Extension of convex mappings of order α of the unit disk in C to convex mappings of the unit ball in Cn

Abstract

For a convex mapping f of order α ∈ [ 0 , 1 ) of the unit disk in the complex plane C , we consider conditions on the parameter β ∈ [ 0 , 1 / 2 ] so that the mapping F ( z ) = ( f ( z 1 ) , [ f ′ ( z 1 ) ] β z ˆ ) , z ˆ = ( z 2 , … , z n ) , is a convex mapping of the Euclidean unit ball in C n .