Several Radius Problems Of Harmonic Mappings And Their Quasi Conformal Extensions

In this paper,we first study the convexity and starlikeness of order a for some classes of harmonic mappings defined in unit disk D,and we consider the corresponding radius problem.Moreover,we research the harmonic quasiconformal homeomorphism of half plane onto itself.The paper is divided into five chapters as follows.In chapter 1,we introduce some basic concepts,the developments of harmonic mappings and haxmonic quasiconformal mappings in the plane and main works of this paper.In chapter 2,for two classes of harmonic mappings defined in D,we establish a sharp radius of fully convex of order a for convolution of them with given coefficient conditions.In chapter 3,denote by LE(f)=zfz-∈Zfz(|∈| = 1)the differential operator of a given harmonic mapping f.Under different coefficient conditions of f,we consider the radii of full convexity and starlikeness of order a of the differential operator,and we obtain some sharp results.Some results improve the related results of Liu et al.In chapter 4,we give several equivalent conditions for the harmonic extension of Beurling-Ahlfors.In addition,by using a sense-preserving homeomorphism defined in the real axis,we obtain a harmonic homeomorphism of the upper half plane to itself.Furthermore,we give a sufficient condition for this harmonic homeomorphism to be quasiconformal and estimate its dilitation.The results improve the related results of Michalski.In chapter 5,we summarize and prospect this paper.