A note on Cantor boundary behavior

For an analytic function f on the open unit disk D and continuous on D ¯ , the Cantor boundary behavior (CBB) is used to describe the curve f ( ∂ 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.