A study of the validity of Oppenheim's inequality for Hurwitz matrices associated with Hurwitz polynomials

In this paper, Hurwitz polynomials, i.e., real polynomials whose roots are located in the open left half of the complex plane, and their associated Hurwitz matrices are considered. New formulae for the principal minors of Hurwitz matrices are presented which lead to ($ i $) a new criteria for deciding whether a polynomial is Hurwitz, ($ ii $) an inequality of a type of Oppenheim's inequality for the Hurwitz matrices up to order $6 $, and ($ iii $) a necessary and sufficient condition for the Hadamard square root of Hurwitz polynomials of degree five to be Hurwitz.