Let $E$ be an infinite compact subset of the complex plane $\mathbb{C}$ such that $\overline{\mathbb{C}}\setminus E$ is simply connected. By Riemann mapping theorem, there exists a unique exterior conformal representation $\Phi$ from $\overline{\mathbb{C}}\setminus E$ onto $\overline{\mathbb{C}}\setminus \{w: |w|\leq 1\}$ satisfying $\Phi(\infty)=\infty$ and $\Phi'(\infty)>0.$ Define $$\Psi(w):=\Phi^{-1}(w),$$ for any $w\in \overline{\mathbb{C}}\setminus \{w: |w|\leq 1\}.$ Let $F$ be a holomorphic function on a neighborhood of $E.$ Define $$f(w):=F(\Psi(w)),$$ when $w\in \{w:1<|w|< r\}$ for some $r>1.$
I would like to have your help about the proof of the following statement:
If the regular part (around $0$) of $f(w)$ is a rational function with at most $m-1$ poles, then $F(z)$ is a rational function with at most $m-1$ poles.
This statement is in a paper without proof and I have no idea how to prove it. Could someone provide me a proof of the statement or give me an idea of the proof? Thank you very much.
Masik