On Some Problems Of Planar Harmonic Mappings

In this thesis, we mainly study the properties of harmonic mappings in the plane,and discuss the problems of the related norm of the Schwarzian derivative.Quasiconformal mappings and harmonic mappings in the plane are the natural generalization of conformal mappings. The theory of harmonic mappings in the plane initially connects with the theory of minimal surface, the complex analysis scholars Clunie and Sheil-Small later point that many classical conclusions of conformal mappings can generalize to harmonic mappings, and make it has the similar conclusions. In recent decades, the research of the theory of harmonic mappings in the plane has been gained rapid development, and permeated through the other branches of mathematical, physics and engineering technology and other fields, and provided a powerful tool for the development of other disciplines.The article includes four chapters:Chapter 1: Preface. This chapter is devoted to the exposition of the basic theory of quasiconformal mappings and Loewner Chains as well as the development history and the research situation of harmonic mappings in the plane. The main results of the dissertation are briefly introduced in this chapter.Chapter 2: The application of the Loewner chains to the criterion of univalent.According to the property that the arbitrary holomorphic functions in the theory of univalent and complex functions can be embedded into the Loewner chains, we proved the univalence criteria of Nehari functions and the general univalence criteria of Chuaqui functions. It has certain internal relation to the univalent of analytic functions and conformal extensions.Chapter 3: The properties of harmonic mappings in the plane. In 2003, Chuaqui,Duren and Osgood defined the Schwarzian derivative and the pre-Schwarzian derivative of the sense-preserving homeomorphism harmonic mappings in the unit disk. By this definition, we obtain some equivalent properties of the related Schwarzian derivative and pre-Schwarzian derivative, and further discuss the distortion properties for harmonic mappings in the plane which are associated with that Nehari’s univalence criteria.Chapter 4: The norm of Schwarzian derivative of harmonic mappings in the plane. We study the properties of the norm of Schwarzian derivative of harmonic mappings which map the unit disk onto a convex domain. In particular, we discuss the norm of Schwarzian derivative of harmonic mappings which map the unit disk onto the interior of a regular hexagon or an octagon inscribed in the unit circle.