The generalized Roper–Suffridge extension operator for some biholomorphic mappings
- Liu, Xiaosong
DOIAbstract
AbstractSuppose f is a spirallike function of type β (or starlike function of order α) on the unit disk D in C. Let Ωn,p1,p2,…,pn={z=(z1,z2,…,zn)′∈Cn:∑j=1n|zj|pj<1}, where 1⩽p1⩽2 (or 0<p1⩽2), pj⩾1, j=2,…,n, are real numbers. In this paper, we prove thatΦn,β2,γ2,…,βn,γn(f)(z)=(f(z1),(f(z1)z1)β2(f′(z1))γ2z2,…,(f(z1)z1)βn(f′(z1))γnzn)′ preserves spirallikeness of type β (or starlikeness of order α) on Ωn,p1,p2,…,pn, where βj∈[0,1], γj∈[0,1pj], and βj+γj⩽1, pj is the same as above, we choose the branches such that(f(z1)z1)βj|z1=0=1,(f′(z1))γj|z1=0=1,j=2,…,n
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