Conformal equivalence of analytic functions on compact sets

In this paper we present a geometric proof of the following fact. Let $D$ be a Jordan domain in $\mathbb{C}$, and let $f$ be analytic on $cl(D)$. Then there is an injective analytic map $\varphi :D\to \mathbb{C}$, and a polynomial $p$, such that $f\equiv p\circ \varphi$ on $D$ (that is, $f$ has a polynomial conformal model $p$).