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Abstract
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 an(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.
Received: 2006-01-30
Revised: 2006-04-13
Published Online: 2007-02-21
Published in Print: 2007-01-29
© Walter de Gruyter
