Nonnegativity set of smallest measure for polynomials with zero weighted mean value on a closed interval

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 $\varphi^{(\alpha)}(t)=(1-t^2)^\alpha$ on the interval $[-1,1]$: $\int_{-1}^1\varphi^{(\alpha)}(t)p_n(t)\,dx=0$. We study the problem on the smallest 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 $\mathcal X(p_n)=\{t\in[-1,1]\colon p_n(t)\ge0\}$ of points of the interval at which the polynomial $p_n\in\mathcal P_n(\varphi^{(\alpha)})$ is nonnegative. In this paper, the properties of an extremal polynomial of this problem are studied and an exact solution is presented for the case of cubic polynomials.