Distribution of the zeros of a polynomial with prescribed lower and upper bounds for its modulus on a compact set

A few years ago, the second named author was asked if he knew the largest open simply connected region G(n) superset of [-1, 1] containing no zero of any polynomial p of degree n is an element of N such that 0 < m <= vertical bar p(x)vertical bar <= M for all x is an element of [-1, 1]. This question is answered in Theorem 3. This required us to first consider a related problem for polynomials on the unit circle, whose solution is given in Theorem 1. The paper contains several other results which are all sharp.