The rate of the smallest value of the weighted measure of the nonnegativity set for polynomials with zero mean value on a closed interval

Let $\mathcal P_n(\alpha)$ be the set of algebraic polynomials $p_n$ of order $n$ with real coefficients and zero weighted mean value with ultraspherical weight $\varphi^{(\alpha)}(t)=(1-t^2)^\alpha$ on the interval $[-1,1]$: $\int_{-1}^1\varphi^{(\alpha)}(t)p_n(t)\,dx=0$. We study the problem on the smallest value $\mu_n=\inf\{m(p_n)\colon p_n\in\mathcal P_n(\alpha)\}$ of the weighted measure $m(p_n)=\int_{\mathcal X(p_n)}\varphi^{(\alpha)}(t)\,dt$ of the set where $p_n$ is nonnegative. The order of $\mu_n$ with respect to $n$ is found: it is proved that $\mu_n(\alpha)\asymp n^{-2(\alpha+1)}$ as $n\to\infty$.