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

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: $\Vert Q_n^{\prime }\Vert ⩽M_n{n}^2\Vert Q_n\Vert$, 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 ${\lim }_{n\to \infty }{M}_n=1/(2j_{\nu })$ where $j_{\nu }$ is the smallest zero of the Bessel function $J_{\nu }(x)$ and $2 ν= \mbox{min}(α, β) - 1$.

12 pages, biblio 15 sources