A New Roper-Suffridge Extension Operator on a Reinhardt Domain

Abstract EN

We introduce a new Roper-Suffridge extension operator on the following Reinhardt domain Ωn, p2,…, pn={z∈ℂn:|z1|2+∑j=2n|zj|pj<1} given by F(z)=(f(z1)+f′(z1)∑j=2najzjpj,(f′(z1))1/p2z2,…,(f′(z1))1/pnzn), where f is a normalized locally biholomorphic function on the unit disc D, pj are positive integer, aj are complex constants, and j=2,…,n.

Some conditions for aj are found under which the operator preserves almost starlike mappings of order α and starlike mappings of order α, respectively.

In particular, our results reduce to many well-known results when all αj=0.