The Stability Region for Schur Stable Trinomials with General Complex Coefficients

Abstract

In this paper, we characterize the stability region for trinomials of the form \(f(\zeta ):=a\zeta ^n + b\zeta ^m +c\), \(\zeta \in \mathbb {C}\), where a, b and c are non-zero complex numbers and \(n,m\in \mathbb {N}\) with \(n>m\). More precisely, we provide necessary and sufficient conditions on the coefficients a, b and c in order that all the roots of the trinomial f belongs to the open unit disc in the complex plane. The proof is based on Bohl’s Theorem (Bohl in Math Ann 65(4):556–566, 1908) introduced in 1908.