Harmonic Quasiconformal Extensions And Some Problems Of Biharmonic Mappings

Harmonic mappings and quasiconformal mappings are generalizations of univa-lent functions,and biharmonic mappings are generalizations of harmonic mappings.In this paper,we mainly study the harmonic quasiconformal extension of the upper half plane and the problems of biharmonic mappings.The main contents are as follows:Beurling and Ahlfors constructed Beurling-Ahlfors extension for the ρ-quasi-symmetric sense-preserving homeomorphism on the real axis,so that it is a quasi-conformal mapping from the upper half plane to itself.By means of Poisson integral formula,Kalaj and Pavlovi’ give the necessary and sufficient conditions for the sense-preserving homeomorphism on the real axis to be extended to the harmon-ic quasiconformal mapping from the upper half plane to its own.In this paper,for a specific class of homeomorphism on the real axis,the expressions of harmon-ic quasiconformal extension from the upper half plane onto itself are given,and their dilatation functions are estimated.Furthermore,the dilatation functions are compared with that under Beurling-Ahlfors extension.The coefficient conditions satisfying the dilatation functions are obtained which are better than those of the Beurling-Ahlfors extension.For the problems of the starlike radius and convex radius of univalent analytic functions and harmonic mappings,many ideal results have been obtained.For bi-harmonic mappings,similar studies are not perfect.In this paper,by establishing coefficient inequalities,we obtain sufficient conditions for the biharmonic function to be fully staxlike and fully convex of order α(0≤α<1).And we apply these coefficient inequalities to further investigate the radii of fully starlikeness and fully convexity of order a of the biharmonic function W which is constructed by Muhan-na.The above results generalize Ponnusamy’s and Qiao’s research on biharmonic mapping W.This thesis consists of three chapters.The first chapter briefly introduces the background,some basic concepts,notations and the main research results of this paper.In the second chapter,we give a extension for the homeomorphism on the real axis,so that it is a quasiconformal mapping from the upper half plane to itself.In the third chapter,we study the related problems of the fully starlikeness and fully convexity of order a of biharmonic mappings.