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

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 $2 \nu= \mbox{min}(\alpha, \beta) - 1$.