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Polynomials with a Prescribed Zero and the Bernstein's Inequality

Published online by Cambridge University Press:  20 November 2018

P. F. Olivier  and
A. O. Watt
P. F. Olivier
Affiliation:
Dalhousie University, Department of Mathematics, Halifax, Nova Scotia, B3H3J5, email:, patrick@cs.dal.ca
A. O. Watt
Affiliation:
Ecole Polytechnique de Thies, B.P. 10 Thies Senegal
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Abstract

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Let be the class of all polynomials/? of degree at most n such that In view of the example zn it follows from Bernstein's inequality for polynomials that at each point z0 of the unitcirle. It was shown by A. Giroux and Q. I. Rahman [2] that if denotes the subclass of polynomials in which vanish at 1, then

where c1 and c2 are constants not depending on n. Here we find the exact value of which has some special significance and also at certain other points of the unit circle.

Keywords

30C1026D0541A17polynomialsBernstein's inequalityChebyshev polynomials
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 3 , 01 June 1993 , pp. 627 - 637
Copyright
Copyright © Canadian Mathematical Society 1993

References

1. Frappier, C., Rahman, Q. I. and St. Ruscheweyh, On polynomials with a prescribed zero, Constr. Approx. 2(1986), 171–177.Google Scholar
2. Giroux, A. and Rahman, Q. I., Inequalities for polynomials with a pre scribed zero, Trans. Amer. Math. Soc. 193(1974), 67–98.Google Scholar