A proper subclass of Maclane's class 𝒜

Abstract

The MacLane's class 𝒜 of analytic functions is the class of nonconstant analytic functions in the unit disk that have asymptotic values at a dense subset of the unit circle. In this paper, we define a subclass ℛ of 𝒜 consisting of those functions that have asymptotic values at a dense subset of the unit circle reached along rectifiable asymptotic paths. We also show that the class ℛ is a proper subclass of 𝒜 by constructing a function f∈𝒜 that admits no asymptotic paths of finite length.