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STUDIA MATHEMATICA  Issue no. 2 / 2022  
Article: 
LINEAR INVARIANCE AND EXTENSION OPERATORS OF PFALTZGRAFF SUFFRIDGE TYPE. Authors: JERRY R. MUIR. 

Abstract: DOI: 10.24193/subbmath. 2022.2.06 Published Online: 20220610 Published Print: 20220630 pp. 295308 VIEW PDF FULL PDF Abstract: We consider the image of a linearinvariant 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 PfaltzgraffSuffridge extension operator. In particular, we determine the order of the linearinvariant 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 linearinvariant 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 RoperSuffridge extension operator introduced by the author. Key words: Linearinvariant family, PfaltzgraffSuffridge extension operator, RoperSuffridge extension operator, convex mapping. Mathematics Subject Classification (2010): 32H02, 32A30, 30C45. 
