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The extreme points of some classes of polynomials

Published online by Cambridge University Press:  14 November 2011

D. A. Brannan  and
J. G. Clunie
D. A. Brannan
Affiliation:
Mathematics Faculty, Open University, Milton Keynes MK7 6AA, England
J. G. Clunie
Affiliation:
Mathematics Faculty, Open University, Milton Keynes MK7 6AA, England
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Synopsis

We study the extreme points of two classes of polynomials of degree at most n:

It turns out that f ∈ Ext if and only if Re f(eiθ) has exactly 2n zeros in [0, 2π). On the other hand, if f∈Hn and 1−|f(eiθ)|2 has 2n zeros in [0, 2π), then either f ∈ Ext Hn or else f(z) = α + βzn where |α|+|β| = l and αβ≠0; if 1−|f(eiθ)|2 has 2m zeros, 2n, then f may or may not belong to Ext Hn.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 101 , Issue 1-2 , 1985 , pp. 99 - 110
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

1Carathéodory, C.. Über den Variabilitätsbereich der Fourier'schen Konstanten von positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo 32 (1911), 193–217.CrossRefGoogle Scholar
2Leeuw, K. de and Rudin, W.. Extreme points and extreme problems in H1. Pacific J. Math. 8 (1958), 467–485.Google Scholar