Given two $N^{th}$ order polynomials $P_0(z)$ and $P_1(z)$, let their roots be $w_k$ and $z_k$ respectively. All the roots of both polynomials lie on the unit circle $\mathcal{U}$. Also $w_i \neq z_j$ for $i,j = 1 ~\text{to}~ N-1$.
Define $P(z) = \alpha P_0(z) + (1-\alpha)P_1(z); \quad 0 < \alpha \leq 1$ -----(1)
I can use Fells's Lemma 1 to prove that all the roots of $P(z)$ also lie on $\mathcal{U}$. However Fell's result is limited for case of convex combination of two polynomials.
I'd like to extend this result to convex combination of multiple polynomials. To do so one approach I'm considering is to look at the companion matrices of polynomial.
Eq. (1) above can be written in an equivalent from in-terms of companion matrices as
$\mathbf{C} = \alpha \mathbf{C}_0 + (1-\alpha)\mathbf{C}_1$
where $\mathbf{C}$ is the companion matrix and its eigenvalues correspond to the roots of the polynomial. I'm trying to show the convex combination of two companion matrices with eigenvalues on $\mathcal{U}$ results in another companion matrix with eigenvalues also n $\mathcal{U}$.
Are there any results towards this problem that could be helpful in proceeding with the proof? I've looked at the Bauer-Fike