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

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), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{{\Lambda _U (z)}} {{\cos \tfrac{{U(z)\pi }} {2}}} \leqslant \frac{4} {\pi }\frac{1} {{1 - \left| z \right|^2 }}$$\end{document} 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U(z) = \tfrac{4} {\pi }\operatorname{Re} \{ \arctan \phi (z)\}$$\end{document}, z ∈ D, with a Möbius transformation φ of D onto itself.