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 ℂ ⁿ. 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.

Keywords

Quasi-convex mapping / growth theorem / covering theorem / distortion theorem / zero of order k + 1

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