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International Journal of Mathematics and Mathematical Sciences
Volume 22 (1999), Issue 3, Pages 463-468
http://dx.doi.org/10.1155/S0161171299224635

A proper subclass of Maclane's class 𝒜

Lebanese American University, Beirut, Lebanon

Received 26 June 1997; Revised 20 April 1998

Copyright © 1999 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 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 is a proper subclass of 𝒜 by constructing a function f𝒜 that admits no asymptotic paths of finite length.