Loewner Chains and the Roper]Suffridge Extension Operator

Let f be a locally univalent function on the unit disc and let a gw0, 1x. We 2 consider the family of operators extending f to a holomorphic map from the unit ball B in Cn to Cn given by Fn, a f. z.s f z1., zX fX z1..a ., where zXs z2 , . . . , zn.. When a s 12 we obtain the Roper]Suffridge extension operator. We show that if fgS then Fn, a f. can be imbedded in a Loewner chain. Our proof shows that if fgSU then Fn,a f. is starlike, and if fgSˆbwith