A note on Cantor boundary behavior

[摘要] For an analytic function f on the open unit disk ID and continuous on (D) over bar, the Cantor boundary behavior (CBB) is used to describe the curve f(partial derivative D) that forms infinitely many fractal-look loops everywhere. The class of analytic functions with the CBB was formulated and investigated in Dong et al. [6]. In this note, our main objective is to give further discuss of the criteria of CBB in Dong et al. [6]. We show that the two major criteria, the accumulation of the zeros of f'(z) near the boundary and the fast mean growth rate of f'(z) near the boundary, do not imply each other. Also we make an improvement of another criterion, which allows us to have more examples of CBB. (C) 2013 Elsevier Inc. All rights reserved.

[发布日期] 2013-12-15

[关键词] Cantor boundary behavior;Infinite Blaschke product;Pre-Schwarzian derivative;Integral means