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Journal of Complex Analysis

Volume 2014, Article ID 378958, 9 pages

http://dx.doi.org/10.1155/2014/378958
Research Article

Weakly Weighted Sharing and Uniqueness of Meromorphic Functions

Korteweg-de Vries Institute for Mathematics, Amsterdam University, P.O. Box 94248, 1090 GE Amsterdam, The Netherlands

Received 9 February 2014; Accepted 7 May 2014; Published 22 May 2014

Academic Editor: Yan Xu

Copyright © 2014 Thamir Alzahary. 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

With the aid of the notion of weakly weighted sharing, we study the uniqueness of meromorphic functions sharing four pairs of small functions. Our results improve and generalize some results given by Czubiak and Gundersen, Li and Yang, and other authors.

1. Introduction and Main Results

In this paper, a meromorphic function means meromorphic in the whole complex plane . We assume the reader is familiar with the standard notion used in the Nevanlinna value distribution theory such as , , , and (see [1, 2]). For any nonconstant meromorphic function , the term denotes any quantity that satisfies as outside a possible exceptional set of finite linear measure.

Let be a nonconstant meromorphic function. A meromorphic function is called a small function of , if . If is a positive integer, we denote by the reduced counting function of the poles of whose multiplicities are less than or equal to and denote by the reduced counting function of the poles of whose multiplicities are greater than or equal to .

Let and be nonconstant meromorphic functions, and let , be two values in . We say that and share the value IM provided that and have the same zeros ignoring multiplicities. In addition, we say that and share the value IM, if and share IM. We say that and share the pair of values IM provided that and have the same zeros ignoring multiplicities.

The following theorem is a well-known and significant result in the uniqueness theory of meromorphic functions and has been proved by Czubiak and Gundersen.

Theorem A (see [3]). Let and be two nonconstant meromorphic functions that share six pairs of values , IM, where whenever and whenever . Then is a Möbius transformation of .

The following example, found by Gundersen, shows that the number “six” in Theorem A cannot be replaced with “five.”

Example 1 (see [4]). Let , . We see that , share , , , and IM, and is not a Möbius transformation of .

Let and be nonconstant meromorphic functions and let be two small meromorphic functions of and . We denote by the reduced counting function of the common zeros of and . We say that and share , if

As in Theorem A and throughout this paper, when and are nonconstant meromorphic functions, we let denote the term which is both and simultaneously.

We denote by the reduced counting function of those -points of , which are not the -points of . We note that and share if and only if and . According to this note, we generalize the definitions of IM and to the weakly weighted IM sharing which is given by the following definition.

Definition 2 (see [5]). Let be a positive integer or infinity, and let , be two small functions of nonconstant meromorphic functions and . We denote by the reduced counting function of those -points of whose multiplicities are less than or equal to , that are not the -points of . If , we say that and share IM.

We note that if and share IM, then and share IM, for all integer . Also, we note that and share if and only if and share IM.

Recently, Li and Yang have proved the following.

Theorem B (see [6]). Let and be two nonconstant meromorphic functions, and let , ( ) be small functions of and , with , whenever . If and share the five pairs , and is not a quasi-Möbius transformation of , then the following identities or inequalities hold: ; ; , , ; , ; ; ; and for if , .

One may ask the following question: is it possible to relax the condition “ and share five pairs of small functions” in Theorem B to the condition “ and share four pairs of small functions?”

The goal of the present paper is to generalize and improve Theorems A and B by using the weakly weighted sharing. We now turn to state our results.

Theorem 3. Let and be two nonconstant meromorphic functions and let , ( ) be small functions of and , with , whenever , and let be five positive integers or infinity with . Suppose that and share IM,   , and is not a quasi-Möbius transformation of . If there exists a number such that then , for all (that means and share IM,   ), and the following identities or inequalities hold:(a) ;(b) ;(c) , , ;(d) , ;(e) ;(f) and , ;(g) and for if , ;(h) and for if and , .The theorem is true, if are interchanged with , respectively.

Remark 4. In Theorem 3, if and share the five pairs , , then from Definition 2 we deduce that condition (2) occurs, and then the properties “(a)–(h)” of Theorem 3 give us the properties “(a)–(g)” of Theorem B. We also see that if and share five distinct small functions , then, from (g) and (h) of Theorem 3, we deduce that is a quasi-Möbius transformation of , and hence, . This result was proved in [7] (entire case ) and [8] (meromorphic case).

From (e) and (f) of Theorem 3, we can immediately obtain the following corollary.

Corollary 5. Let and be two meromorphic functions, and let , ( ) be small functions of and , with , whenever . Suppose that and share the four pairs , for and If there exists a number such that then must be a quasi-Möbius transformation of .

Remark 6. Li and Yang [6, Corollary 2] proved Corollary 5, when and share the five pairs , , and if there exists a number such that . It is evident that Corollary 5 improves Theorem A and the result of Li and Yang [6, Corollary 2].

Obviously, Theorem 3 is a generalization of Theorem B, and Corollary 5 is a generalization of Theorem A and Corollary 2 in [6].

Example 7. Let and be defined as in Example 1, and let , , , , and . It is easy to show that , , and is not a quasi-Möbius transformation of . This shows that the condition “ in Corollary 5” is necessary.

We have the following more general result.

Theorem 8. Let and be two nonconstant meromorphic functions, and let , ( ) be small functions of and , with , whenever , and let be six positive integers or infinity with . Suppose that and share IM, for . If there are two numbers and inside the interval such that and the following two inequalities hold then is a quasi-Möbius transformation of .

We deduce, from Theorem 8, the following corollary which is an improvement to Theorem A.

Corollary 9. Let and be two nonconstant meromorphic functions, and let , ( ) be small functions of and , with , whenever . Suppose that and share the four pairs , . If there are two numbers and inside the interval such that and the following two inequalities hold then is a quasi-Möbius transformation of .

2. Lemmas

In this section, we introduce some lemmas that will be used to prove the main results in this paper.

Lemma 10. Let and be two nonconstant meromorphic functions such that where are small functions of and and at least one of them is not identically zero. If is not a quasi-Möbius transformation of , then .

Proof. Since , then . We note that , because at least one of is not zero. Hence, Since is not a quasi-Möbius transformation of , the right-hand side of the above equation is irreducible. Therefore, by applying Valiron-Mokhonko lemma [9] we get . This completes the proof of Lemma 10.

Lemma 11 (see [10]). Let be a nonconstant meromorphic function and let be distinct small functions of . Then holds for any positive number .

Lemma 12 (see [6]). Suppose that and are nonconstant meromorphic functions, and is a polynomial in and with coefficients being small functions of and . The degree of related to is , and the degree of related to is . Then we have .

3. Proofs of Theorems 3 and 8

If (or ), then, for , and we let , , , and , where is any complex number such that , ( ). We can take , (or , if ), and ( ) in our proofs instead of , , and ( ), respectively. Therefore, we assume that none of and ( ) is infinity.

3.1. Proof of Theorem 3

It is not difficult to find six small functions ( ) of and that are not all zero and six other small functions ( ) of and that are not all zero, such that Consequently, the following two functions: satisfy and , for ( ).

Suppose that . For all , we denote by the reduced counting function of those -points of whose multiplicities are greater than or equal to , that are not the -points of in . By using Lemmas 11 and 12, we have and the inequality (14) gives us In the same method as the above, it can be shown that and this gives us It follows from (15) and (17) that which means that , and hence . Since (this follows from the assumption of Theorem 3), then, from the last inequality, we deduce that , which means that Consequently, we deduce (a) from (15), (17), and (19), and then from (14), we get (b). Also, (14) gives us From (a), (14), and (19), we deduce the following relation:

Let us now prove . We first assume that there exits such that . Then from (21), we have where is the counting function of the points that are zeros of , and is the counting function of those -points of whose multiplicities are not equal to .

It follows from the preceding equation and (21) that we have Let us consider two cases.

Case 1. Suppose that there are two values of , ( ) that are finite. Suppose that and . Let be a quasi-Möbius transformation such that , . Since is not a quasi-Möbius transformation of , then , and then by (a), we have It follows from (19)–(23), (a), and (b) that which is impossible.

By using the above method, it can be proved that there are no other two values of , ( ) that are simultaneously finite. Therefore, this case cannot occur.

Case 2. Suppose that there is such that and , for all with .

Case  2.1. Suppose that . That is, , for all with , and hence, . As in Case 1, let be a quasi-Möbius transformation such that , . Since is not a quasi-Möbius transformation of , then . From (23) and (a), we see From (19)–(23) and (26) we have which gives Thus, from (28), we deduce that That means On the other hand, by using (23)–(29) we get It follows from (19), (23), (30), (31) and by using (a) and (b), we see thus, the inequality (32) gives us Similarly, we have From (28), (29), (a), and (b), we get which yields From (21) and (36) (remembering ), we have , which is impossible. That means that . In the same method as the above, we can prove that , . This proves that and share IM, for all .

Case  2.2. Suppose that . We do as in Case 2.1 to get From (19)–(23), (37), (a), and (b), we get and from (38), we deduce that Similarly, we have and from (40) and (b), it follows that which yields From (21) and (42) (remembering ), we have , which is impossible. That means that . Therefore, from (19) and (20), we get (e). By using the idea which is used in Case 1, then, from (a), (b), and (e) and for all , we observe In the same way, for all , and , we get It is obvious that (c) follows from (43) and (44). It follows from (c) that, for all , From the last two inequalities, we deduce (d).

Let . We select six small functions ( ) of and that are not all zero such that the following two functions satisfy for with . Since is not a quasi-Möbius transformation of , it follows from Lemma 10 that if , then , and this contradicts (a). So, . It follows from (a) and (b), and by applying Lemma 12 to the function , that which yields and from this and (e), we get and it follows from (49) and Lemma 11 that, for all , we get From (49) and (50), we get (f).

Suppose that , for . By using (a), (b), and Lemma 11, we have which gives From (52) and (c), we obtain which yields From (52) and (54), we get (g).

By using the same technique as in the proof of (g), we prove (h). If are interchanged with , respectively, then from (14)–(19), we deduce that Theorem 3 is clear. This completes the proof of Theorem 3, when .

Now, assume that . Therefore, by Lemma 10, we deduce that . If the possibility , we deduce that , which is impossible. That means and are not simultaneously zero, and hence . We use the same way when to show that Theorem 3 is clear. This proves Theorem 3.

3.2. Proof of Theorem 8

Assume that is not a quasi-Möbius transformation of . Let and be defined as in (12) and (13). Suppose that . Note that Since , for all , it follows from (55), and by applying Lemmas 11 and 12, that which implies that Of course, if , then according to Lemma 10, we get , which means that (57) remains clear when . Similarly, we have Symmetrically, by using (13) we have

Suppose that . Then, by Lemma 10, we deduce that , and (otherwise, we have ). It follows from (59) that and the inequality (60) tells us that ; that is, ; this together with the inequality (5), we get a contradiction. So, . In the same way, we prove that . From (58), we have which yields In the same method as the above and by using (59), we can show that From (62) and (63), we obtain , which is a contradiction with (5). That means that should be a quasi-Möbius transformation of . This proves Theorem 8.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author thanks the referee for his/her helpful suggestions concerning the presentation of this paper.

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