A holomorphic function \(f\) on a neighborhood of the closure of the unit disk \(\mathbb D=\{z:|z|<1\}\) and a polynomial \(p\) are said to be conformally equivalent if \(f=p\circ\phi\) with an injective holomorphic function \(\phi:\mathbb D\to\mathbb C\). In this case, \(p\) is called a conformal model for \(f\) on \(\mathbb D\). According to the Riemann mapping theorem, \(\mathbb D\) can be replaced by any Jordan domain. The main result of the paper states that, under additional assumptions on the behavior of \(f\) on the boundary \(\partial\mathbb D=\mathbb T\) of \(\mathbb D\), a conformal model also works in the meromorphic setting, in which case the polynomial \(p\) must be replaced by a rational map \(r\).
Theorem 2: Let \(D\) be a Jordan domain with analytic boundary, and let \(f\) be a meromorphic function on some open neighborhood of the closure of \(D\). Assume that \(f\) has no critical point on \(\partial D\) and that \(f(\partial D)\) is a Jordan curve whose bounded face contains 0. Then there is an injective holomorphic function \(\phi:D\to\mathbb C\) and a rational map \(r\) of degree \(\max(M,N)\) such that \(f=r\circ\phi\) on \(D\), where \(M\) and \(N\) are the number of zeros and the number of poles of \(f\) in \(D\), counting multiplicity. The rational map \(r\) may be chosen so that all of its zeros and poles are in \(\phi(D)\), except for a pole of multiplicity \(M-N\) at \(\infty\) if \(M\geq N\) or a zero of multiplicity \(N-M\) at \(\infty\) if \(N\geq M\).
The degree of \(r\) in Theorem 2 is optimal. However, the authors show that, in the analytic setting, the polynomial in the conformal model for \(f\) cannot be taken in general to have the smallest possible degree \[ N(f,D):=\sup_w\,n(w), \] where \(n(w)\) is the number of preimages of \(w\) under \(f\) in \(D\), counted with multiplicity.
For a smooth closed Jordan curve \(\Gamma\subset\mathbb C\), denote by \(\Omega_-\) and \(\Omega_+\), respectively, the bounded and unbounded faces of \(\Gamma\). Let \(\phi_-:\mathbb D\to\Omega_-\) and \(\phi_+:\mathbb D_+\to\Omega_+\) be conformal maps, \(\mathbb D_+:=\{z:|z|>1\}\), \(\phi_+(\infty)=\infty\), \(\phi'_+(\infty)>0\). The map \(k_{\Gamma}:=\phi_+^{-1}\circ\phi_-:\mathbb T\to\mathbb T\) is the fingerprint or conformal welding diffeomorphism of \(\Gamma\). Theorem 2 is applied to characterize the welding diffeomorphisms of polynomial pseudo-lemniscates, that is analytic Jordan curves of the form \(p^{-1}(\Gamma)\), where \(p\) is a polynomial and \(\Gamma\) is an analytic Jordan curve.