Estimates of the growth of orthogonal polynomials whose weight is bounded away from zero

It is proved that for any $\varepsilon>0$ and any point $x_0$ in the interval $(-1,1)$ there exists a weight function $\rho(x)$ on $[-1,1]$ with $\rho(x)\geqslant1$, $x\in[-1,1]$, such that the following inequalities hold for the corresponding orthonormal polynomials $p_n(x)$:
$$ |p_n(x_0)|\geqslant n^{1/2-\varepsilon},\qquad n\in\Lambda, $$
where $\Lambda$ is some infinite sequence of positive integers.
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