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A variational method for the construction of convergent iterative sequences

Part of: Geometric function theory

Published online by Cambridge University Press:  09 April 2009

Zalman Rubinstein
Zalman Rubinstein
Affiliation:
Department of Mathematics, University of Colorado, Boulder, Colorado 80309, U.S.A. Department of Mathematics, University of Haifa, Mount Carmel, Haifa, Israel
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Abstract

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Convergent iterative sequences are constructed for the polynomials fm = z + zm, m ≧ 2, with initial point the lemniscate {z: |fm (z)| ≦1}. In the particular case m = 2 convergent iterative sequences are constructed also for f-1m, (z) with an arbitrary initial point. The method is based on a certain variational principle which allows reducing the problem to the well known situation of an analytic function mapping a simply connected domain into a proper subset of itself and possessing a fixed point in the domain.

MSC classification

Secondary: 30C10: Polynomials
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 1 , August 1986 , pp. 51 - 58
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Baker, I. N. and Rubinstein, Z., ‘Simultaneous iteration by entire or rational functions and their inverses’, J. Austral. Math. Soc. Ser. A 34 (1983), 364–367.CrossRefGoogle Scholar
[2]Marden, M., ‘Geometry of polynomials’, (Mathematical Surveys Number 3, Amer. Math. Soc., Providence, R.I., 1966).Google Scholar
[3]Walter, Peter L., ‘Iterated complex radicals’, The Mathematical Gazette 67 (1983), 269–273.Google Scholar