Advanced versions of the inverse function theorem for holomorphic mappings

Abstract

This opus is dedicated to the bright memory of the distinguished mathemati-
cian Gabriela Kohr. Her contribution to analysis of one and several complex
variables brought new knowledge into the modern theory as well as new colors
to the subject. During our meetings with Gabriela at various conferences she
always proposed some interesting and often nonstandard questions related to
classical issues as well as new directions. It is worth to mentioned her excel-
lent book [48] together with Ian Graham on classical and modern problems in
Geometric Function Theory in complex spaces (see also, [51] and [52]).
One of the well presented subjects in this book among others is the so-
called Bloch constant which is closely connected to the classical Inverse Function
Theorem and related topics. We will try here to describe some aspects of those
problems. For the readers convenience the manuscript contains a quit long list
of references and remarks related to the sources of the results mentioned in the
sequel.