Loewner Chains and the Roper–Suffridge Extension Operator

Abstract Let f be a locally univalent function on the unit disc and let α ∈ [0,  1 2 ]. 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 =  1 2 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| π 2 then Φn, α(f) is a spirallike map of type β. In particular we obtain a new proof that the Roper–Suffridge operator preserves starlikeness. We also obtain the radius of starlikeness of Φn, α(S) and the radius of convexity of Φn, 1/2(S). We show that if f is a normalized univalent Bloch function on U then Φn, α(f) is a Bloch mapping on B. Finally we show that if f belongs to a class of univalent functions which satisfy growth and distortion results, then Φn, α(f) satisfies related growth and covering results.