Semigroups in geometrical function theory (Q2743225)

This is an interesting book concerning several applications of semigroups in the geometric function theory. The primary focus of the book is to investigate classical theorems with new meanings which are related to applications of evolution equations to the geometry of domains in the complex plane. The book is divided into five chapters including also a preface and a detailed list of bibliography. In the first part are introduced basic notions and results about holomorphic functions of a complex variable and metric spaces and fixed points principles. In chapter 1 there is studied the Wolff-Denjoy theory on the unit disc. For this purpose, it is presented the Schwarz-Pick lemma with several applications. Next are discussed several results about the boundary behaviour of holomorphic self-mappings and fixed points of holomorphic self-mappings. Another look at the Wolff-Denjoy theory is the using of the hyperbolic metric of a domain. The chapter 2 is devoted to the study of the hyperbolic geometry on the unit disc and fixed points. In the next chapter is presented the generation theory on the unit disc. For example, several flow invariance conditions for the class of holomorphic functions are obtained in Section 3.5. Chapter 4 is devoted to give a connection of the iterating theory of functions in one complex variable and the asymptotic behaviour of solutions of ordinary differential equations governed by the evolution problem. The most interesting chapter is the last chapter which is devoted to show some relationships between semigroups and the geometry of domains in the complex plane. The authors study those univalent functions on the unit disc whose images are starlike or spirallike domains. Another objective of this chapter is to study the dynamics of starlike or spirallike domains when the origin is placed out on the boundary. The book ends with a list of bibliography containing 160 titles. NEWLINENEWLINENEWLINEThe book is well written and elegantly structured. Also its subject is interesting and the figures included are suggestive. It is useful to students and postgraduate students who are interested in complex dynamic systems, as well as to all researchers specialized in complex analysis and differential equations.