# holomorphic that send the unit circle to the triangle $0,1,i$

Let $f$ be holomorphic on the unit open disk, and continuous on the closure of that disk, such that $\forall \theta \in \mathbb{R}$ the value $f(e^{i \theta})$ lives on the boundary of the triangle with vertices $0,1,i$. What can I say about the values that $f$ takes and lives on the triangle? For example , there is some $|z_0|<1$ , such that $f(z_0)= \frac{1}{2} (1+i)$ or $f(z_0) = \frac{1}{10} (1+i)$? Maybe I can use Rouche, and the characterization of the elements of the convex hull, written as a linear combination. But I don't know how

-