Loewner chains associated with the generalized Roper–Suffridge extension operator on some domains

In this paper, we consider the generalized Roper–Suffridge extension operator defined by Φn,β2,γ2,...,βn,γn(f )(z) = f (z1), f (z1) z1 β2 f (z1) γ2 z2, . . . , f (z1) z1 βn f (z1) γnzn for z = (z1, z2, . . . , zn) ∈ Ωp1,p2,...,pn, where 0 βj 1, 0 γj 1 − βj , pj > 1, and we choose the branch of the power functions such that ( f (z1) z1 )βj |z1=0 = 1 and (f (z1))γj |z1=0 = 1, j = 1, 2, . . . , n, Ωp1,p2,...,pn = (z1, z2, . . . , zn) ∈ Cn: n j=1 |zj |pj < 1 . We prove that the set Φn,β2,γ2,...,βn,γn(S(U)) can be embedded in Loewner chains and give the answer to the problem of Liu Taishun. We also obtain that the operator Φn,β2,γ2,...,βn,γn(f ) preserves starlikeness or spirallikeness of type α on Ωp1,p2,...,pn for some suitable constants βj , γj, where S(U) is the class of all univalent analytic functions on the unit disc U in the complex plane C with f (0) = 0 and f (0) = 1. © 2007 Elsevier Inc. All rights reserved.