This reference details valuable results that lead to improvements in existence theorems for the Loewner differential equation in higher dimensions, discusses the compactness of the analog of the Caratheodory class in several variables, and studies various classes of univalent mappings according to their geometrical definitions. It introduces the infinite-dimensional theory and provides numerous exercises in each chapter for further study. The authors present such topics as linear invariance in the unit disc, Bloch functions and the Bloch constant, and growth, covering and distortion results for starlike and convex mappings in Cn and complex Banach spaces.
Table of Contents
Univalent functions: elementary properties of univalent functions; subclasses of univalent functions in the unit disc; the Loewner theory; Bloch functions and the Bloch constant; linear invariance in the unit disc; univalent mappings in several complex variables and complex Banach spaces; univalence in several complex variables; growth, covering and distortion results for starlike and convex mappings in Cn and complex Banach spaces; Loewner chains in several complex variables; Bloch constant problems in several complex variables; linear invariance in several complex variables; univalent mappings and the Roper-Suffridge extension operator.
Graham, Ian; Kohr, Gabriela