On the least measure of the nonnegativity set of an algebraic polynomial with zero weighted mean value on a segment

Let $\mathcal P_n(\varphi^{(\alpha)})$ be the set of algebraic polynomials $P_n$ of order $n$ with real coefficients and zero weighted mean value with respect to the ultraspherical weight $\varphi^{(\alpha)}(x)=(1-x^2)^\alpha$ on the interval $[-1,1]$: $\int_{-1}^1\varphi^{(\alpha)} P_n(x)\,dx=0$. We study the problem about the least possible value $\inf\{\mu(P_n)\colon P_n\in\mathcal P_n(\varphi^{(\alpha)})\}$ of the measure $\mu(P_n)=\int_{\mathcal X(P_n)}\varphi^{(\alpha)}(t)\,dt$ of the set $\mathcal X(P_n)=\{x\in[-1,1]\colon P_n(x)\ge0\}$ of points of the interval at which the polynomial $P_n\in\mathcal P_n(\varphi^{(\alpha)})$ is nonnegative. In this paper, the problem is solved for $n=2$ and $\alpha>0$. V. V. Arestov and V. Yu. Raevskaya solved the problem for $\alpha=0$ in 1997; in this case, an extremal polynomial has one interval of nonnegativity such that one of its endpoints coincides with one of the endpoints of the interval. In the case $\alpha>0$, we find that an extremal polynomial has two intervals of nonnegativity with endpoints $\pm1$.