Schwarz-Pick Systems of Pseudometrics for Domains in Normed Linear Spaces

Abstract

This chapter presents a study on Schwarz–Pick systems of pseudometrics for domains in normed linear spaces. The chapter presents an elementary account of systems of assigning pseudometrics to domains in normed linear spaces so that the Schwarz–Pick inequality holds for any holomorphic mapping of one domain into another and so that the pseudometric assigned to the open unit disc of the complex plane is the usual Poincaré metric. The chapter focuses on the Caratheodory and Kobayashi systems of pseudometrics. Of all systems, these assign the smallest and largest pseudometrics, respectively, to a given domain. The chapter begins with the definition of an infinitesimal Finder pseudometrica, which is used to measure lengths of curves, and construct an associated pseudometric called the integrated form of ∝. The chapter presents examples on the Carathéodory–Reiffen–Finsler (CRF)-pseudometrics, which are a Schwarz–Pick system. These pseudometrics are used together with the contraction mapping theorem to obtain the Earle–Hamilton fixed point theorem and a corresponding inverse function theorem. The chapter also discusses the basic properties of Schwarz–Pick systems and provides a sufficient condition for the completeness of a domain with respect to a pseudometric assigned to it by such a system. By considering differentiability of pseudometrics, it is shown that the Caratheodory pseudometric, the infinitesimal CRF-pseudometric, and its integrated form all assign the same lengths to curves.