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Annales Polonici Mathematici
Article . 2004 . Peer-reviewed
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A proof of the Livingston conjecture for the fourth and the fifth coefficient of concave univalent functions

Authors: Wirths, Karl-Joachim;

A proof of the Livingston conjecture for the fourth and the fifth coefficient of concave univalent functions

Abstract

The special class of ``concave'' functions denoted by Co(\(p\)) consists of functions \(f(z)\) which are analytic in the unit disc \(D\) except for a simple pole at \(p\), \(0 0\) for all \(z \in D\), \(P(0) = 1\), and \(P(p) = (1 + p^2)/(1 - p^2)\). The author introduces a representation of \(P(z)\) in terms of a function \(\omega(z)\) analytic in \(D\) with \(\omega(D) \subset \overline D\). A very clever method of chasing coefficients then completes the proof. As the author observes, this simple representation seems to offer an opening into proving the conjecture in general.

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Keywords

Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.), Coefficient problems for univalent and multivalent functions of one complex variable, concave univalent functions, Livingston conjecture

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
6
Average
Top 10%
Average
bronze