Functions of bounded boundary rotation

Classes of functions U _k, which generalize starlike functions in the same manner that the class V _k of functions with boundary rotation bounded by kπ generalizes convex functions, are defined. The radius of univalence and starlikeness is determined. The behavior of f _α(z) = ∫ ₀ ^z [f'(t)]^α dt is determined for various classes of functions. It is shown that the image of |z|<1 under V _kfunctions contains the disc of radius 1/k centered at the origin, and V _k functions are continuous in |z|≦1 with the exception of at most [k/2+1] points on |z|=1.