An extremal problem for algebraic polynomials with zero mean value on an interval

Let $\mathscr P_n^0(h)$ be the set of algebraic polynomials of degree $n$ with real coefficients and with zero mean value (with weight $h$) on the interval $[-1,1]$:
$$ \int_{-1}^1h(x)p_n(x)dx=0; $$
here $h$ is a function which is summable, nonnegative, and nonzero on a set of positive measure on $[-1,1]$. We study the problem of the least possible value
$$ i_n(h)=\inf\{\mu(p_n):p_n\in\mathscr P_n^0\} $$
of the measure $\mu(p_n)=\operatorname{mes}\{x\in[-1,1]:p_n(x)\ge0\}$ of the set of points of the interval at which the polynomial $p_n\in\mathscr P_n^0$ is nonnegative. We find the exact value of $i_n(h)$ under certain restrictions on the weight $h$. In particular, the Jacobi weight
$$ h^{(\alpha,\beta)}(x)=(1-x)^\alpha(1+x)^\beta $$
satisfies these restrictions provided that $-1<\alpha,\beta\le0$.