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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

An extremal problem for polynomials with a prescribed zero


Authors: Q. I. Rahman and Frank Stenger
Journal: Proc. Amer. Math. Soc. 43 (1974), 84-90
MSC: Primary 30A06
DOI: https://doi.org/10.1090/S0002-9939-1974-0333123-0
MathSciNet review: 0333123
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\mathcal{P}_{n,b}}$ denote the class of all polynomials $ {p_n}(z)$ of degree at most $ n$ in $ z$ which satisfy $ {\max _{\vert z\vert = 1}}\vert{p_n}(z)\vert = 1$, and $ \vert{p_n}(1)\vert = b,0 \leqq b < 1$. Let $ c \in (0,n]$, and set

$\displaystyle {\mu _b}(c,n) = \mathop {\sup }\limits_{{p_n} \in {\mathcal{P}_{n,b}}} \{ \mathop {\min }\limits_{\vert z\vert = 1 - c/n} \vert{p_n}(z)\vert\} .$

Upper estimates for $ {\mu _b}(c,n)$ are obtained.

References [Enhancements On Off] (What's this?)

  • [1] N. G. de Bruijn, Inequalities concerning polynomials in the complex domain, Nederl. Akad. Wetensch., Proc. 50 (1947), 1265–1272 = Indagationes Math. 9, 591–598 (1947). MR 0023380
  • [2] Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
  • [3] S. Bernstein, Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d'une variable réelle, Gauthier-Villars, Paris, 1926.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1974-0333123-0
Keywords: Extremal problem, polynomials with a prescribed zero, Bernstein theorem
Article copyright: © Copyright 1974 American Mathematical Society