Non-trivial Univalent Bilateral Harmoic Mapping

The paper studies planar non-trivial univalent bilateral harmonic mappings with the form f(z)= ??{?z+2iarg(?-e-?z)}+?.where ?,?,?,? are constants with the condition????0,|e-?z|<|?|.This paper gives the necessary and sufficient conditions for the mapping and its inverse mapping which are K-quasiconformal harmonic mappings,and calculate the coefficients of the non-trivial univalent bilateral harmonic mapping in the unit disk.Plane harmonic mapping theory was originally linked with minimal surface theory,then,complex analysis of scholars J.Clunie and T.Sheil-S-mall applying conformal mapping theory and though to harmonic mapping,they work for univalent harmonic mapping.Studying on the inverse mapping of univalent harmonic mapping,as early as 1945,Choquet recorded the "simple example" ?= f(z)defined byu = x,tan v tany = tanh x,where z = x+iy and w = u+ iv.And Jacques Deny had shown this to be essentially the only nontrivial example of a harmonic mapping with harmonic inverse.Hengartner and Schober mentioned that the inverse of a univalent harmonic mapping will seldom be harmonic.However,no proof appeared in print until 1987.When is the inverse of a harmonic mapping also harmonic?Certainly this is true when the given mapping is analytic or affine.Are there other example?In 1987,when EdgarReich studied a more general problem and characterized the class of harmonic mappings f that admit a non-affine harmonic mapping g such that the composition g o f is again harmonic.He also obtain the general situation of the inverse mappings of the preserving univalent harmonic mappings,that is,harmonic mappings f with the form f(z)=?{?z+2iarg(?-e-?z)}+?.The above recovered the Choquet-Deny result,in other words,Reich can proved that the "simple example" met the above conclusion.We refer to the harmonic mappings of the above form as non-trivial univalent bilateral harmonic mappings,and the research content of this paper is that explain the nature of this mappings.The text is divided into three chapters.Chapter 1:Preface.We give some of the notations and concepts used in this article.also introduce the related concepts of harmonic mapping,quasiconformal mapping and non-trivial univalent bilateral harmonic mappings.We also briefly review the background of the non-trivial univalent bilateral harmonic mapping theory and the statement our main results.Chapter 2:K-quasiconformal harmonic mappings and its inverse mappings.We combine Reich and Zhang Zhaogong to prove the theorem of Reich with the proof method of formal type f(z)=?{?z+2iarg(?-e-?z)}+?and f(z)= A[?z+?+log(1-e-az-?)-log(1-e-?z-?)]+B for the non-trivial univalent bilateral harmonic mappings.The research background of K-quasiconformal mappings is considered,and the properties of K-quasiconformal map-pings are presented to demonstrate the necessary and sufficient conditions for the univalent harmonic mappings and its inverse mappings are K-quasiconformal harmonic mappings in a single connected region.Chapter 3:Coefficients of non-trivial univalent bilateral harmonic mappings.Def-inition given by Zhang and Liu in a single connected region of the univalent harmonic mappings is the non-trivial univalent bilateral harmonic mappings under the background of the necessary and sufficient conditions,the univalent harmonic mappings are in the same way we draw with the conditions of the type f(z)= ?{?z+ 2iarg(?-e-?z)}+?,and be able to further study some coefficient estimation of the non-trivial univalent bilateral harmonic mappings.