Complex Analysis, find all analytic functions

Question

Find all analytic functions $f: \mathbb{C} \rightarrow \mathbb{C}$ such that $|f(z)-1| + |f(z)+1| =4 $ for all $z \in \mathbb{C} $ and $f(0) = \sqrt{3} i$

I understand that the given equation represents an ellipse. I get the ellipse equation to be: $\frac{x^2}{4} + \frac{y^2}{3} = 1$. But, I am not able to find the exact form for $f(z)$.

Answer 1

Hint: Note that $|2f(z)| = |f(z)+1 + f(z) - 1|\le |f(z)+1|+|f(z)-1|=4$, so that $|f(z)|\le 2$ for all $z\in \mathbb C$. What can you say about entire and bounded functions?

Answer 2

That condition says that $f(\mathbb{C})$ is a subset of an ellipse. In particular, it is a bounded set. So…