On the \(\Gamma\)-stability of systems of differential equations in the Routh-Hurwitz and the Schur-Cohn cases (Q1815277)

Let \(\Gamma\) denote a pathwise-connected region in the complex plane and \(S\) a set of polynomials of degree \(n\). The author says that the set \(S\) is \(\Gamma\)-stable if all polynomials from \(S\) have their zeros inside \(\Gamma\). He obtains necessary and sufficient conditions for \(\Gamma\)-stability in the following two particular cases: (i) \(\Gamma\) is the imaginary axis, and \(S\) is the set of all convex combinations of (real or complex) polynomials which satisfy the Routh-Hurwitz conditions; (ii) \(\Gamma\) is the unit circle, and \(S\) is the set of all convex combinations of polynomials which satisfy the Schur-Cohn conditions.