Parametric representation and extension operators for biholomorphic mappings on some Reinhardt domains§

Abstract

In this article we obtain a generalization of the Pfaltzgraff–Suffridge extension operator, denoted by Φ n,p , on some Reinhardt domains in C n . We prove that if f∈ S 0(Bn ) and p≥ 2n/(n+1) then where . In particular, if f is starlike on the Euclidean unit ball B n then is starlike on Ω n,p . We also give examples of starlike mappings on Ω n,p . Moreover, we study certain convexity properties associated with the operator Φ n,p . Further, we prove that the Roper–Suffridge extension operator Ψn preserves starlikeness of order 1/2. In the last section we obtain certain subordination results associated with the operator Φ n,p .