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 $$f(\zeta ):=a\zeta ^n + b\zeta ^m +c$$ f ( ζ ) : = a ζ n + b ζ m + c , $$\zeta \in \mathbb {C}$$ ζ ∈ C , where a, b and c are non-zero complex numbers and $$n,m\in \mathbb {N}$$ n , m ∈ N with $$n>m$$ 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.