N.S. Payuchenko. Markov’s weak inequality for algebraic polynomials on a closed interval ... P. 160-166

For a real algebraic polynomial $P_n$ of degree $n$, we consider the ratio $M_n(P_n)$ of the measure of the set of points from $[-1,1]$ where the absolute value of the derivative exceeds $n^2$ to the measure of the set of points where the absolute value of the polynomial exceeds 1. We study the supremum $M_n=\sup M_n(P_n)$ over the set of polynomials $P_n$ whose uniform norm on $[- 1,1]$ is greater than 1. It is known that $M_n$ is the supremum of the exact constants in Markov's inequality in the class of integral functionals generated by a nondecreasing nonnegative function. In this paper we prove the estimates $1+3/(n^{2}-1)\le M_n \le 6n+1$ for $n\ge2$.

Keywords: Markov's inequality, algebraic polynomials, Lebesgue measure, weak-type inequalities

Received April 2, 2019

Revised May 13, 2019

Accepted May 20, 2019

Funding Agency: This work was supported by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Nikita Slavich Pauchenko, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: aueiyo@gmail.com

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Cite this article as: N.S.Pauchenko. Markov’s weak inequality for algebraic polynomials on a closed interval, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 2, pp. 160–166.