单位圆到凸区域上的调和拟共形映照

Harmonic Quasiconformal Mappings of the Unit Disk Onto Convex Domains

设F(x)=p(x)e^(ir(x))为单位圆周到约当凸曲线Γ上的保向同胚映照.本文证明:若ess inf|F'(x)|>0且对于一切的φ∈R有|F(φ+x)+F(φ-x)-2F(φ)|≤M|x|~α,这里α>1,M为正常数,则ω=P[F](z)为单位圆到凸区域Ω=int(Γ)上为调和拟共形映照.

Let F（x） = f（e^（ix）） =ρ（x）e^（iγ（x）） be an orientation preserving homeomorphism ofthe unit circle onto the convex Jordan curveΓ.In this paper,we prove that if essinf ｜F′（x）｜0,and ｜F（φ＋ x） ＋ F（φ-x）-2F（φ）｜≤M｜x｜~α for every x,φ∈R,whereα1 and M is a positiveconstant,then w = P[F]（z） is a harmonic quasiconformal mapping of unit disk ontoΩ= int（Γ）.