On The Studies Of Certain Properties Of P-harmonic Mappings In The Unit Disk

A2p-times continuously differentiable complex-valued function F=u+iv in a domain D C C is p-harmonic if F satisfies the p-harmonic equation△pF:=△(△p-1)F＝0,where△represents the Laplacian operatorEspecially, if p=1(resp. p=2), then the mapping F is called harmonic (resp. biharmonic).The main aim of this thesis is to study certain properties of p-harmonic mappings, such as Landau-Bloch theorem of p-harmonic mappings, equivalent modulus of harmonic mappings, Girela-Pelaez's problem for harmonic mappings and so on. This thesis consists of five 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 properties of harmonic mappings defined in the unit disk. Firstly, we study the coefficient estimate and Landau-Bloch's theorem of bounded harmonic mappings, the relationship between the linear connectivity and the univalency of harmonic mappings, and harmonic Bloch spaces. The obtained results improve the corresponding known ones. Secondly, we discuss the properties of harmonic Lipschitz spaces and the equivalent mod-ulus for K-quasiregular harmonic mappings. Consequently, we generalize the corresponding results of Dyakonov published in Acta Math, in1997. At last, the harmonic Hardy space is defined and the related properties are investigated. As a consequence, we show that the answer to the open problem raised by Girela and Pelaez is affirmative for harmonic mappings. In addition, we establish co-efficient estimates and a distortion theorem for mappings in harmonic Hardy spaces. In Chapter three, we investigate the composition and Goodman-Saff:s con-jecture for biharmonic mappings in the unit disk. We mainly show that the answer to Goodman-Saff's conjecture is affirmative for a class of univalent bi-harmonic mappings which contains the set of all harmonic univalent mappings.In Chapter four, we study the Bloch constant, Landau-Bloch's theorem and the region of variability for p-harmonic mappings in the unit disk. Our results generalize the corresponding ones obtained by Colonna, Abdulhadi and Muhanna.In Chapter five, we discuss some properties on the solutions of a certain elliptic PDE in the unit disk. The study is a generalization of the corresponding ones in analytic functions and harmonic mappings. We first prove that the answer to Girela and Pelaez's problem for solutions of this class of PDEs is also affirmative. In the meanwhile, for such a solution, we show that if it admits some bounded Dirichlet-type energy integral, then it belongs to Hp spaces and has a harmonic majorant.