Correspondence between spirallike functions and starlike functions

Let λ be a real number with -π/2 < λ < π/2. In order to study λ-spirallike functions, it is natural to measure the angle according to λ-spirals. Thus we are led to the notion of λ-argument. This fits well the classical correspondence between λ-spirallike functions and starlike functions. Using this idea, we extend deep results of Pommerenke and Sheil-Small on starlike functions to spirallike functions. As an application, we solved a problem given by Hansen in 6.