The Univalent Stability And Relative Problems For Harmonic Mappings And Their Shear Functions

Complex valued harmonic mapping can be known as a generalization of analytic function. Its has a close relation with univalent function and quasiconformal mapping. On the topics of the estimation of the univalent radius and Bloch constants for bound harmonic mapping, the distortion theory for univalent harmonic mapping, the univalent harmonic mappings to be harmonic quasiconformal mappings and so on, many researchers recently made deeply researches on them. A lot of beautiful results were obtained. As a result, it is a very interesting topic to investigate the properties for univalent harmonic mappings. In this paper, we mainly discuss the problems of the univalency and the stability for harmonic mapping and its shear function, the estimation of the univalent radius and Bloch constants for certain kinds of biharmonic mappings, etc. Main contents are as follows.We, firstly, consider the problem of the univalency for a class of harmonic mappings with their images to be linearly connected domain. Based on the researches of Chuaqui and Hernandez, Huang, Sh.Chen and so on, by introducing parameters α and β, and using shear construction, we consider the relations between the univalency and their image domains with linearly connectivity for harmonic mapping fa(z)=h(z)+ag(z) and their shear functions Fp(z)=h(z)+βg(z) in the unit disk. Some better results are obtained. As an application, we obtain one necessary and sufficient condition to judge one kind of harmonic mappings to be harmonic quasiconformal mappings.We find out that no any univalent harmonic mapping has the univalent stability. Thus, it is very meaningful to study this issue by introducing parameters. Which kinds of harmonic mappings have the univalent stability? How to judge the univalent stability for a given harmonic mapping? Based on the examination for the family of SHU and SHCC introduced by Hernandez and Martin recently, we define the stability for harmonic quasiconformal mappings, denoted by SHK(r), one necessary and sufficient condition for the functions to be SHU and SHK(r) is obtained. In addition, one sufficient condition is obtained to determine whether fλ(z)=h(z)+λg(z) has the close to convex stability for a given sense persevering locally univalent harmonic mapping f(z)=h(z)+g(z).Many researches pay close attention to the study of the properties for bound harmonic mappings. Especially, to consider the problems of the univalency radius and Bloch constants estimation for bounded harmonic mappings. In chapter4, using the coefficient estimation inequalities and the precisely Schwarz lemma in the unit disk for harmonic mappings and others methods, we estimate the univalent radius and the Bloch constants for biharmonic mappings F(z)and LF(z), where L is a differential operator. Our results improve the one latest made by Liu Mingsheng and Sh.Chen, etc.