Another look at the Landau theorem

Abstract

Let \(f\) be the analytic self-map of the open unit disk \(\mathbf{D}\) in complex plane with \(f(0)=0\) and \(f'(0)\neq 0\). The classical Landau theorem shows that the image \(f(\mathbf{D})\) contains a schlicht disk. In this note, it is proved that the injectivity of \(f\) in the Landau theorem can be strengthened to be starlike. In particular, it provides a way to construct starlike functions from bounded analytic functions. Furthermore, we obtain a new version of the Landau theorem for vector-valued analytic functions from the perspective of modulus functions.