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Forum Mathematicum

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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

Publication History

Received:
2006-01-30
Revised:
2006-04-13
Published Online:
2007-02-21

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 a n(f), n ⩾ 2, fCo(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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