Generalized theorems of Liénard and Shepherd

The paper considers a real polynomial $p(x)=a_0+a_1x+\dots+a_nx^n$ ($a_0>0$) for which there hold inequalities $\Delta_1>0, \Delta_3>0,\dots$ or $\Delta_2>0, \Delta_4>0$, where $\Delta_1,\Delta_2,\dots,\Delta_n$ are the Hurwitz determinants for polynomial $p(x)$. It is proven that polynomial $p(x)$ can have, in the right half-plane, only real roots, where the quantity of positive roots of polynomial $p(x)$ equals the quantity of changes of sign in the system of coefficients $a_0,a_2,\dots,a_n$, when $n$ is even, and $a_0,a_2,\dots,a_{n-1},a_n$, when $n$ is odd. From the proven theorem, in particular, there follows the Liénard and Shepherd criterion of stability.