The MacLane class and the Eremenko-Lyubich class

TL;DR

This paper explores the properties of certain classes of complex functions, strengthening previous results by relaxing conditions and demonstrating the existence of functions outside a specific class using advanced techniques.

Contribution

It extends known results by replacing local univalence with bounded critical values and shows a function in the Eremenko-Lyubich class not belonging to class  using Bishop's technique.

Findings

Replaced local univalence condition with bounded critical values.

Constructed a function in the Eremenko-Lyubich class outside .

Strengthened understanding of the structure of these function classes.

Abstract

In 1970 G. R. MacLane asked if it is possible for a locally univalent function in the class $\mathcal{A}$ to have an arc tract. This question remains open, but several results about it have been given. We significantly strengthen these results, in particular replacing the condition of local univalence by the more general condition that the set of critical values is bounded. Also, we adapt a recent powerful technique of C. J. Bishop in order to show that there is a function in the Eremenko-Lyubich class for the disc that is not in the class $\mathcal{A}$.