The Schwarzian And Pre-Schwarzian Derivatives Of Harmonic Mappings In The Plane

In this thesis,we mainly introduce new definitions of pre-Schwarzian derivative and Schwarzian derivative of harmonic mappings in the plane,and discuss some properities of the new Schwarzian derivative and pre-Schwarzian derivative.Besides,we connect the new pre-Schwarzian derivative with radial John disks,and estimate the norm of the new Schwarzian derivative of harmonic mapping which maps the unit disk onto the regular n-gon.This thesis includes three chapters.Chapter 1,Preface.The first chapter is the introduction.In the first,section,we first introduce the development history of the Schwarzian derivative,and then introduce the two definitions of the Schwarzian derivative of plane harmonic mappings.One is that Chuaqui,Duren and Osgood give a definition of the Schwarzian derivative of harmonic mappings.This definition re.quires the dilatation of the harmonic mapping to be a analytic function,but this definition guarantees that the corresponding Gauss curvature is greater than or equal to zero,which guarantees that the harmonic mapping can be promoted to a minimal surface.The second is that Hernandez and Martin use the Jacobia-n determinant to give another definition of Schwarzian derivative of harmonic mappings.This definition does not require the dilatat.ion of the harmonic mapping to be a analytic function,but the corresponding Gauss curvature is less than or equal to zero,it can not guarantee the harmonic mapping can be promoted to a minimal surface.In the third chapter,we give a new definition of Schwarzian derivative of plane harmonic mappings.This Schwarzian derivative does not require the dilatation of the harmonic mapping to be a analytic function,but the Gauss curvature is greater than or equal to zero,which ensures that the harmonic mapping can be promoted to a minimal surface.In the second and third section of this chapter,we mainly introduce basic properties and relevant conclusions of the two definitions.Chapter 2.The pre-Schwarzian dereivative of harmonic mappings and John disks.In this chapter,we first give a new definition of pre-Schwarzian derivative of harmonic mappings,namely:(?)in which Ph is defined as the pre-Schwarzian derivative of analytic function h.The main work of this chapter is to study the properties of the pre-Schwarzian derivative of the harmonic mapping and to estimate its norm.Besides,this chapter also focuses on some applications of this pre-Schwarzian derivative.Such as,we using this pre-Schwarzian derivative give two sufficient conditions and two necessary conditions f(U)to to be a radial John disk.Therefore,in this chapter,we also introduce some basic properties and theories of John disk.Chapter 3,The Schwarzian derivative of harmonic mappings and its norm.In this chapter,first give a new definition of Schwarzian derivative of harmonic mappings,namely:sf=Sh+?/1+|?|2(?"-?'h"/h')-3/2(?'?/1+|?|2)2,in which Sh is defined as the Schwarzian derivative of analytic function h,and ? is the dilatation of the harmonic mapping f.anddiscuss some properties of the Schwarzian derivative.At last,we estimate that the upper bound of the norm ||Sf| of the Schwarzian derivative of harmonic mapping which maps the unit disk onto the regular n-gon is 8/3.