Asymptotics of sharp constants of Markov-Bernstein inequalities in integral norm with Jacobi weight

TL;DR

This paper investigates the asymptotic behavior of the sharp constants in Markov-Bernstein inequalities within the $L^2$ space weighted by Jacobi weights, revealing their limit in terms of Bessel function zeros under certain conditions.

Contribution

It provides the first detailed analysis of the asymptotics of sharp constants in these inequalities for Jacobi weights, linking them to zeros of Bessel functions.

Findings

Limit of sharp constants is 1/(2 j_ν) as n approaches infinity.

The limit holds when |α - β| < 4.

The result connects polynomial inequalities to special functions.

Abstract

The classical A. Markov inequality establishes a relation between the maximum modulus or the $L^{\infty}\left([-1,1]\right)$ norm of a polynomial $Q_{n}$ and of its derivative: $\|Q'_{n}\|\leqslant M_{n} n^{2}\|Q_{n}\|$, where the constant $M_{n}=1$ is sharp. The limiting behavior of the sharp constants $M_{n}$ for this inequality, considered in the space $L^{2}\left([-1,1], w^{(\alpha,\beta)}\right)$ with respect to the classical Jacobi weight $w^{(\alpha,\beta)}(x):=(1-x)^{\alpha}(x+1)^{\beta}$, is studied. We prove that, under the condition $|\alpha - \beta| < 4 $, the limit is $\lim_{n \to \infty} M_{n} = 1/(2 j_{\nu})$ where $j_{\nu}$ is the smallest zero of the Bessel function $J_{\nu}(x)$ and $2 \nu= \mbox{min}(\alpha, \beta) - 1$.