On The Studies Of Harmonic, Biharmonic And P-harmonic Mappings

Analytic functions are important objects in the study of complex analysis. As a generalization of analytic functions, planar harmonic mappings attract more and more attention. In 1952, Heinz used these mappings to study the Gauss curvature of nonparametric minimal surfaces (cf. [1]). In the study of harmonic mappings, a landmark paper is [2] in which Clunie and Sheil-Small proved that many of the classical results for analytic functions have analogues in the case of harmonic mappings, and so the study of analytic functions becomes a source of problems for the study of harmonic mappings. As a generaliza-tion of harmonic mappings, biharmonic mappings arise in many physical study, particularly, in fluid dynamics and elasticity problems. As a generalization of harmonic mappings and biharmonic mappings, in [3], the authors introduced the p-harmonic mappings F, where p≥1. Obviously, when p=1 (resp.2), F is harmonic (resp. biharmonic).The main aim of this thesis is to investigate properties of harmonic, bi-harmonic and p-harmonic mappings. First, we determine the extreme points and support points of several classes of harmonic (resp. p-harmonic) mappings; Then, we discuss the starlikeness and convexity for some p-harmonic mappings; After that, we study the Schwarzian derivatives and the affine and linear invari-ant families of biharmonic mappings, and also the subordination of p-harmonic mappings is discussed; Finally, we discuss the existence of neighborhoods for p-harmonic mappings.This thesis consists of six chapters. It is arranged as follows.In Chapter one, we mainly introduce the background of our research and state our main results.In Chapter two, we discuss the extreme points of weak subordination fami-lies of harmonic mappings and generalize the weak form of the conjecture about extreme points of subordination families of analytic functions to the case of har- monic mappings, which was raised by Abu-Muhanna and Hallenbeck in [4]. Our obtained result is a partial answer to this problem.In Chapter three, we introduce the Schwarzian derivatives for biharmonic mappings, and obtain several necessary and sufficient conditions for their Schwarzian derivatives to be analytic. Then we prove several estimates related to the Ja-cobian of the functions in the affine and linear invariant families of biharmonic mappings.In Chapter four, we mainly consider the subordination of p-harmonic map-pings. First, a characterization for p-harmonic mappings to be subordinate is obtained; Second, we investigate the relation of integral means of subordinate p-harmonic mappings. Our result is a generalization of the corresponding one in Schaubroeck [5] to the case of p-harmonic mappings; Third, two classes of ex-treme points of closed convex hulls of the corresponding subordination families are determined; Finally, we discuss the subordinate sequences for p-harmonic mappings and obtain the relation between the convergence of these sequences and the convergence of the corresponding sequences of their partial derivatives.In Chapter five, by using coefficient inequalities, we introduce two classes of univalent p-harmonic mappings. Then we investigate their starlikeness and convexity, determine the extreme points and support points, and discuss the existence of neighborhoods of these mappings.In Chapter six, we introduce two classes of p-harmonic mappings and con-sider the properties of these mappings. First, we discuss the starlikeness and convexity of the mappings in these classes; Second, we give the characteriza-tions of two corresponding subclasses; Third, we determine the extreme points of these subclasses; Finally, we consider the determination of the support points of these classes and the existence of neighborhoods of the corresponding map-pings.