Suppose \(f, g: \DD\to \Omega\) are holomorphic with \(f\) injective and \(f(0) = g(0)\).
Show that \[ \Forall 0 < r < 1,\qquad g\qty{\theset{\abs{z} < r}} \subseteq f\qty{\theset{\abs{z} < r}} .\]
The first part of this problem asks for a statement of the Schwarz lemma.
The inclusion requires \(f(\DD)=\Omega\). For \(\Omega=\CC\), \(f(z)=z\) and \(g(z)=2z\), the image \(g\qty{\theset{\abs z<r}}\) is the disk of radius \(2r\), which is not contained in \(f\qty{\theset{\abs z<r}}\).
Solution
Assume that \(f\) is a bijection \(\DD\to\Omega\); the remark gives a counterexample without this hypothesis.
1. \(F\coloneqq \inverseof{f}\circ g\) is a holomorphic map \(\DD\to\DD\) with \(F(0)=0\).
The inverse \(\inverseof{f}\colon\Omega\to\DD\) of the bijective holomorphic map \(f\) is holomorphic, and \(g(\DD)\subseteq\Omega\), so \(F\) is defined and holomorphic on \(\DD\) with values in \(\DD\). Since \(g(0)=f(0)\), we have \(F(0)=\inverseof{f}(f(0))=0\).
2. \(\abs{F(z)}\le\abs{z}\) for all \(z\in\DD\).
This is the Schwarz lemma applied to \(F\), which step 1 permits.
3. Q.E.D.
Let \(0<r<1\) and \(\abs{z}<r\). By step 2, \(w\coloneqq F(z)\) satisfies \(\abs{w}<r\), so \(g(z)=f(F(z))=f(w)\in f\qty{\theset{\abs{w}<r}}\).