This paper introduces the
concept of ω-swirly domains as a generalization of the notion of starlike domains.
After analyzing certain geometric properties of this concept, some analytical
properties of conformal mappings of the unit disc onto w-swirly domains are
developed. The main theorem determines the maximum radius rω for which
{z : |z| < rω} is mapped onto an ω-swirly domain by all schlicht functions on the unit
disc; thus it gives an estimate of the twistedness of the images of subdiscs centered at
the origin under conformal mappings of the unit disc. Comparisons are then made
between the concept of ω-swirly and other geometric notions in the theory of
conformal mapping.
