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Authors: JERRY R. MUIR.
DOI: 10.24193/subbmath. 2022.2.06
Published Online: 2022-06-10
Published Print: 2022-06-30
pp. 295-308
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Abstract: We consider the image of a linear-invariant family $mathcal{F}$ of normalized locally biholomorphic mappings defined in the Euclidean unit ball $B_n$ of $C^n$ under the extension operator Φn,m,β[f](z,w)=mleft(f(z),[Jf(z)]βwmright),(z,w)∈Bn+m⊆Cn×Cm,Φn,m,β[f](z,w)=mleft(f(z),[Jf(z)]βwmright),(z,w)∈Bn+m⊆Cn×Cm, where $eta in C$, $Jf$ denotes the Jacobian determinant of $f$, and the branch of the power function taking $0$ to $1$ is used. When $eta=1/(n+1)$ and $m=1$, this is the Pfaltzgraff--Suffridge extension operator. In particular, we determine the order of the linear-invariant family on $B_{n+m}$ generated by the image in terms of the order of $mathcal{F}$, taking note that the resulting family has minimum order if and only if either $eta in (-1/m,1/(n+1)]$ and the family $mathcal{F}$ has minimum order or $eta=-1/m$. We will also see that order is preserved when generating a linear-invariant family from the family obtained by composing $mathcal{F}$ with a certain type of automorphism of $C^n$, leading to consequences for various extension operators including the modified Roper--Suffridge extension operator introduced by the author.
Key words: Linear-invariant family, Pfaltzgraff-Suffridge extension operator, Roper-Suffridge extension operator, convex mapping.
Mathematics Subject Classification (2010): 32H02, 32A30, 30C45.
