  We prove that every function that is meromorphic on the closure of an
analytic Jordan domain and sufficiently well-behaved on the boundary is
conformally equivalent to a rational map whose degree is smallest possible. We
also show that the minimality of the degree fails in general without the
boundary assumptions. As an application, we generalize a theorem of Ebenfelt,
Khavinson and Shapiro by characterizing fingerprints of polynomial
pseudo-lemniscates.
