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 byΦn,β2,γ2,…,βn,γn(f)(z)=(f(z1),(f(z1)z1)β2(f′(z1))γ2z2,…,(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:∑j=1n|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.