Some Questions Concerning Normal Families And Uniqueness

In this thesis, two important results are proved. One is the normality criterion concerning differential polynomials:Let F be a family of meromorphic functions in a domain D, and let k, n≥k+2 be positive integers. Let a≠0 be a finite complex number. If for each f∈F, the zeros of f have multiplicity at least n, and G(f)=a(?)f=a, G(f)= b(?)f= b in D for every pair of functions f,g∈F, where G(f)=P(f(k))+H(f) is a differential polynomial of f, then F is normal in D.The other is the uniqueness theorem concerning shared values:Let f(z) and g(z) be two either nonconstant entire functions or meromorphic functions with only one pole at the same time. Let n,κbe two positive integers with n> 3κ+12.1f (fn)(κ) and (gn)(κ) share z CM,(fn)(κ) and (gn)(κ) share 0 IM,then either f(z)= c1ecz2 g(z)=c2e-cz2,where c1,c2 and c are three constants satisfying (-1)κ(c1c2)nc2=-1,or f=tg for a constant t such that tn=1.This thesis consists of six chapters. The first chapter introduces the thesis' study-ing work, studying purpose, background and so on. The second chapter introduces Nevanllina's value distribution theory. The third chapter outlines the basic knowledge and classic results. In the fifth and sixth chapters, we consider the above two results, including their backgrounds, proofs, and so on. In last chapter, we give some open questions.