A note on Cantor boundary behavior

For an analytic function ff on the open unit disk DD and continuous on D¯, the Cantor boundary behavior (CBB) is used to describe the curve f(∂D)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)f′(z) near the boundary and the fast mean growth rate of f′(z)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.