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 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 .

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