Abstract

A mapping ϕ ⁣:[−1,1]→[0,∞)\phi\colon [-1,1]\rightarrow [0,\infty) is a curved majorant for a polynomial pp in one real variable if ∣p(x)∣≤ϕ(x)|p(x)|\leq \phi(x) for all x∈[−1,1]x\in[-1,1]. If Pnϕ(R){\mathcal P}_n^\phi({\mathbb R}) is the set of all one real variable polynomials of degree at most nn having the curved majorant ϕ\phi, then we study the problem of determining, explicitly, the best possible constant Mnϕ(x)\mathcal{M}^\phi_{n}(x) in the inequality ∣p′(x)∣≤Mnϕ(x)∥p∥,|p'(x)| \le \mathcal{M}^\phi_n(x)\|p\|, for each fixed x∈[−1,1]x\in[-1,1], where p∈Pnϕ(R)p\in {\mathcal P}_n^\phi ({\mathbb R}) and ∥p∥\|p\| is the sup norm of pp over the interval [−1,1][-1,1]. These types of estimates are known as Bernstein type inequalities for polynomials with a curved majorant. The cases treated in this manuscript, namely ϕ(x)=1−x2\phi(x) = \sqrt{1-x^2} or ϕ(x)=∣x∣\phi(x) = |x| for all x∈[−1,1]x\in[-1,1] (circular and linear majorant respectively), were first studied by Q. I. Rahman [``On a problem of Tur{á}n about polynomials with curved majorants'', Trans. Amer. Math. Soc. 163 (1972) 447–455]. In that reference the author provided, for each n∈Nn\in{\mathbb N}, the maximum of Mnϕ(x)\mathcal{M}^\phi_n(x) over [−1,1][-1,1] as well as an upper bound for Mnϕ(x)\mathcal{M}^\phi_n(x) for each x∈[−1,1]x\in[-1,1], where ϕ\phi is either a circular or a linear majorant. Here we provide sharp Bernstein inequalities for some specific families of polynomials having a linear or circular majorant by means of classical convex analysis techniques (in particular we use the Krein-Milman approach)

Author information

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Gustavo A. Muñoz-Fernández

Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain

gustavo\_fernandez@mat.ucm.es

Viktor M. Sánchez

Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain

victorms@mat.ucm.es

Juan B. Seoane-Sepúlveda

Dep. de Análisis Matemático, Universidad Complutense de Madrid, Plaza Ciencias 3, 28040 Madrid, Spain

jseoane@mat.ucm.es

Keywords

  • Bernstein and Markov inequalities
  • trinomials
  • extreme points

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Suggested citation

G. A. Muñoz-Fernández, V. M. Sánchez, J. B. Seoane-Sepúlveda. “Estimates on the Derivative of a Polynomial with a Curved Majorant Using Convex Techniques.” Journal of Convex Analysis 17 (2010), No. 1, 241–252. https://doi.org/10.68381/jca17018

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Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.