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}
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\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}
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\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.
