Conditions for the Schur stability of segments of polynomials of the same degree

One of the most interesting problems in the robustness analysis of systems is the investigation of conditions that guarantee the stability of segments of polynomials. In this paper, given two polynomials \(P_1(z)\) and \(P_2(z)\) of the same degree, we give conditions on the polynomial \(H(z)=z^{n}[P_1(z)P_2(\frac{1}{z})-P_1(\frac{1}{z})P_2(z)]\) such that the Schur stability of the segment \([P_1, P_2]\) is implied by the Schur stability of both \(P_1(z)\) and \(P_2(z)\).