Аннотация:
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.
