Loewner Chains and the Roper–Suffridge Extension Operator

Abstract

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|<π2 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.