α-normal Functions And Harmonic α-normal Mappings

Normal family is one of the core concepts of complex function theory,which provides a tool to study meromorphic functions.In particular,it has obtained many results on meromorphic functions through Bloch’s principle,and these results have promoted the research and development of normal family theory.Based on the close relationship between normal families and normal functions,the properties of normal functions and α-normal functions have been studied extensively.The thesis studies the properties of α-normal functions,harmonic normal mappings,harmonic α-normal mappings and harmonic α-normal-type mappings.This thesis consists of four chapters.It is arranged as follows.In Chapter one,the background and status of the thesis are introduced.In Chapter two,existence of extreme points of α-normal function is discussed.The application of α-normal unitα-valued set,which give a necessary condition for a little α-normal function to be an extreme point of the unit ball of the class of little α-normal functions,and further discuss the constructure of the α-normal unitα-valued set of a little α-normal function.In Chapter three,the existence of extreme points and support points of harmonic normal mappings are discussed.By harmonic normal unitα-valued set,we have a necessary condition for a little harmonic normal mapping to be an extreme point of the unit ball of the class of little harmonic normal mappings;necessary conditions for a little harmonic normal mapping to be a support point of the unit ball of the of little harmonic normal space.In Chapter four,the following properties about harmonic α-normal mappings and harmonicα-normal-type mappings are discussed: affine invariance,linear invariance,inclusion relations,relations with uniformly locally univalent harmonic mappings and subordination principle.Thus,the properties of harmoni α-Bloch mapping and harmonic α-Bloch-type mapping are generalized.