THE ROPER-SUFFRIDGE EXTENSION OPERATOR AND ITS APPLICATIONS TO CONVEX MAPPINGS IN C2

The purpose of this paper is twofold. The first is to investigate the Roper-Suffridge extension operator which maps a biholomorhic function f on D to a biholomorphic mapping F on Omega(n, p2), ... , (pn) (D) = {(z(1), z(0)) is an element of D x Cn-1 : Sigma(n)(j-2) vertical bar z(j)vertical bar(pj) < 1/lambda(D)(z(1))}, p(j) >= 1, where z(0) = (z(2), ... , z(n)) and lambda(D) is the density of the Poincare metric on a simply connected domain D subset of C. We prove this Roper-Suffridge extension operator preserves epsilon-starlike mapping: if f is epsilon-starlike, then so is F. As a consequence, we solve a problem of Graham and Kohr in a new method. By introducing the scaling method, the second part is to construct some new convex mappings of domain Omega(2, m) = {(z(1), z(2)) is an element of C-2 : vertical bar z(1)vertical bar(2) + vertical bar z(2)vertical bar(m) < 1} with m >= 2, which can be applied to discuss the extremal point of convex mappings on the domain. This scaling idea can be viewed as providing an alternative approach to studying convex mappings on Omega(2, m). Moreover, we propose some problems.