Suppose we have two monic polynomials with real coefficients \begin{align*} & p_1(t) = t^n + a_{n-1} t^{n-1} + \dots + a_0 \\ & p_2(t) = t^n + b_{n-1} t^{n-1} + \dots + b_0. \end{align*} Further assume $p_1(t)$ and $p_2(t)$ are both Hurwitz stable, i.e., the roots are all lying on the open left half plane of $\mathbb C$; equivalently, real parts of all roots are $<0$.
I am wondering whether or not there is some $M < 0$ (existence of such $M$ is enough) such that if the real parts of all roots smaller than $M$, then the convex combination of the two polynomials is Hurwitz stable.
Intuitively, I am thinking by making real parts of roots "negative" enough, the convex combination would stay in the left half plane.
p.s. Routh-Hurwitz Theorem gives sufficient condition to determine whether a polynomial is Hurwitz stable. But the rule seems complicated.