Researches On Normal Families And Uniqueness Of Meromorphic Functions

In this thesis, by using the Nevanlinna theory and uniqueness theory, we prove two results:One is under certain conditions, the small function of f is equivalent to the small function of f(k).The other is the uniqueness theorem concerning shared values:Let n, k, m and l be four positive integers. Let f(z) and g(z) be two either nonconstant entire functions or meromorphic functions with m,l poles respectively (ignoring multiplicities). If n> max{3k+12, k+m+l+3},(fn)(k) and (gn)(k) share z CM,(fn)(k) and (gn)(k) share0IM, then either f(z)=c1ecz2, g(z)=c2e-cz2or f(z)≡t·g(z), where c1, c2, c and t are four complex numbers satisfying4n2(c1C2)nc2=-11or tn=1respectively.This thesis consists of six chapters. The first chapter introduces the development process of uniqueness theory and normal families theory of meromorphic functions, and our research work, purpose and so on. The second chapter introduces the basic Nevanlinna theory, including some basic definitions and signs, the two Nevanlinna’s fundamental theorems and their other forms, lemma on the logarithmic derivative, Hayman inequality and so on. The third chapter outlines the basic knowledge and clas-sic results of the normal families theory and the uniqueness theory, including Marty normal criterion and Montel normal criterion, the various promotion forms about five-value theorem and four-value theorem and so on. The fourth chapter introduces some normal criterion concerning Hayman conjecture. In the fifth and sixth chapters, we consider the above two results, including their backgrounds, proofs and so on.