| The Pre-Schwarzian derivative and Schwarzian derivative of univalent analytic functions is an important tool for studying the geometric properties of univalent analytic functions,which play an important role in the study of the univalent criterion of analytic functions,quasiconformal extension and Teichmüller theory.Univalent harmonic mapping is a generalization of univalent analytic functions,the definition of its Schwarzian derivatives has many forms,in this paper,we consider a definition given by Hernandez and Martin,and use this Schwarzian derivative to study the quasiconformal extension of harmonic mapping and its related results.The main contents are as follows:Let’s introduce the research background and related preparations.We introduce the relevant knowledge needed in the paper,including Carleson measure,the definition and basic properties of analytic function spaces,and the definition and properties of Schwarzian derivative of univalent analytic function and univalent harmonic mapping in Chapter 1.We consider a class of univalent harmonic mapping with quasiconformal extension to the whole complex plane and the complex dilation satisfies the p-Carleson measure,we give the equivalent characterizations of its Pre-Schwarzian derivative and Schwarzian derivative in Chapter 2.We consider a class of univalent harmonic mapping with the image domain f(D)being an M-linearly connected domain and the modulus of complex dilation being less than 1/1+M,we give an equivalent characterization of its Pre-Schwarzian derivative and Schwarzian derivative.At the same time,we also consider univalent harmonic mapping with complex dilation satisfyies K-Carleson type measure,similar equivalent characterizations are obtained,and the results of conformal mapping are generalized accordingly in Chapter 3.We first give a Carleson measure characterization of harmonic Bloch mapping,and then discuss the uniformally locally univalent of harmonic mapping in Chapter 4. |
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