The first special subclass of the family of univalent functions was that of convex functions introduced by Study. These are the functions which map |z|<1 onto convex domains, they were also studied by Gronwall, Löwner and others. Next was the class of the starlike functions introduced by Alexander and later by Nevanlinna and others. There are number of powerful methods developed by several authors, the notion of an univalent function is readily extended to that of a multivalent function, a multivalent function defined in a certain domain being one which in that domain takes no value more than p times. Various authors applied to multivalent functions analogous of the elementary methods for univalent functions. The univalent functions represent the classical case of the theory and this we are going to study in this book in detail.