- Let \(f ( z )\) be a holomorphic function on \(D : = \{ z \in \mathbb { C } : | z | < 1 \} , | f ( z ) | < 1 , f ( \alpha ) = 0\) for some \(| \alpha | < 1\) . Show that for \(z \in D\)
\[ | f ( z ) | \leq | { \frac { z - \alpha } { 1 - { \bar { \alpha } } z } } | . \]
Solution
Let \(\varphi_\alpha(z)=\frac{z-\alpha}{1-\bar\alpha z}\), an automorphism of \(D\) with \(\varphi_\alpha(\alpha)=0\).
1. \(g\da f\circ\varphi_\alpha^{-1}\) is a holomorphic self-map of \(D\) with \(g(0)=0\).
\(g\) is a composition of holomorphic maps, \(\abs g<1\) because \(\abs f<1\), and \(g(0)=f(\alpha)=0\).
2. Q.E.D.
By step 1 and the Schwarz lemma, \(\abs{g(w)}\le\abs w\) on \(D\). Taking \(w=\varphi_\alpha(z)\) gives \(\abs{f(z)}\le\abs{\varphi_\alpha(z)}=\abs{\frac{z-\alpha}{1-\bar\alpha z}}\).