Barth, Karl F. and Rippon, Philip J.
Infinitely many asymptotic values of locally univalent functions.
Annales Academiae Scientiarum Fennicae Mathematica, 28 pp. 303–314.
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McMillan and Pommerenke showed that a locally univalent meromorphic function in the disc, without Koebe arcs, has at least three asymptotic values in each boundary arc. The modular function shows that the number three is best possible. We show that if satisfies certain further conditions, each of which narrowly excludes the modular function, then the number of asymptotic values in each boundary arc must be infinite.
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