On The Studies Of Certain Properties Of Polyharmonic Mappings

A 2p-times continuously differentiable complex-valued function f in a do-main Ω in the complex plane C is a polyharmonic mapping or p-harmonic map-ping if it satisfies the following partial differentiable equation △pf=△(△p-1)f=0, where △ denotes the Laplacian operator z= x+iy ∈Ω and p is a positive integer. Especially, if p= 1 (resp. p= 2), then f is called harmonic (resp. biharmonic).The main aim of this thesis is to study the properties of polyharmonic mappings defined in the unit disk D, such as partial sums, convolution char-acterizations, the existence of extreme points, starlikeness, convexity, Landau type theorem, three circle theorem, Lipschitz continuity and so on. This thesis consists of four chapters and the arrangement is as follows.In Chapter one, we provide the background on our research and the state-ment of our main results.In Chapter two, we investigate the coefficient estimates, the Fekete-Szego problem and the close-to-convex radii of partial sums on a class of close-to-convex harmonic mappings.In Chapter three, we discuss the convolution characterizations of a class of biharmonic mappings BH0(φk,σ,α, b), and then find the coefficient characteri-zations and the extreme points on its subclass T BH0((φk; σ, α; b).In Chapter four, we discuss the properties of polyharmonic mappings. Firstly, we obtain the coefficient estimates on a class of polyharmonic map-pings, and then apply the obtained results to establish two Landau type theo-rems. Secondly, we study the three circle theorem, the Schwarz lemma and the Lipschitz continuity on some classes of polyharmonic mappings. At last, we get some geometry properties, such as starlikeness, convexity and so on.