The Schwarz-Pick lemma and Julia lemma for real planar harmonic mappings

Abstract

The classical Schwarz-Pick lemma and Julia lemma for holomorphic mappings on the unit disk D are generalized to real harmonic mappings of the unit disk, and the results are precise. It is proved that for a harmonic mapping U of D into the open interval I = (−1, 1), $$\frac{{\Lambda _U (z)}} {{\cos \tfrac{{U(z)\pi }} {2}}} \leqslant \frac{4} {\pi }\frac{1} {{1 - \left| z \right|^2 }}$$ holds for z ∈ D, where ΛU(z) is the maximum dilation of U at z. The inequality is sharp for any z ∈ D and any value of U(z), and the equality occurs for some point in D if and only if \(U(z) = \tfrac{4} {\pi }\operatorname{Re} \{ \arctan \phi (z)\}\), z ∈ D, with a Mobius transformation φ of D onto itself.