Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 859-873
Syracuse University, Department of Mathematics
Syracuse, NY 13244, U.S.A.
The Open University, Walton Hall, Department of Mathematics and Statistics
Milton Keynes MK7 6AA, U.K.; phil.rippon 'at' open.ac.uk
University of Liverpool,
Department of Mathematical Sciences
Liverpool L69 7ZL, U.K.;
david.sixsmith 'at' open.ac.uk
Abstract. In 1970 MacLane asked if it is possible for a locally univalent function in the class 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 Bishop in order to show that there is a function in the Eremenko–Lyubich class for the disc that is not in the class A.
2010 Mathematics Subject Classification: Primary 30D40.
Key words: MacLane class, Eremenko–Lyubich class, locally univalent, asymptotic value, arc tract, level curve, quasiconformal folding.
Reference to this article: K. F. Barth, P. J. Rippon and D. J. Sixsmith: The MacLane class and the Eremenko–Lyubich class. Ann. Acad. Sci. Fenn. Math. 42 (2017), 859-873.
https://doi.org/10.5186/aasfm.2017.4252
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