Univalence And Linear Connetivity Of Harmonic And Biharmonic Mappings

In the theory of complex analysis, the univalence theory for complex-value mappings in complex plane is an important and fundamental problem. In the past two decades, The univalence of harmonic mappings, biharmonic mappings, P-harmonic mappings, logarithmic harmonic mappings and so on were studied by many scholars from different ways.For example, Shao-lin Chen, S. Ponnusamy, A. Rasila, and X. Wang studied the relationship between harmonic mappings and their horizontal shear mappings to discriminate the univalent of harmonic mappings.In this paper, we study the relation between the quasi-conformal harmonic mappings and the linear connectivity of their shear mappings> the relation between biharmonic mappings and their harmonic parts.In Chapter I, we introduce the basic concepts, known results, research problem and then we give the main results that are studied in the paper.In Chapter II, we consider that if given f(z)=h(z)+g(z) is a locally univalent harmonic mapping on the unit disk D={z||z|<1}, and its shear mapping h(z)-g(z) is univalent with linearly connected image domain.we study the univalence and the linear connectivity of h(z), Fγ(z)= h(z)+γg(z) under some restrict conditions of the dilatation of f(z). The results that we obtain show that parts of them are better than that made by Shao-lin Chen.In Chapter Ⅲ, under the conditions that when F(z)=|z|g(z)+h(z) is a biharmonic mapping on the unit disk D={z||z|<1}, with 0<c≤||hz(z)|-|h2-(z)|, |gz(z)|+gz-(z)|≤ ∧, z∈D.we consider the relation between h(z) and F(z) for their univalence and linear connetivity of h(D) and F(D), we prove that if h(z) is univalent and h(D) is linear connected domain, so are F(z) and F(D), and vice versa. We also give examples to illustrate the relations.The similar theorems are extended to the biharmonic mappings and P-harmonic mappings.