Dynamics of spirallike functions

In this paper, we study functions that are spirallike with respect to a boundary point of the unit disc by using dynamical systems which determine such functions. Namely, we prove that the differential equation muh(z) = (z - tau)(1 - (tau) over barz)h'(z)p(z), with the initial condition h(0) = 1, defines a univalent function h on the open unit disc A whose image h(A) is spirallike with respect to a boundary point if and only if tau is an element of partial derivativeDelta, Re p > 0, the angular limit anglelim(z-tau)(1 - (tau) over barz)p(z) = B > 0, and \mu/beta - 1 < 1, mu not equal 0. Furthermore, we consider those perturbations of the above-mentioned dynamical system which define functions spirallike with respect to interior points of the unit disc (the centers of spirallikeness). We examine the dynamics and stability of the process which results when these points are pushed back to the boundary. In this way, we show that each function spirallike with respect to a boundary point is a complex power of a function starlike with respect to the same boundary point. This enables us to get some new representations of such functions as well as to point out some estimates of their growth in the spirit of M. S. Robertson.