Loewner Chains and the Roper–Suffridge Extension Operator

Let f be a locally univalent function on the unit disc and let α∈[0,12]. We consider the family of operators extending f to a holomorphic map from the unit ball B in Cn to Cn given by Φn,α(f)(z)=(f(z1),z′(f′(z1))α), where z′= (z2,…,zn). When α=12 we obtain the Roper–Suffridge extension operator. We show that if f∈S then Φn,α(f) can be imbedded in a Loewner chain. Our proof shows that if f∈S* then Φn,α(f) is starlike, and if f∈Ŝβ with |β|