Abstract
Let 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 Lund et al. (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 |f′(z)| 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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
