J Austral Math Soc Ser A 58 pp1--14, 1995.

Markoff Type Inequalities for Curved Majorants

A. K. Varma, T. M. Mills and Simon J. Smith

Abstract

Let p_n(x) be a real algebraic polynomial of degree n, and consider the L_p norms on I = [-1, 1]. A classical result of A. A. Markoff states that if ||p_n||_¥ £ 1, then ||p¢_n||_¥ £ n². A generalization of Markoff's problem, first suggested by P. Turan, is to find upper bounds for ||p_n^(j)||_p if |p_n(x)| £ y(x), x Î I. Here y(x) is a given function, a curved majorant. In this paper we study extremal properties of ||p¢_n||₂ and ||p¢¢_n||₂ if p_n(x) has the parabolic majorant |p_n(x)| £ 1 - x², x Î I. We also consider the problem, motivated by a well-known result of S. Bernstein, of maximising ||(1 - x²)p¢¢_n||₂ if ||p_n||_¥ £ 1.

1991 AMS Subject Classification: 26D10, 26D15

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Authors

A. K. Varma

Department of Mathematics, University of Florida, Gainesville, Florida 32611, USA.

T. M. Mills

Department of Mathematics, La Trobe University, Bendigo, P. O. Box 199, Bendigo, Victoria 3550, Australia.

Simon J. Smith

Department of Mathematics, La Trobe University, Bendigo, P. O. Box 199, Bendigo, Victoria 3550, Australia.