On properties of Hurwitz polynomials (Q1922461)

A polynomial with real coefficients is called Hurwitz if all its roots have negative real parts. Fix a monic Hurwitz polynomial \(F(\lambda)= \lambda^n+ \sum_{j=1}^{n-1} a_j\lambda^j\) of degree \(n\). Using the well-known determinantal criterion for the property of being a Hurwitz polynomial, it is not difficult to see that there is \(t_1>0\) such that for every \(t>t_1\) the polynomial \(\lambda^{n+1}+ tF(\lambda)\) is Hurwitz; moreover, for any polynomial \(L(\lambda)\) of degree at most \(n\) with real coefficients, there exists \(t_2>0\) such that the polynomial \(\lambda^{n+1}+ tF(\lambda)+ L(\lambda)\) is Hurwitz for every \(t>t_2\). In the reviewed paper upper bounds for \(t_1\) and \(t_2\) are derived.