Linear invariance and extension operators of Pfaltzgraff–Suffridge type
DOI:
https://doi.org/10.24193/subbmath.2022.2.06Keywords:
Linear-invariant family, Pfaltzgraff–Suffridge extension operator, Roper–Suffridge extension operator, convex mapping.Abstract
We consider the image of a linear-invariant family F of normalized locally biholomorphic mappings defined in the Euclidean unit ball Bn of Cn under the extension operator
Φn,m,β [f ](z, w) = (f (z), [Jf (z)]βw\, (z, w) ∈ Bn+m ⊆ Cn × Cm,
where β ∈ C, Jf denotes the Jacobian determinant of f , and the branch of the power function taking 0 to 1 is used. When β = 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 Bn+m generated by the image in terms of the order of F, taking note that the resulting family has minimum order if and only if either β ∈ (−1/m, 1/(n+1)] and the family F has minimum order or β = −1/m. We will also see that order is preserved when generating a linear-invariant family from the family obtained by composing F with a certain type of automorphism of Cn, leading to consequences for various extension operators including the modified Roper–Suffridge extension operator introduced by the author.
Mathematics Subject Classification (2010): 32H02, 32A30, 30C45.
Received 9 December 2021; Accepted 29 December 2021.
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