Cantor Boundary Behavior of Analytic Functions

  • Xin-Han Dong
  • Ka-Sing Lau
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


Let\(\mathbb{D}\) be the open unit disc and let \(\partial \mathbb{D}\) be the boundary of \(\mathbb{D}\). For f(z) analytic in \(\mathbb{D}\) and continuous on \(\overline{\mathbb{D}}\), it follows from the open mapping theorem that \(\partial f(\mathbb{D}) \subset f(\partial \mathbb{D})\). These two sets have very rich and intrigue geometric properties. When f(z) is univalent, then they are equal and there is a large literature to study their boundary behaviors. Our interest is on the class of analytic functions f(z) for which the image curves \(f(\partial \mathbb{D})\) form infinitely many loops everywhere, they are not univalent of course. We formulate this as the Cantor boundary behavior. We give sufficient conditions for such property, making use of the distribution of the zeros of f and the mean growth rate of f . Examples includes the complex Weierstrass functions, and the Cauchy transform of the canonical Hausdorff measure on the Sierpiski gasket.


Connected Domain Jordan Curve Blaschke Product Iterate Function System Sierpinski Gasket 
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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsHunan Normal UniversityChang ShaChina
  2. 2.Department of MathematicsThe Chinese University of Hong KongHong KongChina

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