Weakly Weighted Sharing and Uniqueness of Meromorphic Functions

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 M？bius 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 M？bius 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