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Let $D$ be the open unit disc. The Schwarz Lemma states that if $f: D \to D$ with $f(0) = 0$, then $|f(z)| \leq |z|$ for all $z \in D$ among other things. Can this extend to the case when $f: D \to \overline{D}$, where $\overline{D}$ is the closed unit disc?

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up vote 8 down vote accepted

Holomorphic non-constant maps are open, which means that $f(D)$ must be an open subset of the plane, if $f$ is not the constant map. If you know that $f(D) \subseteq \overline{D}$, this means that you actually have $f(D) \subseteq D$, so the Lemma applies.

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Alternatively, the maximum modulus theorem implies that if $|f(z)|=1$ for some some $z\in D$, then $f$ is constant. – Jonas Meyer Dec 4 '11 at 18:25

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