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)).
In this paper we prove a high order Schwarz-Pick lemma for holomorphic mappings between unit balls in complex Hilbert spaces.In addition,a Schwarz-Pick estimate for high order Fréchet derivatives of a holomorphic function f of a Hilbert ball into the right half-plane is obtained.
DAI ShaoYu1,2,CHEN HuaiHui1 & PAN YiFei3,4 1Department of Mathematics,Nanjing Normal University,Nanjing 210097,China
In this paper we establish characterizations of α-Bloch functions on the unit ball without use of derivative, which are stronger, more precise and general than those obtained by Nowak and Zhao.
CHEN HuaiHui School of Mathematics and Computer Science, Nanjing Normal University, Nanjing 210097, China
