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Computational Methods and Function Theory 6 (2006), No. 1, 51--58
Copyright Heldermann Verlag 2006

Cases of Equality for Refinements of Bernstein's Inequality

Richard Fournier
fournier@crm.umontreal.ca , Département de Mathématiques et de Statistique, Université de Montréal, C.P. 6128, succ. Centre-ville, Montréal, Québec, H3C 3J7, Canada.

Frédéric Lesage
lesage@dms.umontreal.ca , Département de Mathématiques et de Statistique, Université de Montréal, C.P. 6128, succ. Centre-ville, Montréal, Québec, H3C 3J7, Canada.

[Abstract-pdf] [Abstract-ps]

We identify extremal polynomials for a class of known refinements of Bernstein's polynomial inequality. For example, the inequality
|zp'(z) - p(z) + p(0)| +|p(0)| ≤ (n-1) max|u|≤ 1 |p(u)|, |z| ≤ 1
holds for any polynomial p of degree n ≥ 2, and equality holds for some z, |z|=1, if and only if $p$ is a monomial of degree n or else p = 0.

Keywords: Polynomials, extremal polynomials, Bernstein's inequality.

MSC 2000: 41A17, 30A10.

[FullText-pdf (236 K)] [FullText-ps (399 K)]