Estimates for polynomials in the unit disk with varying constant terms

Stephan Ruscheweyh; Magdalena Wołoszkiewicz

Stephan Ruscheweyh ; Magdalena Wołoszkiewicz

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 65 (2011), / Harvested from The Polish Digital Mathematics Library

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Résumé

Let $∥ \cdot ∥$ be the uniform norm in the unit disk. We study the quantities $M_{n} (α) : = inf (∥ z P (z) + α ∥ - α)$ where the infimum is taken over all polynomials $P$ of degree $n - 1$ with $∥ P (z) ∥ = 1$ and $α > 0$ . In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that ${inf}_{α > 0} M_{n} (α) = 1 / n$ . We find the exact values of $M_{n} (α)$ and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.

Publié le : 2011-01-01
EUDML-ID : urn:eudml:doc:289725

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     author = {Stephan Ruscheweyh and Magdalena Wo\l oszkiewicz},
     title = {Estimates for polynomials in the unit disk with varying constant terms},
     journal = {Annales Universitatis Mariae Curie-Sk\l odowska, sectio A -- Mathematica},
     volume = {65},
     year = {2011},
     language = {en},
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Stephan Ruscheweyh; Magdalena Wołoszkiewicz. Estimates for polynomials in the unit disk with varying constant terms. Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica, Tome 65 (2011) . http://gdmltest.u-ga.fr/item/bwmeta1.element.ojs-doi-10_17951_a_2011_65_2_169-178/

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