Study On Some Problems Of The Uniqueness Of Meromorphic Functions

The modern science is developing faster and faster. Especially by the development and application of the computer technology, many fields in science have made great progress. The development of the pure mathematics also influenced by that, such as chaos theory, image theory, complexdynamical system and so on. The theory of the Nevanlinna's value distribution is one of the important branches in the complex variable function theory and its study has important meaning and application. From the modern value distribution theory created by Nevanlinna, it has developed for a long time, and make some progress in the complex variable function theory. This paper is studying the uniqueness of the meromorphic function which belongs to the theory of value distribution. In this field, there are some important problems need to be resolved like the problem of 1CM+3IM.In this paper, we mainly study two problems about the uniqueness of meromorphic functions in the value distribution theory, and obtain the following results.Theorem 3.6 Let f( z) be a nonconstant meromorphic function satisfying(?), let n be an integer and let a( z), b( z) be two distinct small functions related to f( z). Let (?) where(?) are small functions related to f( z). If f( z) and F(z) share a( z) and b( z) CM almost, then (?).Theorem 4.4 Let f( z) be a nonconstant meromorphic function of finite order, let n be a positive integer, and let a(z) and b(z) be two distinct small functions related to f( z). Let (?) be nonzero complex numbers, let (?) be distinct finite values, and let (?) If f( z) and F( z) share a(z) and b(z) CM almost, and(?),then (?)In this paper, we obtain two results above. The first one is about meromorphic functions with their linear differential polynomial, and it makes a progress of the result in Wang Jianping's paper in 2002. The second has studied the shift and difference of meromorphic functions, which makes a progress of the result got by Liu Dan et.al in 2014.