POLYNOMIALS LEAST DEVIATING FROM ZERO IN \(L^p(-1;1)\), \(0 \le p \le \infty \), WITH A CONSTRAINT ON THE LOCATION OF THEIR ROOTS

Abstract

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.