xuan,²and Hua Wang¹

Academic Editor: Farrukh Mukhamedov

Received20 Feb 2014

Accepted01 May 2014

Published15 May 2014

Abstract

By using Tsuji's characteristic, we investigate uniqueness of meromorphic functions in an angular domain dealing with the shared set, which is different from the set of the paper (Lin et al., 2006) and obtain a series of results about the unique range set of meromorphic functions in angular domain.

1. Introduction

The purpose of this paper is to deal with the uniqueness problem of meromorphic functions sharing one set in an angular domain by using Tsuji’s characteristic. Thus, the notation and theory of Nevanlinna (see [1, 2]) about meromorphic function are basis for readers.

We use to denote the open complex plane, to denote the extended complex plane, and to denote an angular domain.

In 1929, Nevanlinna (see [3]) first investigated the uniqueness of meromorphic functions in the whole complex plane and obtained the well-known theorem-5 theorem of two meromorphic functions sharing five distinct values.

Theorem 1 (see [3]). If and are two nonconstant meromorphic functions that share five distinct values , , , , and in , then .

After his theorems, the uniqueness problems of meromorphic functions sharing values in the whole complex plane attracted many investigations (see [2]). In 2004, Zheng [4] studied the uniqueness problem under the condition that five values are shared in some angular domain in . In recent years, there are many results on the uniqueness of meromorphic function in an angular domain sharing values and sets (see [5–16]). Zhang [15], Zheng [17], Cao and Yi [18], Xu and Yi [19], and Xuan [20] continued to investigate the uniqueness of meromorphic functions sharing five values and four values, and Lin et al. [8] and Lin et al. [7] investigated the uniqueness of meromorphic and entire functions sharing sets in an angular domain. To state their results, we need the following basic notations and definitions of meromorphic functions in an angular domain (see [1, 4, 17]).

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

Let and be two nonconstant meromorphic functions in . If , we say and share the set (counting multiplicities) in . If , we say and share the set (ignoring multiplicities) in . In particular, when , where , we say and share the value in if , and we say and share the value in if . When , we give the simple notation as before, , and so on (see [19]).

In 2006, Lin et al. [7] dealt with the uniqueness problem on meromorphic functions sharing three finite sets in an angular domain and obtained the following theorems.

Theorem 2 (see [7, Thereom 1]). Let , , and , where is an integer and , are two nonzero constants, such that the algebraic equation has no multiple roots. Assume that is a meromorphic function of lower order in and for some . Then, for each with , there exists an angular domain with and such that if the conditions and hold for a meromorphic function of finite order or, more generally, with the growth satisfying either or where is a set of finite linear measures, then .

In 2011, Chen and Lin [21] further investigated the uniqueness of meromorphic functions sharing three finite sets in an angular domain and obtained the following result.

Theorem 3 (see [21, Thereom 1]). Let and be defined as in Theorem 2, and let be an integer. Assume that is a meromorphic function of lower order in and and that is a meromorphic function of finite order or, more generally, with the growth satisfying either or condition (4). Then, for each with , there exists an angular domain with and condition (3), such that if , then .

In 2010, Zheng [16] proved the following theorem by using Tsuji’s characteristic to extend the five- theorem of Nevanlinna’s to an angular domain. Tsuji’s characteristic will be introduced in Section 2.

Theorem 4 (see [16]). Let and be both meromorphic functions in an angular domain with , and let be transcendental in Tsuji’s sense. Assume that are distinct complex numbers. If , then .

2. Main Results

In this paper, we will focus on the uniqueness problem of shared set of meromorphic functions in an angular domain by using Tsuji’s characteristic. In fact, we will study the uniqueness of meromorphic functions in an angular domain sharing one set of the form , where and let be a complex number satisfying , and obtain the following results.

Theorem 5. Let and be both meromorphic functions in an angular domain with , and let be transcendental in Tsuji’s sense. If and is an integer , then .

A set is called a unique range set for meromorphic functions in an angular domain , if for any two nonconstant meromorphic functions and the condition implies . We denote by the cardinality of a set . Thus, from Theorem 5, we can get the following corollary.

Corollary 6. There exists one finite set with , such that any two meromorphic functions and in an angular domain which are transcendental in Tsuji’s sense must be identical if .

Theorem 7. Let and be both meromorphic functions in an angular domain with , and let be transcendental in Tsuji’s sense. If , , , and is an integer , then .

Corollary 8. There exists one finite set with , such that any two analytic functions and in which are transcendental in Tsuji sense must be identical if .

Theorem 9. Let and be both meromorphic functions in an angular domain with , and let be transcendental in Tsuji’s sense. If and is an integer , then .

A set is called a unique range set with weight 1 for meromorphic functions in , if for any two nonconstant meromorphic functions and the condition implies . Thus, from Theorem 9, we can get the following corollary.

Corollary 10. There exists one finite set with , such that any two meromorphic functions and in an angular domain which are transcendental in Tsuji’s sense must be identical if .

Theorem 11. Let and be both meromorphic functions in an angular domain with , and let be transcendental in Tsuji’s sense. If , , , and is an integer , then .

From Theorem 11, we can get the corollary as follows.

Corollary 12. There exists one finite set with , such that any two analytic functions and in which are transcendental in Tsuji sense must be identical if .

We found that the conclusions of Theorems 5–11 and Corollaries 6–12 hold for transcendental functions in Tsuji sense.

Thus, a question arises naturally, whether the conclusions of these theorems and corollaries hold for a general function in an angular domain.

For the above question, we can get the following theorem.

Theorem 13. Let the assumptions of Theorems 5–11 and Corollaries 6–12 be given with the exception that is transcendental in Tsuji sense. Assume that, for some and , where , , and is the number of poles of in . Then .

3. Preliminaries and Some Lemmas

In this section, we will introduce some notations of Tsuji’s characteristic in an angular domain (see [16, 22]). For meromorphic function in an angular domain and , we define where are the poles of in appearing often according to their multiplicities and then Tsuji characteristic of is We denote by the number of poles of in , then when pole occurs in the sum only once, and we denote it by . For meromorphic function in and for all complex numbers , if then is called transcendental with respect to the Tsuji characteristic [16], and we have the Tsuji deficiency of as follows: for , is defined by the above formula with and in place of and , and is defined by the above formula with in place of . If no confusion occurs in the context, then we simply write for and for . is called the Tsuji deficiency of at and if , then is said to be a Tsuji deficient value of . In addition, from [16], we have the following properties of this Tsuji’s characteristic: and from [16, Lemma ], the fundamental inequalities hold for distinct points , where denotes a set of with finite linear measure. It is not necessarily the same for every occurrence in the context.

Remark 14. In fact, from (12) and [16], we can get that the form holds for distinct points , where satisfies (13) and is the counting function of the zeros of in where does not take any of the values .

Lemma 15 (see [16, Lemma ]). Assume that is a meromorphic function in . Then for , one has where is a constant independent of and .

For sake of simplicity, we omit the subscript in all notations and use , , , , and instead of , , , , and , respectively.

By a similar discussion as in [23], one can obtain a standard and Valiron-Mohonko type result in as follows.

Lemma 16 (also see [16, Theorem ]). Let be a meromorphic function in . Then for all irreducible rational function in with coefficients meromorphic and small with respect to in , one has where is stated as in (13) and is the degree of in .

Lemma 17. Suppose is a nonconstant meromorphic function in . Then where is stated as in (13).

Proof. Since then from properties of , we have that is, Since then from (20) and (21), we can get the conclusion of this lemma.

Next, we will give two main lemmas of this paper as follows.

Lemma 18. Let and be transcendental meromorphic functions in in Tsuji sence satisfying , and let be distinct nonzero complex numbers. If where , , is the counting function which only counts simple zeros of the function in , and is some set of of infinite linear measure, then where are constants with .

Proof. Set
Suppose that ; from Lemma 15 and (13), we have where and . Since and by an elementary calculation, we can conclude that if is a common simple zero of and in , then . Thus, from (11), we have where . The poles of in can only occur at zeros of and in or poles of and in . Moreover, only has simple zeros in . Hence, from (26), we have where is the reduced counting function for the zeros of in where does not take one of the values .
Since then from (27) and (28), we have From Remark 14, we have where is a set of of finite linear measure, and it need not be the same at each occurrence. From (29)-(30), it follows for that since From (31)-(32), we have, for , From (22) and (33), since , are transcendental in Tsuji sense in , we have Thus, we can get a contradiction. Therefore, ; that is, For the above equality, by integration, it follows that where and .

Lemma 19. Let and be meromorphic functions in and transcendental in Tsuji sense, if and satisfy , and let be distinct nonzero complex numbers. If where , , and are stated as in Lemma 18, then where are constants with .

Proof. Let be stated as in the proof of Lemma 18; since , it follows that Similar to argument as in Lemma 18, we have, for , From (37) and (40), since , are transcendental in Tsuji sense in , it follows that Thus, we can get a contradiction. Therefore, ; that is, For the above equality, by integration, we have where and .

The following result can be derived from the proof of Frank-Reinders’ theorem in [24].

Lemma 20. Let and Then is a unique polynomial for transcendental meromorphic functions; that is, for any two transcendental meromorphic functions and in Tsuji sense, implies .

Lemma 21 (see [16, Lemma ]). Let be a meromorphic function in , for any real number , . Then for , one has where is a constant depending on , , , and is the number of poles of in .

4. Proofs of Theorems 5 and 7

4.1. The Proof of Theorem 5

From the definition of , we have , , and where , are polynomials of degrees and 2, respectively. We also see that and have only simple zeros.

Let and be defined as and . Since , we have . From (47) and (48), we have where and are the zeros of and in , respectively.

From (46), we have From Lemma 16, we have . Thus, combining (49) and (50), by Lemmas 17 and 18 and , we have Similarly, we can obtain Thus, by Lemma 18, we have For the above equality, by integration, it follows that where and . Since is nonempty and , we have , . Hence where , .

Two cases will be considered as follows.

Case 1 (). From the definition of and (55), we can see that every zero of in has a multiplicity of at least . Here, three following subcases will be discussed.

Subcase 1 (). From (47), we have where are distinct values. It follows that We can see that it has values satisfying the above inequality. Thus, from (21) and , we can get a contradiction.

Subcase 2 (). From (47), we have where , . It follows that every zero of in has a multiplicity of at least 2 and every zero of in has a multiplicity of at least . Then, by Remark 14, we have Since is transcendental in Tsuji sense in and , we can get a contradiction.

Subcase 3 (). By using the same argument as in Subcase 1 or Subcase 2, we can get a contradiction.

Case 2 (). If , from (55), we have ; that is, From (48) and (60), we have Since , from (46), it follows that has at least distinct zeros. Then, by Remark 14, we have

By applying Lemma 18 to (60) and from (62), since and is transcendental in Tsuji sense in , we can get a contradiction.

Thus, we have and ; that is, . Notting the form of , we can get that . Then, by Lemma 20, we get .

Therefore, the proof of Theorem 5 is completed.

4.2. The Proof of Theorem 7

Since and , it follows that By applying (60), from (51) and (52), since , we get Then, from Lemma 18, we have , where and . Thus, by using the same argument as in Theorem 5, we can prove the conclusion of Theorem 7.

5. Proofs of Theorems 9 and 11

5.1. The Proof of Theorem 9

Since , we have . From (46)–(48), we get where are the distinct zeros of . And from (50) and (65), by Lemma 17, we have Then from (49) and (66), since and , we have Similarly, we get Thus, by Lemma 19, we have where and . By using arguments similar to that in proof of Theorem 5, we have .

Therefore, this completes the proof of Theorem 9.

5.2. The Proof of Theorem 11

Since and , it follows that By applying (70), from (67) and (68), since , we get Then, from Lemma 19, we have , where and . Thus, by using the same argument as in Theorem 5, we can prove the conclusion of Theorem 11.

Hence, the proof of Theorem 11 is completed.

6. The proof of Theorem 13

Since condition implies that is transcendental in Tsuji sense, then the conclusions of Theorem 13 can be obtained easily from Theorems 5–11 and Corollaries 6–12.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees for reading the paper very carefully and making a number of valuable and kind comments which improved the presentation. The first author was supported by the NSF of China (11301233, 61202313), the Natural Science Foundation of Jiang-Xi Province in China (20132BAB211001), and the Foundation of Education Department of Jiangxi (GJJ14644) of China. The second author is supported in part by Beijing Municipal Research Foundation for the Excellent Scholars Ministry (2011D005022000009); Science and Technology Research Program of Beijing Municipal Commission of Education (SQKM201211417011); Signal and Information Processing Academic Subject of BUU.

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Copyright © 2014 Hong-Yan Xu 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.

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