The generalized Roper–Suffridge extension operator ✩

Let f (z) be a normalized convex (starlike) function on the unit disc D. Let Ω = {z ∈ Cn: |z1|2 + |z2|p2 +···+|zn|pn < 1}, where z = (z1, z2, . . . , zn), z1 ∈ D, (z2, . . . , zn) ∈ Cn−1, pi 1, i = 2, . . . ,n, are real numbers. In this note, we prove that Φ(f )(z) = (f (z1), f (z1)1/p2z2, . . . , f (z1)1/pnzn) is a normalized convex (starlike) mapping on Ω, where we choose the power function such that (f (z1))1/pi |z1=0 = 1, i = 2, . . . , n. Some other related results are proved.  2003 Elsevier Inc. All rights reserved.