Comments on the paper
T. Koornwinder, A. Kostenko and G. Teschl, Jacobi polynomials,
Bernstein-type inequalities and dispersion estimates for the discrete Laguerre operator,
Adv. Math. 333 (2018), 796-821;
slightly extended version in arXiv:1602.08626v3 [math.CA].

It was shown in [1, Theorem 2] that the Bernstein-type inequality (5.1) holds for $\alpha,\beta\ge0$.

It was shown in [2, Theorem 1] that formulas (4.8) and (4.9) hold for
$\alpha,\beta\ge0$..

[1] J. M. Carnicer, E. Mainar and J. M. Pe\~na,
    Stability properties of disk polynomials,
    Numer. Algorithms 87 (2021), 119-135.

[2] Yu-Tian Li,
    The Koornwinder-Kostenko-Teschl conjecture for Jacobi polynomials and the
    discrete Laguerre phase transition,
    arXiv:2608.30486v1 [math.CA], 2026.