The Stability Of Univalent Harmonic Mappings And The Related Problems

Complex valued harmonic mapping theory generalizes analytic function theory.It has a close relation with univalent function theory, quasiconformal mapping theory and Teichmüller space. So it is of significance to study harmonic mapping theory and related problems. Recently, many researchers are made on the stability of univalent harmonic mappings, distortion theorem, Landau theorem and Bloch theorem for bound harmonic mappings, Schwarz derivatives of harmonic mappings. In this paper,we mainly study the construction of stable univalent convex harmonic mappings and starlike mappings, and their coefficient estimates, and distortion theorem, the growth estimate and covering problem for special univalent harmonic mappings, etc. Main contents are as follows:Firstly, we introduce the research background of the stable univalent harmonic mapping, then we consider the problem of the construction of stable univalent convex harmonic mappings and starlike mappings. Based on the researches of Rodrgo Hernández and Maria J. Martín, Huang and so on, we construct one class of stable univalent convex and starlike harmonic mappings in D { zz(27)(28)1|||} by the way of shear constructure with k- convex analytic functions. Our results improve and generalize the results obtained by R. Hernández, etc.Secondly, we study the coefficient estimate, distortion theorem, the growth estimate and the covering problem for univalent harmonic mappings. Based on Dominika Klimek and Andrej Michalski who studied the cases when the analytic parts(?) is the identity mapping, a convex mapping or a starlike mapping, we study the same problems under the cases in which analytic parts are convex or starlike of order β, we also estimate their coefficient, distortion growth and covering domain for(?)Some results are sharp. And the results obtained generalize those got by Dominika Klimek and Andrej Michalski, etc.Thirdly, we study the coefficient estimate, distortion theorem, growth order and covering problem for locally univalent harmonic mappings. By the sharp results are obtained by Qin Deng, we define some harmonic mappings with their analytic part h(?) to be univalent analytic function with negative coefficients and obtain some sharp results for their coefficient and distortion estimates of the co-analytic parts g(s). In addition, we give some interesting results on the growth order and covering range for(?) Also, some of results are sharp.