On The Studies Of Some Properties Of Analytic Functions And P-harmonic Mappings

A2p-times continuously differentiable complex-valued function F=u+iv in a domain D∈C is p-harmonic if F satisfies the p-harmonic equation where△represents the Laplacian operator Especially, if p=1, then the mapping F is called harmonic. Obviously, all analytic functions are harmonic.The main aim of this dissertation is to discuss some properties of analytic functions and p-harmonic mappings. It consists of three chapters, and the arrangement is as follows.In Chapter one, we mainly introduce the background of our research and state our main results.In Chapter two, we introduce two classes of analytic functions K[A, B; λ,6], S[A, B; λ,6], and discuss their properties. First, by using convolution, we obtain a necessary and sufficient condition for an analytic function to be in K[A, B; λ, b](resp. S[A, B; λ, b]). Then we apply this necessary and sufficient condition to discuss the neighborhood inclusion property and determine the radii of starlike-ness and convexity of analytic functions of complex order with totally monotone coefficients. Our results generalize the corresponding ones obtained by Silver-man, Silvia, Telage in1978and Sheil-Small, Silvia in1989.In Chapter three, we introduce three classes of p-harmonic mappings Hp,q0, Hp,q1and THp*. First, by using coefficient inequalities, we obtain a sufficient condition for univalent p-harmonic mappings to be starlike and convex. Then we apply this sufficient condition to estimate the orders and radii of starlikeness and convexity of mappings in Hp,q0. Then, we discuss the starlikeness, convexity, covering theorems of mappings in Hp,q1. At last, we determine the extreme points in THp*. The obtained results are generalizations of the corresponding ones obtained by Ganczar in2001.