Self-inversive polynomials of odd degree

Abstract

If the coefficients of a self-inversive polynomial P(z)=∑ k=0 m A k z k ∈ℂ[z] of odd degree m≥3 satisfy the inequality $$|A_{m}|\ge \cos \frac{\pi }{2(m+1)}\inf_{\scriptstyle{c,d\in\mathbb{C}}\atop\scriptstyle{|d|=1}}\sum _{k=0}^{m}|cA_{k}-d^{m-k}A_{m}|,$$ then all zeros of P are on the unit circle and they are simple.