Markov’s weak inequality for algebraic polynomials on a closed interval
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 $n\ge2$.