Let λG(z)|dz|be the hyperbolic metric on a simply connected proper domain G ⊂ C containing the origin, and let||·||j be the Banach norms of Cnj for j=1, 2, …, k.This note is to prove that if f is a normalized biholomorphic convex function on G, then
ΦN,1/p1,…,1/pk(f)(z)=F1/p1,…,1/pk(z)=f(z1), (f'(z1))1/p1z, …, (f'(z1))1/pkw)
is a normalized biholomorphic convex mapping on
ΩN={(z1, z, …, w) ∈ C×Cn1×…×Cnk:||z||1p1 + … +||w||kpk<1/λG (z1)},
where N=1 + n1 + … + nk and the branch is chosen such that (f'(z1))1/pj|z1=0=1, j=1, …, k. Applying to the Roper-Suffridge extension operator, we obtain a new convex mappings construction of an unbounded domain and a refinement of convex mappings construction on a Reinhardt domain, respectively.
Danli ZHANG
,
Huiming XU
,
Jianfei WANG
. CONVEX MAPPINGS ASSOCIATED WITH THE ROPER-SUFFRIDGE EXTENSION OPERATOR[J]. Acta mathematica scientia, Series B, 2019
, 39(6)
: 1619
-1627
.
DOI: 10.1007/s10473-019-0612-9
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