Let f = B 1 /B 2 be a ratio of finite Blaschke products having no critical points having no critical points on ∂D. Then f has finitely many critical level curves (level curves containing critical points of f ) in the disk, and the non-critical level curves of f interpolate smoothly between the critical level curves. Thus, to understand the geometry of all the level curves of f , one need only understand the configuration of the finitely many critical level curves of f . In this paper we show that in fact such a function f is determined not just geometrically but conformally by the configuration of its critical level curves. That is, if f 1 and f 2 have the same configuration of critical level curves, then there is a conformal map φ such that f 1 ≡ f 2 •φ. We then use this to show that every configuration of critical level curves which could come from an analytic function is instantiated by a polynomial. We also include a new proof of a theorem of Bôcher (which is an extension of the Gauss-Lucas theorem to rational functions) using level curves.Corollary: 6.2 For every finite Blaschke product B of degree n, there is some n degree polynomial p such that the set G := {z : |p(z)| < 1} is connected, and some conformal map φ : D → G such that B ≡ p • φ on D.

Abstract. We prove that every meromorphic function on the closure of an analytic Jordan domain which is 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 pseudolemniscates.The starting point of this paper is the following conjecture on the behavior of holomorphic functions on the closed unit disk.Conjecture 1. Let f be holomorphic on a neighborhood of the closure of the unit disk D. Then there is an injective holomorphic function φ : D → C and a polynomialIn this case, we say that f and p are conformally equivalent and that p is a conformal model for f on D. Note that D can be replaced by any Jordan domain, by the Riemann mapping theorem.The above was conjectured by the first author in [9], motivated by questions arising from the study of level curve configurations of meromorphic functions. The case where f is a finite Blaschke product follows from the work of Ebenfelft, Khavinson and Shapiro [1] on the characterization of the so-called fingerprints of polynomial lemniscates. Proofs of this special case were also given independently by the first author in [9] using level curves, and by the second author in [11] using a simple Riemann surfaces welding argument combined with the uniformization theorem.Conjecture 1 was proved in full generality by George Lowther and David Speyer on the internet mathematics forum math.stackexchange.com, using Lagrange interpolation and the inverse mapping theorem (see [8]). Our main motivation for this paper comes from the observation that the proof in [11] of the finite Blaschke product case can easily be generalized to obtain a proof of Conjecture 1, with some additional assumptions on the behavior of the function f on ∂D. Although this approach yields a slightly weaker result than that of Lowther and Speyer, it has considerable advantages. For instance, the argument also works in the meromorphic setting, in which case the polynomial p must be replaced by a rational map Date: February 24 2015.

SummaryThe incidence of malocclusion was recorded for 4 years. Malocclusion occurred only in animals less than 2 months old. The incidence was significantly reduced (P > 0.001) by breeding from animals without affected siblings: it is suggested that malocclusion in this colony has a genetic basis.

In this paper we present a geometric proof of the following fact. Let D be a Jordan domain in C, and let f be analytic on cl(D). Then there is an injective analytic map φ : D → C, and a polynomial p, such that f ≡ p • φ on D (that is, f has a polynomial conformal model p).