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A proof of the Livingston conjecture
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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