Suppose \(f\) is holomorphic and \(\abs{f(z)}\leq 1\) for \(\Re(z) > 0\) with \(f(1) = 0\). Find an upper bound for \(\abs{f(2)}\).
Solution
Use \[ \phi(z)={z-1\over z+1}, \] which maps the right half-plane biholomorphically onto \(\DD\) and satisfies \(\phi(1)=0\). Set \[ F=f\circ\phi^{-1}:\DD\to\DD. \] Then \(F(0)=f(1)=0\), so Schwarz’s lemma gives \(|F(w)|\le|w|\). Since \[ \phi(2)={1\over3}, \] we obtain \[ |f(2)|=|F(\phi(2))|=|F(1/3)|\le {1\over3}. \]