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

Let P-n (phi((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 phi((alpha))(t) = (1-t(2))(alpha) on the interval [-1,1]: integral(1)(-1) phi((alpha))(t)p(n)(t)dx = 0. We study the problem on the smallest possible value inf{mu(p(n)) : p(n) is an element of P-n (phi((alpha)))} of the measure mu(p(n)) = integral(chi)(p(n)) phi((alpha))(t) dt of the set chi(p(n)) = {t is an element of[-1, 1] : p(n)(t) >= 0} of points of the interval at which the polynomial p(n) is an element of P-n (phi((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.