Properties Of Harmonic Bloch Mappings And Normal Harmonic Mappings

As a generalization of analytic functions,planar harmonic mappings attract more and more attention.In 1984,Clunie and Sheil-Small proved that many classical results for analytic functions have analogues in the case of harmonic mappings.This thesis studies the properties of harmonic mappings,and mainly focuses on properties of harmonic Bloch mappings and normal harmonic mappings.This thesis consists of three chapters.It is arranged as follows.In Chapter one,background and status of the research are introduced.In Chapter two,existence of extreme points and support points of harmonic Bloch mappings and little harmonic Bloch mappings are discussed.First,in terms of harmonic Bloch unit-valued set,a necessary condition for a little harmonic Bloch mapping to be an extreme point of the unit ball of the normalized little harmonic Bloch space is discussed,and an example is constructed to show that this necessary condition is not a sufficient condition for a little harmonic Bloch mapping to be an extreme point of the unit ball of the normalized little harmonic Bloch space.Second,it is proved that a harmonic Bloch mapping is a support point of the unit ball of the normalized harmonic Bloch space if and only if the harmonic Bloch unit-valued set is not empty,a characterization for the support points of the unit ball of harmonic Bloch space is also given.In Chapter three,properties of normal harmonic mappings are discussed.First,characterizations for normal harmonic mappings are investigated,specially,five-point theorem of normal harmonic mappings is proved.Second,maximum principle of normal harmonic mappings is given,which is a generalization of the classical maximum principle for harmonic mappings.By using maximum principle of normal harmonic mappings,convergence of sequences for sensepreserving normal harmonic mappings is studied,and asymptotic values and angular limits of harmonic Bloch mappings are also discussed.