Some Results About Value Distribution Theory And Normal Family For Meromorphic Functions

In this paper, we mainly study the value distribution and normal family theory for meromor-phic functions, and get some new results. These results deeply improved the former theorems. In this paper, the most important result is to get a new Picard type theorem concerning elliptic function.1. one result value distribution theory for meromorphic functions.In Chapter2, we continue to study Picard type theorem and get a new Picard type theorem concerning elliptic function, Which is the most important result of the paper. Specifically, we prove the following result:Let f be a nonconstant meromorphic function on C and h be a nonconstant elliptic function. Then if all zeros of f are multiple except finitely many and T0(γ, h)=o{T0(γ,f)} as γâ†'∞, then f’=h has infinitely many solutions (including the possibility of infinitely many common poles of f and h).2. one result about normal families for meromorphic functionsIt’s an important topic in the theory of normal family that considering the normal family with shared values or shared functions. In Chapter3, we studies the normal families when they share some values, we get one normal criteria. Our main result is as followLet ψ≠0be a meromorphic function in domain D, all of whose poles are simple; Let F be a family of meromorphic functions in domain D such that, for each f∈F, f≠0in D. If, for each f∈F and each g∈F, f′and g’share ψ in D, then F is normal in D.3. one result about quasinormal families for meromorphic functions.In Chapter4, we continue to study the theory of families of meromorphic functions in plane domains, all of whose zeros are multiple, and get one quasinormal criteria. Our main result is as followsLet F be a family of meromorphic functions in domain D, all of whose zeros are multiple; H(z) be a nonconstant meromorphic function in domain D, and there exists v∈N such that, for each a∈C, n(r,1/H(z)-a)≤v.If, for each f∈F,f′(z)≠H’(z), then F is quasinormal of order v in D.