On The Study Of The Properties Of A Class Of Harmonic Mappings And A Class Of Biharmonic Mappings

Suppose D denotes a domain in the complex plane C.For a twice continu-ously differentiable function u,if ?u=0,then u is called a harmonic function,where ? stands for Laplace operator,i.e.?=?2/?x2+?2/?g2.For a four-times continuously differentiable function u,if ?2u=?(?u)=0,then u is said to be a biharmonic function.Let F=u+iv be a complex-valued function.If both u and v are harmonic,then F is called a harmonic mapping:if both u and v are bi-harmonic,then F is called a biharmonic mappingThe main aim of this thesis is to study the related properties of harmonic mappings and biharmonic mappings.This thesis consists of three chapters and the arrangement is as followsIn Chapter 1,we provide the background of our studied problems and the statement of our main resultsIn Chapter 2,we introduce a class of harmonic mappings SHg0(n,?,?)and study their properties.Our results are as follows:(1)We establish some coefficient characterizations for harmonic mappings to be in SHg0(n,?,?);(2)By applying the coefficients estimates,we get the distortion theorem and the existence of extreme points of SHg0{n,?,?).In Chapter 3,We introduce a class of biharmonic mappings SBHD(n,?,??,?),and investigate their related properties,such as the coefficient estimates,dis-tortion theorem,convex combination and existence of extreme points.