Dynamical approach to starlike and spirallike functions

This chapter is devoted to showing some relationships between semigroups and the geometry of domains in the complex plane. Mostly we will study those univalent (one-to-one correspondence) functions on the unit disk whose images are starshaped or spiralshaped domains. Several important aspects, however, had to be omitted, e.g. convex and close-to-convex functions (see, for example, [57, 55]), and other different classes of univalent functions. We have selected the forthcoming material according to the guiding principle that the demonstrated methods may be generalized to higher dimensions. For example, the celebrated Koebe One Quarter Theorem states that the image of a univalent function h on ∆ normalized by the condition h(0) = 0 and h’(0) = 1 contains a disk of radius 1/4. This theorem is no longer true at higher dimensions. Nevertheless, the dynamical approach analogues of the Koebe theorem have been recently established and used for subclasses of starlike (or spirallike) functions (see, for example [141, 109, 26, 56, 14]).