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 E r,ρ such that
holds for every z ∈ D, w = ρe iα and harmonic mapping F with F(D) ⊂ D and F(z) = w, where Δ(z, r) is the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρe iα and w′ ∈ e iα E r,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 H z (u) on the unit disk D such that
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\(\left( {e^{i\theta } Fz\left( z \right) + e^{ - i\theta } F_{\bar z} \left( z \right)} \right)\).
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Chen, H. The Schwarz-Pick lemma for planar harmonic mappings. Sci. China Math. 54, 1101–1118 (2011). https://doi.org/10.1007/s11425-011-4193-x
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DOI: https://doi.org/10.1007/s11425-011-4193-x