In 2010 Ebenfelt, Khavinson, and Shapiro posted to the Arxiv a paper (Two dimensional shapes and lemniscates) which analyzed the fingerprints of smooth shapes, and showed that for any finite Blaschke products $B$, there is an injective analyic map $\phi:\mathbb{D}\to\mathbb{C}$ such that $B\circ\phi$ is a polynomial (it was not presented this way, but their Theorem 3.1 may be so interpreted). In this case we call the polynomial $B\circ\phi$ a conformal model for $B$.
This result has been expanded upon and improved in various ways (see footnotes below). My question is two-fold.
Questions
- Is the theorem of Ebenfelt, Khavinson, and Shapiro the first of this type (guaranteeing the existence of a nice conformal model for $f$ under assumptions on $f$)?
- What are the uses of this kind of result? (I know it is common to precompose $f$ with a conformal map $\phi$ in order to work on a better domain $D$, but I am curious about the other side, when one seeks to work with a conformal model which is in someway preferable to $f$.)
Improvements/Alternate proofs of EKS's Theorem 3.1
We will call functions $f$ and $g$ related by the equation $g=f\circ\phi$ (for a conformal map $\phi$) conformally equivalent.
1) 2013: G. Lowther : Proof on this website that every function which is analytic (meromorphic is a pretty easy extensions) on a connected compact set is conformally modelled by a rational function (analytic proof using approximating rational functions, in response to conjecture by the OP): Conjecture: Every analytic function on the closed disk is conformally a polynomial.
2) 2013(doctoral thesis)/2015(published): T. Richards (the OP): Proof of EKS's Theorem 3.1 using topological proof depending on an enumeration of critical level curve configurations: http://arxiv.org/abs/1310.7122
3) 2014 : M. Younsi : Showing that ratios of finite Blaschke products are conformally modeled on $\mathbb{D}$ by rational functions, using conformal welding : http://arxiv.org/pdf/1406.3545v2.pdf
4) 2015: T. Richards (the OP): Showing all functions analytic on the closed disk are conformally modelled by polynomials, again using analysis of critical level curve configurations: http://arxiv.org/abs/1505.02671
5) 2015: T. Richards and M. Younsi : Showing that if $f$ on $E$ is meromorphic, and $f(\partial E)$ is a Jordan curve, then $f$ is conformally modeled by a rational function $r$ having smallest possible degree : http://arxiv.org/pdf/1506.05061v1.pdf