ON STABILITY OF A POLYNOMIAL

A polynomial, p(z) = a(0)z(n) + a1z(n-1) + ... + a(n-1)z + a(n); with real coefficients is called a stable or a Hurwitz polynomial if all its zeros have negative real parts. We show that if a polynomial is a Hurwitz polynomial then so is the polynomial q(z) = a(n)z(n)+a(n-1) z(n-1)+ ... +a1z+ a(0) (with coefficients in reversed order). As consequences, we give simple ratio checking inequalities that would determine unstability of a polynomial of degree 5 or more and extend conditions to get some previously known results.