A New Roper-Suffridge Extension Operator on a Reinhardt Domain

We introduce a new Roper-Suffridge extension operator on the following Reinhardt domain Ω n , p 2 , … , p n = { z ∈ ℂ n : | z 1 | 2 + ∑ j = 2 n | z j | p j < 1 } given by F ( z ) = ( f ( z 1 ) + f ′ ( z 1 ) ∑ j = 2 n a j z j p j , ( f ′ ( z 1 ) ) 1 / p 2 z 2 , … , ( f ′ ( z 1 ) ) 1 / p n z n ) , where f is a normalized locally biholomorphic function on the unit disc D , p j are positive integer, a j are complex constants, and j = 2 , … , n . Some conditions for a j 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 α j = 0 .