Growth and distortion theorems on subclasses of quasi-convex mappings in several complex variables

Abstract

In this paper, the refining growth and covering theorems for f are established, where f is a quasi-convex mapping of order α and x = 0 is a zero of order k + 1 of f(x) − x. As an application, we obtain the upper and lower bounds on the distortion theorem of f(x) defined on the unit polydisc of ℂn. The upper bound of the distortion theorem for f(x) defined on the unit ball of a complex Banach space is also given. Our results extend the growth and distortion theorems for convex functions of one complex variable to quasi-convex mappings of several complex variables.