On Certain Mean Values of Polynomials on the Unit Interval

Authors: Dryanov D.; Lubinsky Q.D.I.S.

Source: Journal of Approximation Theory, Volume 101, Number 1, November 1999 , pp. 92-120(29)

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Abstract:

For any continuous function f: [-1, 1] map Copf and any p isin (0, infin ), let Verbar f Verbar _p colone (2^-1 int ¹_-1 |f(x)|^p dx)^1/p; in addition, let Verbar f Verbar colone max_-1x1 |f(x)|. It is known that if f is a polynomial of degree n, then for all p>0, Verbar f Verbar les C_pn^2/p Verbar f Verbar _p,where C_p is a constant depending on p but not on n. In this result of Nikolski breve inodot (1951), which was independently obtained by Szegö and Zygmund (1954), the order of magnitude of the bound is the best possible. We obtain a sharp version of this inequality for polynomials not vanishing in the open unit disk. As an application we prove the following result. If f is a real polynomial of degree n such that f(-1)=f(1)=0 and f(z)0 in the open unit disk, then for p>0 the quantity Verbar f prime Verbar / Verbar f Verbar _p is maximized by polynomials of the form c(1+x)^n-1 (1-x), c(1+x)(1-x)^n-1, where c isin Ropf \{0}. This extends an inequality of Erd odblac s (1940). Copyright 1999 Academic Press.