Landau's constants

Definition

Let $\mathbb{D}$ be the unit disc, and let $\mathcal{F}$ be the set of holomorphic functions $f \colon \mathbb{D} \to \C$ with $f'(0) = 1$. For $f \in \mathcal{F}$:

1. $L_f$ is the supremum of the radii $r$ such that $f(\mathbb{D})$ contains an open disc of radius $r$.
2. $B_f$ is the supremum of the radii $r$ of the open discs $f(\Omega)$, where $\Omega \subseteq \mathbb{D}$ is open and $f$ is injective on $\Omega$.

Landau's constants are the three numbers $$B = \inf_{f \in \mathcal{F}} B_f, \qquad L = \inf_{f \in \mathcal{F}} L_f, \qquad A = \inf \set{L_f : f \in \mathcal{F},\ f \text{ injective}}.$$ The number $B$ is the Bloch constant, $L$ is the Landau constant, and $A$ is the univalent Landau constant.