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

  • 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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Keywords: Extremal problem, polynomials with a prescribed zero, Bernstein theorem
Article copyright: © Copyright 1974 American Mathematical Society