Univalent polynomials and Koebe’s one-quarter theorem

Abstract

The famous Koebe 1 4 theorem deals with univalent (i.e., injective) analytic functions f on the unit disk D. It states that if f is normalized so that f(0) = 0 and f ′ (0) = 1, then the image f(D) contains the disk of radius 1 4 about the origin, the value 1 4 being best possible. Now suppose f is only allowed to range over the univalent polynomials of some fixed degree. What is the optimal radius in the Koebe-type theorem that arises? And for which polynomials is it attained? A plausible conjecture is stated, and the case of small degrees is settled.