Polynomials least deviating from zero in $L^p(-1;1) $, $ 0 \le p \le \infty $, with a constraint on the location of their roots

We study Chebyshev's problem on polynomials that deviate least from zero with respect to $L^p$-means on the interval $[-1;1]$ with a constraint on the location of roots of polynomials. More precisely, we consider the problem on the set $\mathcal{P}_n(D_R)$ of polynomials of degree $n$ that have unit leading coefficient and do not vanish in an open disk of radius $R \ge 1$. An exact solution is obtained for the geometric mean (for $p=0$) for all $R \ge 1$; and for $0<p<\infty$ for all $R \ge 1$ in the case of polynomials of even degree. For $0<p<\infty$ and $R\ge 1$, we obtain two-sided estimates of the value of the least deviation.