Loewner Chains and the Roper–Suffridge Extension Operator

Let f be a locally univalent function on the unit disc and let α∈[0,12]. We consider the family of operators extending f to a holomorphic map from the unit ball B in Cn to Cn given by Φn,α(f)(z)=(f(z1),z′(f′(z1))α), where z′= (z2,…,zn). When α=12 we obtain the Roper–Suffridge extension operator. We...

Full description

Saved in:
Bibliographic Details
Published in:Journal of mathematical analysis and applications 2000-07, Vol.247 (2), p.448-465
Main Authors: Graham, Ian, Kohr, Gabriela, Kohr, Mirela
Format: Article
Language:English
Subjects:
convexity
Exact sciences and technology
Functions of a complex variable
Loewner chains
Mathematical analysis
Mathematics
Operator theory
radius of convexity
radius of starlikeness
Sciences and techniques of general use
spirallikeness
starlikeness
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Let f be a locally univalent function on the unit disc and let α∈[0,12]. We consider the family of operators extending f to a holomorphic map from the unit ball B in Cn to Cn given by Φn,α(f)(z)=(f(z1),z′(f′(z1))α), where z′= (z2,…,zn). When α=12 we obtain the Roper–Suffridge extension operator. We show that if f∈S then Φn,α(f) can be imbedded in a Loewner chain. Our proof shows that if f∈S* then Φn,α(f) is starlike, and if f∈Ŝβ with |β|
ISSN:0022-247X
1096-0813
DOI:10.1006/jmaa.2000.6843