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 Phi(n),(beta 2),(gamma 2) ,..., (beta n),(gamma n) (f) (z) = (f (z(1)), (f(z(1))/z(1))(beta 2) (f' (z(1)))(gamma 2) z(2) ,..., (f(z(1))/z(1))(beta) (f'(z(1)))(gamma n) z(n)) for z = (z(1),z(2),...,z(n)) epsilon Omega(p1),(p2) ,..., p(n), where 0 <= beta(j) <= 1, 0 <= gamma(j) <= 1 - beta(j), p(j) > 1, and we choose the branch of the power functions such that (f(z(1))/z(1))(beta j) |(z1=0) = 1 and (f'(z(1)))(gamma j) |(z1=0) = 1, j = 1,2,...,n, Omega(p1),(p2) ,..., p(n) = {(z(1), z(2) ,..., z(n)) epsilon C-n: Sigma(n)(j=1) vertical bar z(j)vertical bar p(j) < 1}. We prove that the set on Phi(n,beta 2,gamma 2) ,..., (beta n,gamma n) (S(U)) can be embedded in Loewner chains and give the answer to the problem of Liu Taishun. We also obtain that the operator Phi(n,beta 2,gamma 2) ,..., (beta n,gamma n) (f) preserves starlikeness or spirallikeness of type alpha on Omega(p1,p2) ,..., (pn) for some suitable constants beta(j), gamma(j), 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. (c) 2007 Elsevier Inc. All rights reserved.