mathimatikoi.org

Let $\mathcal{F}$ be the set of holomorphic maps $f:\mathbb{D} \rightarrow \mathbb{D}$ where $\mathbb{D}$ is the unit disk $\mathbb{D}=\{z \in \mathbb{C}: |z|<1\}$ such that $f(0)=0$ and $f\left(\frac{1}{2} \right) = \frac{1}{3}$. Determine the value

$$\mathcal{V}=\sup \left\{ \left| f \left(\frac{i}{2}\right) \right| : f \in \mathcal{F} \right\}$$