Coefficient Estimates For Harmonic Mappings

Let D denote a domain in the complex plane C.A twice continuously differentiable complex-valued function F = u + iv in ID is called harmonic if F satisfies the Laplacian equation:?F = 0,where ? =(?),Obviously,harmonic mappings are also a gen-eralization of analytic functions,the harmonic mapping on the complex plane is getting more and more attention.In 1933,Fekete and Szego gets an upper bound estimate of |a3-?a22| of a class of analytic functions,sole mathenaticians have also studied the Fekete-Szego problem of the classes of univalent analytic function and obtained a lot of meaningful results.Since then,the Fekete-Szego problem of harmonic mappings have attracted much attention.The main aim of this thesis is to study the coefficient estimates of harmonic mappings and biharmonic mappings.This thesis consists of four chapters and its arrangement is as follows.In Chapter one,we introduce the background on our research and the statement of our main results.In Chapter two,firstly,we introduce the Fekete-Szego coefficient estimates for certain classes of analytic functions associated with q-difference operator.Secondly,we discuss the Fekete-Szego coefficient estimates for the class of har-monic mappings SCV? associated with q-difference operator.We demonstrate that the Theorems 1.2.1 and 1.2.2.Thirdly,some corollaries related to them are given.In Chapter three,firstly,we introduce the Fekete-Szego coefficient esti-mates for certain classes of analytic functions associated with fractional q-difference operator.Secondly,we discuss the Fekete-Szego coefficient estimates for the class of harmonic mapping Mq,?,?,?(?)associated with fractional q-difference operator.We demonstrate that the Theorems 1.3.1 and 1.3.2.Third-ly,some corollaries related to them are given.In Chapter four,at first,we introduce the classes of harmonic mappings BSHL0(n,?),BSHD0(n,?)and Salagean operator.Next,we discuss the coeffi-cient estimates for the classes of biharmonic mappings BSHL0(n,?),TBSHL0(n,?),BSHD0(n,?)and TBSHD0(n,?).We demonstrate that the Theorems 1.4.1?1.4.4.