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A proof of the Livingston conjecture
Citation Information: Forum Mathematicum. Volume 19, Issue 1, Pages 149–157, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: 10.1515/FORUM.2007.007, February 2007
- Published Online:
Let D denote the open unit disc and f : D → be meromorphic and injective in D. We further assume that f has a simple pole at the point p ∈ (0, 1) and an expansion
In particular, we consider functions f that map D onto a domain whose complement with respect to is convex. Because of the shape of f (D) these functions will be called concave univalent functions with pole p and the family of these functions is denoted by Co(p).
It is proved that for fixed p ∈ (0, 1) the domain of variability of the coefficient a
n(f), n ⩾ 2, f ∈ Co(p), is determined by the inequality
This settles two conjectures published by A. E. Livingston in 1994 and by Ch. Pommerenke and the authors of the present article in 2004.
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