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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An extremal problem for polynomials with a prescribed zero
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by Q. I. Rahman and Frank Stenger PDF
Proc. Amer. Math. Soc. 43 (1974), 84-90 Request permission

Abstract:

Let ${\mathcal {P}_{n,b}}$ denote the class of all polynomials ${p_n}(z)$ of degree at most $n$ in $z$ which satisfy ${\max _{|z| = 1}}|{p_n}(z)| = 1$, and $|{p_n}(1)| = b,0 \leqq b < 1$. Let $c \in (0,n]$, and set \[ {\mu _b}(c,n) = \sup \limits _{{p_n} \in {\mathcal {P}_{n,b}}} \{ \min \limits _{|z| = 1 - c/n} |{p_n}(z)|\} .\] Upper estimates for ${\mu _b}(c,n)$ are obtained.
References
  • 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 23380
  • Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
  • 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
  • © Copyright 1974 American Mathematical Society
  • 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