Semigroups in Geometrical Function Theory

This manuscript provides an introduction to the generation theory of nonlinear one-parameter semigroups on a domain of the complex plane in the spirit of the Wolff-Denjoy and Hille-Yoshida theories. Special attention is given to evolution equations reproduced by holomorphic vector fields on the unit disk. A dynamic approach to the study of geometrical properties of univalent functions is emphasized. The book comprises six chapters. The preliminary chapter and chapter 1 give expositions to the theory of functions in the complex plane, and the iteration theory of holomorphic mappings according to Wolff and Denjoy, as well as to Julia and Caratheodory. Chapter 2 deals with elementary hyperbolic geometry on the unit disk, and fixed points of those mappings which are nonexpansive with respect to the Poincare metric. Chapters 3 and 4 study local and global characteristics of holomorphic and hyperbolically monotone vector-fields, which yield a global description of asymptotic behaviour of generated flows. Various boundary and interior flow invariance conditions for such vector-fields and their parametric representations are presented. Applications to univalent starlike and spirallike functions on the unit disk are given in Chapter 5. The approach described may also be useful for higher dimensions.