On the minimum moduli of normalized polynomials

Abstract

Consider any complex polynomial p_n(z)=1+$\mathop \Sigma \limits_{j = l}^n$ a_jz^j which satisfies $\mathop \Sigma \limits_{j = l}^n$|a_j|=1, and let Γ_n denote the supremum of the minimum moduli on |z|=1 of all such polynomials p_n(z). We show that

$$1 - \frac{1}{n} \leqslant \Gamma _n \leqslant \sqrt {1 - \frac{1}{n}} , for all n \geqslant 1.$$

If the coefficients of p_n(z) are further restricted to be positive numbers and if $\tilde \Gamma _n$ denotes the analogous supremum of the minimum modulion |z|=1 of such polynomials, we similarly show that

$$1 - \frac{1}{n} \leqslant \tilde \Gamma _n \leqslant \sqrt {1 - \frac{3}{{\left( {2n + 1} \right)}}} , for all n \geqslant 1.$$

We also include some recent numerical experiments on the behavior of Γ_n, as well as some related conjectures.

Research supported by the Air Force Office of Scientific Research, the Department of Energy, and the Alexander von Humboldt Stiftung.