Cantor boundary behavior of analytic functions
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AbstractLet A(D) be the space of analytic functions on the open disk D and continuous on D. Let ∂D be the boundary of D, we are interested in the class of f∈A(D) such that the image f(∂D) is a curve that forms loops everywhere. This fractal behavior was first raised by Lundetal.(1998)[21] in the study of the Cauchy transform of the Hausdorff measure on the Sierpinski gasket. We formulate the property as the Cantor boundary behavior (CBB) and establish two sufficient conditions through the distribution of zeros of f' (z) and the mean growth rate of {pipe}f' (z) {pipe} near the boundary. For the specific cases we carry out a detailed investigation on the gap series and the complex Weierstrass functions; the CBB for the Cauchy transform on the Sierpinski gasket will appear elsewhere. © 2012 Elsevier Ltd.
All Author(s) ListDong X.-H., Lau K.-S., Liu J.-C.
Journal nameAdvances in Mathematics
Volume Number232
Issue Number1
PublisherAcademic Press
Place of PublicationUnited States
Pages543 - 570
LanguagesEnglish-United Kingdom
KeywordsAnalyticity, Blaschke product, Boundary behavior, Cantor set, Conformal, Fractal, Growth rate, Lacunary series, Simply connected, Univalence, Weierstrass function, Zeros

Last updated on 2021-11-01 at 01:52