The generalized Roper-Suffridge extension operator

Let f (z) be a normalized convex (starlike) function on the unit disc D. Let Omega={z is an element of C-n:\z(1)\(2)+\z(2)\(p2)+...+\z(n)\(pn)<1}, where z=(z(1), z(2),...,z(n)), z(1) ∈ D (z(2),...,z(n)) ∈ Cn-1, p(i)≥1 i=2,...,n, are real numbers. In this note, we prove that Φ(f)(z)=(f(z(1)), f'(z(1))(1/p 2) z(2),...,f'(z(1))(1/p n) z(n)) is a normalized convex (starlike) mapping on Ω, where we choose the power function such that (f'((z)1))(1/p i)|z(1=0)=1, i=(2),..., n. Some other related results are proved. (C) 2003 Elsevier Inc. All rights reserved.