The MacLane class and the Eremenko–Lyubich class

Accepted for publication by Ann. Acad. Sci. Fenn. Math (2017). Available on the arXiv. This is a joint work with Phil Rippon and Karl Barth.

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, and this question remains open despite several partial results. Here we significantly strengthen these results by introducing new techniques associated with the Eremenko-Lyubich class for the disc. 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 $\mathcal{A}$.