The Schwarz-Pick lemma for planar harmonic mappings

Abstract

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 thatis the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρe iα and w ∈ e iα Er,ρ, there exists a harmonic mapping F such that F (D) ⊂ D, F (z) = w and F (z ) = w for some z ∈ ∂Δ(z, r). (II) The author establishes a Finsler metric Hz(u) on the unit disk D such that H F (z) (e iθ Fz(z) + e −iθ F z (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(e iθ Fz(z) + e −iθ F z (z)).

Schwarz–Pick type estimates of pluriharmonic mappings in the unit polydisk

“…We refer to [5,6,7,8,9,10,13,20,23,28] for further discussion on this topic. In this paper, we generalize Theorem A to higher dimensional case, and give the estimate for the partial derivatives of arbitrary order.…”

A Schwarz Lemma for Harmonic Mappings Between the Unit Balls in Real Euclidean Spaces

“…In this section, we will introduce two main lemmas, which are important for the proof of Theorem 3 and which extend the related lemmas proved by Chen in [2]. Lemma 1 constructs a bijection (R, I) from R m × R + onto the upper half ball {(a, b) : a ∈ R m , b ∈ R, |a| 2 + b 2 < 1, b > 0}, which will be used to construct u a,b,r in Theorem 3 for the case that b > 0.…”

“…However, the same problem also exists in harmonic mappings. The work in the following by Chen [2] seems to be the first result of this kind of study for harmonic mappings in the complex plane.…”

“…Let D be the unit disk in the complex plane C. Denote the disk {z ∈ C : |z| < r} by D r ; its closure is the closed disk D r . For 0 < r < 1 and 0 ≤ ρ < 1, Chen [2] constructed a closed domain E r,ρ and proved that Theorem A. Let 0 ≤ ρ < 1, α ∈ R and 0 < r < 1 be given.…”

“…So (1.2) can be regarded as considering the region of F (B 2 r ) when F ∈ Ω 2,2 regardless of F (0) = 0 or F (0) = 0. In [2], the most important theorem for the proof of Theorem A is the theorem as follow, which is the motivation for our study of the extremal mapping. The mappings U a,b,r and F a,b,r in the following theorem are defined in [2].…”