Let $E_r \subset \mathbb{C}$ be the Bernstein ellipse of radius $r \geq 1$, i.e. the ellipse with focal points $-1,1$, semi-major axis $\frac{r + r^{-1}}{2}$ and semi-minor axis $\frac{r - r^{-1}}{2}$. I need a function $f : E_r \to \mathbb{C}$ that satisfies
- $f$ is analytic except at a finite number of points $z_k \subset E_r$ where $f$ has simple poles,
- $|f| = 1$ on $[-1,1]$, and
- $f$ has no zeros (inside $E_r$).
Does such an $f$ exist? If so, is it unique? How do I construct it? Any help would be greatly appreciated!
Edit: I realised the last condition is redundant. It can always be satisfied by adding a constant and scaling.