- Show that there is no holomorphic function f in the unit disc D that extends continuously to \(\partial D\) such that \(f ( z ) = 1 / z \ \mathrm { f o r } \ z \in \partial \mathbb { D }\)
Solution
Suppose such an \(f\) existed. Define \[ h(z)=zf(z)-1. \] Then \(h\) is holomorphic in \(\mathbb D\) and continuous on \(\overline{\mathbb D}\). On the boundary, \(f(z)=1/z\), so \(h(z)=0\) whenever \(|z|=1\). By the maximum modulus principle, \[ |h(z)|\le \max_{|\zeta|=1}|h(\zeta)|=0 \] for every \(z\in\mathbb D\). Hence \(h\equiv0\). But \[ h(0)=-1, \] a contradiction. Therefore no such \(f\) exists.