New developments on Turán’s extremal problems for polynomials. (English) Zbl 0911.26008
Govil, N. K. (ed.) et al., Approximation theory. In memory of A. K. Varma. New York, NY: Marcel Dekker. Pure Appl. Math., Marcel Dekker. 212, 433-447 (1998).
The well-known inequality of P. Turán [Compos. Math. 7, 89-95 (1939; Zbl 0021.39504)] asserts that if \(P\) is an algebraic polynomial of degree at most \(n\) having all its zeros in \([-1,1]\), then
\[
\| P'\| _\infty>\frac{\sqrt n}6\| P\| _\infty,
\]
where \(\| f\| _\infty:=\max\limits_{-1\leq t\leq 1}| f(t)| \). Here the constant \(\frac{\sqrt n}6\) is not best possible and Turán’s result was sharpened by J. Eröd [Mat. Fiz. Lapok 46, 58-83 (1939; Zbl 0021.39505)]. The paper under review is a survey on Turán type inequalities in the \(L^r\) norms. Most of them (in the \(L^2\) norm) were obtained by Professor A.K. Varma to memory of whom the above paper is dedicated.
For the entire collection see [Zbl 0890.00042].
For the entire collection see [Zbl 0890.00042].
Reviewer: W.Pleśniak (Kraków)
MSC:
26C05 | Real polynomials: analytic properties, etc. |
41A17 | Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) |