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 $f^{\prime }(0)\ne 0$. 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.