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Markoff type inequalities for curved majorants

Part of: Inequalities

Published online by Cambridge University Press:  09 April 2009

A. K. Varma ,
T. M. Mills  and
Simon J. Smith
A. K. Varma
Affiliation:
Department of Mathematics, University of Florida, Gainesville, Florida 32611, USA
T. M. Mills
Affiliation:
Department of Mathematics, La Trobe University, Bendigo P.O. Box 199, Bendigo, Victoria 3550, Australia
Simon J. Smith
Affiliation:
Department of Mathematics, La Trobe University, Bendigo, P.O. Box 199, Bendigo Victoria 3550, Australia
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Abstract

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Let pn(x) be a real algebraic polynomial of degree n, and consider the Lp norms on I = [−1, 1]. A classical result of A. A. Markoff states that if ‖pn‖. ∞ ≤ 1, then ‖P′n‖∞ ≤ n2. A generalization of Markoff's problem, first suggested by P. Turán, is to find upper bounds for ‖pn(J)‖p if ∣pn(x)∣≤ ψ(x)x ∈ I. Here ψ(x) is a given function, a curved majorant. In this paper we study extremal properties of ‖p′n‖2 and ‖p″n‖2 if pn(x) has the parabolic majorant ∣p(x)∣≤ 1 − x2, x ∈ I. We also consider the problem, motivated by a well-known result of S. Bernstein, of maximising ‖(1 − x2)

MSC classification

Secondary: 26D05: Inequalities for trigonometric functions and polynomials 26D10: Inequalities involving derivatives and differential and integral operators 26D15: Inequalities for sums, series and integrals
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 58 , Issue 1 , February 1995 , pp. 1 - 14
Copyright
Copyright © Australian Mathematical Society 1995

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