Quasiconformality And Some Geometrical Properties For Harmonic Mappings

Harmonic mapping as the generalization of conformal mapping,in recent years,has attracted great attention from the domestic and foreign mathematicians to study its Schwarz lemma,Lipschitz property,quasiconformal extension,dilatation estimate and so on.In this thesis,we mainly studied the following two parts:(1).the Lipschitz and co-Lipschitz property for harmonic mappings with its holomorphic part is M-linearly connected.(2).The Lipschitz property with respect to pseudo-hyperbolic metric for harmonic Bloch mappings.In the first part,assume that f=h+g is a sense-preserving harmonic mapping of the unit disk D,where h and g are analytic in D.Under the assumption that its holomorphic part h is univalent and h(D)is a M-linearly connected domain,we obtained the necessary and sufficient conditions forfto be bi-Lipschitz.Furthermore,we proved that f(D)is aM1-linearly connected domain,where M 1 is a constant related toM and ωf.Moreover,ltt Tθ=h+eiθg,where θ∈R.Then we proved that under the normal conditionsh(0)=g(0)=h’(0)-1=0,Tθis univalent and Tθ(D)is a M2-linearly connected domain,where M2 is a constant related to M and ωf.In the second part,using the the property that a harmonic mapping composite a Mobius transformation keeps the invariant of the norm in Bh space(resp.Bh*space),together with some restriction on the module of the Mobius transformation,we obtained the Lipschitz property with respect to pseudo-hyperbolic metric for harmonic Bloch mappings.Our results improved the corresponding results in[8].Moreover,under the assumption of quasiregular for this kind of mappings,we first found the equivalent relationship between the norms ofBhspace and Bh*space.Then,we obtained the Lipschitz property with respect to pseudo-hyperbolic metric for harmonic quasiregular mappings.