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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Brüdern, Jörg / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Neeb, Karl-Hermann / Noguchi, Junjiro / Shahidi, Freydoon / Sogge, Christopher D. / Wienhard, Anna

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Volume 19, Issue 1 (Jan 2007)

Issues

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

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.

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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Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Martin Chuaqui and Christian Pommerenke
Pacific Journal of Mathematics, 2014, Volume 270, Number 2, Page 319
[2]
Bappaditya Bhowmik and Karl-Joachim Wirths
Results in Mathematics, 2014, Volume 65, Number 1-2, Page 1
[3]
Farit G. Avkhadiev and Karl-Joachim Wirths
Computational Methods and Function Theory, 2008, Volume 8, Number 1, Page 107
[4]
F. G. Avkhadiev
Lobachevskii Journal of Mathematics, 2012, Volume 33, Number 3, Page 208
[5]
Bappaditya Bhowmik
Mathematische Nachrichten, 2012, Volume 285, Number 5-6, Page 606
[6]
Bappaditya Bhowmik
Archiv der Mathematik, 2010, Volume 95, Number 6, Page 575
[7]
Bappaditya Bhowmik and Saminathan Ponnusamy
Analysis, 2008, Volume 28, Number 3/2008
[8]
B. Bhowmik, S. Ponnusamy, and K. -J. Wirths
Siberian Mathematical Journal, 2009, Volume 50, Number 4, Page 609
[9]
M. Darus and R. W. Ibrahim
Lobachevskii Journal of Mathematics, 2008, Volume 29, Number 4, Page 221

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