Let $z_1$ and $z_2$ be two distinct points in the unit sphere of $\mathbb{C}$, i.e., $z_1 \neq z_2$ and $|z_1|=|z_2|=1$.
I'd like to construct the bounded uniformly continuous holomorphic function $f : \mathbb{D} \rightarrow \mathbb{C}$ ($\mathbb{D}$ is the open unit disk of $\mathbb{C}$) satisfying:
(1) $\| f \|_{\mathbb{D}}=\sup\{|f(z)|:|z| < 1\} \leq 1$,
(2) $f(z_1)=0$ and $f(z_2)=1$.
I think there exists such a holomorphic function but I couldn't construct that one.