Quasiconformal Extensions And Coefficient Estimates For Harmonic Mappings

Harmonic mappings and quasiregular mappings are two classes of natural generalization of analytic functions. They are closely related with complex dynamical theory, Tcichmuller theory, value distribution theory, complex differential equations theory. Theory of quasiconformal extensions, univalent criterions and coefficient estimations for harmonic mappings are important problems in the theories of quasiconformal mappings and harmonic mappings. Recently, on the topic of these problems, many researchers made deep research and obtained some plentiful results. In this paper, we mainly discuss the problems of the quasiconformal extensions for harmonic mappings and coefficient estimations for some subclasses of harmonic mappings.Firstly, utilizing a complex parameter λ ∈ D, we give a criterion of stably quasiconformal extensions for the harmonic mappings f=h+g and obtain a sharper estimation for their maximal dilatations. Our results unify the methods of quasiconformal extensions of analytic functions by Beeker with the one of harmonic mapping by Hernandez and Martin. We also study the quasiconformal extensions and the maximal dilatation estimations for harmonic Teichmuller mappings f=h+ah. Based on the results of Chen, Hernandez and Martin, we give a criterion of quasiconformal extension for this class of mappings. By building the modulus growth theorem of a class of Mobiustransformations, we obtain the sharp estimation of its maximal dilatation estimation.Secondly, we study coefficient estimation problems of harmonic mappings. Under different conditions such as bounded modulus and bounded gradient, Chen, Liu, Wang, Huang and so on made deep research. Krushkal gave the coefficient estimation for the analytic functions which can be quasiconformal extended. In this paper, we generalized Krushkal’s result to harmonic mappings which can be stably quasiconformal extended.Thirdly, for the normal univalent harmonic mappings SHO(S), Ponnusamy and Kailiraj showed the coefficient conjecture by Clunie and Sheil-Small. We also study the coefficient estimation problems of this class of mappings, use the second Bcltrami coefficients to give a result which is asymptotically sharp to the analytic case. Moreover, we also give the growth and covering theorem for this class of harmonic mappings.