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Forum Mathematicum

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Volume 19, Issue 1 (Jan 2007)


A proof of the Livingston conjecture

Farit G Avkhadiev
  • Chebotarev Research Institute, Kazan State University, 420008 Kazan, Russia.
  • Email:
/ Karl-Joachim Wirths
  • Institut für Analysis und Algebra, TU Braunschweig, 38106 Braunschweig, Germany.
  • Email:
Published Online: 2007-02-21 | DOI: https://doi.org/10.1515/FORUM.2007.007


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.

About the article

Received: 2006-01-30

Revised: 2006-04-13

Published Online: 2007-02-21

Published in Print: 2007-01-29

Citation Information: Forum Mathematicum, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/FORUM.2007.007. Export Citation

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