A characterization of convexity

Let $\Omega$ be a simply connected domain in the complex plane, and let $\phi : \mathbb{D} \to \Omega$ be a conformal mapping from the open unit disk $\mathbb{D}$ onto $\Omega$. It is well known that $\Omega$ is convex if and only if

$$\mathrm{Re} \left( \frac{z \phi''(z)}{\phi'(z)} \right) \ge -1, \quad z \in \mathbb{D}.$$

We show that this relation is, in fact, inherited by a single Blaschke factor, and that when considering the family of all finite Blaschke products, entirely new intrinsic properties of $\phi$ are revealed.

This is an ongoing research with T. Ransford, Oliver Roth and Annika Moucha.