摘要
The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 ρ < 1, the author constructs a closed convex domain Er,ρ such that F((z,r)) eiαEr,ρ = {eiαz : z ∈ Er,ρ} holds for every z ∈ D, w = ρeiα and harmonic mapping F with F(D)D and F(z) = w, where △(z,r) is the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρeiα and w ∈ eiαEr,ρ, there exists a harmonic mapping F such that F(D) D, F(z) = w and F(z ) = w for some z ∈ △(z,r). (II) The author establishes a Finsler metric Hz(u) on the unit disk D such that HF(z)(eiθFz(z) + e-iθFz(z)) ≤1 /(1- |z|2)holds for any z ∈ D, 0 θ 2π and harmonic mapping F with F(D)D; furthermore, this result is precise and the equality may be attained for any values of z, θ, F(z) and arg(eiθFz(z) + e-iθFz(z)).
The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. （I） For any 0 〈 r 〈 1 and 0 ρ 〈 1, the author constructs a closed convex domain Er,ρ such that F（（z,r）） eiαEr,ρ = {eiαz ： z ∈ Er,ρ} holds for every z ∈ D, w = ρeiα and harmonic mapping F with F（D）D and F（z） = w, where △（z,r） is the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρeiα and w ∈ eiαEr,ρ, there exists a harmonic mapping F such that F（D） D, F（z） = w and F（z ） = w for some z ∈ △（z,r）. （II） The author establishes a Finsler metric Hz（u） on the unit disk D such that HF（z）（eiθFz（z） ＋ e-iθFz（z）） ≤1 /（1- ｜z｜2）holds for any z ∈ D, 0 θ 2π and harmonic mapping F with F（D）D; furthermore, this result is precise and the equality may be attained for any values of z, θ, F（z） and arg（eiθFz（z） ＋ e-iθFz（z））.
基金
supported by National Natural Science Foundation of China (Grant No.10671093)
