A stable polynomial is one with zeros in the upper half plane or lower half plane. Interlacing polynomials are polynomials with only real zeros, where between every two zeros of one polynomial lies a zero of the other polynomial, in the sense that they can be ordered from least to greatest. Two interlacing polynomials automatically must be within a degree of each other. We now state two important theorems, (see Rahman & Schmeisser, page 197-199).
Hermite-Bielher - Given two real-valued polynomials, $f$ and $g$, then $f(x)+g(x) w$ is stable for every $w\in\mathbb{C}$, if and only if, $f$ and $g$ have real interlacing zeros.
Hermite-Kakeya - Given two real-valued polynomials, $f$ and $g$, then $f(x)+g(x) r$ has only real zeros for every $r\in\mathbb{R}$, if and only if, $f$ and $g$ have real interlacing zeros.
Hence, we put these theorems together,
Hermite-Bielher-Kakeya - Given two real-valued polynomials, $f$ and $g$, then $f(x)+g(x)w$ is stable for every $w\in\mathbb{C}$, if and only if, $f(x)+g(x) r$ has only real zeros for every $r\in\mathbb{R}$.
I think this generalizes, but I am struggling with finding a proof.
Hermite-Bielher-Kakeya (Generalization) - Given real-valued polynomials, $\{f_k\}_{k=0}^n$, then $\sum_{k=0}^n f_k(x)w^k$ is stable for every $w\in\mathbb{C}$, if and only if, $\sum_{k=0}^n f_k(x) r^k$ has only real zeros for every $r\in\mathbb{R}$.
One direction is obvious, since it is a weaker condition. I am interested in the other direction. Let's simplify the statement to the simplest case possible.
Hermite-Bielher-Kakeya (Generalization) (Small Case) - Given three real-valued polynomials, a quadratic, $Q_2(x)$, a linear, $Q_1(x)$, and a constant, $Q_0(x)$, then $Q_2(x)w^2+Q_1(x)w+Q_0(x)$ is stable for every $w\in\mathbb{C}$, if and only if, $Q_2(x)r^2+Q_1(x)r+Q_0(x)$ has only real zeros for every $r\in\mathbb{R}$.
I currently already have a proof of this particular case, however my proof is ridiculous complicated, spanning about 14 pages of geometric arguments and using various relations to multi-variate polynomials and some operator theory. In comparison the Hermite-Bielher and Hermite-Kakeya theorems have very tiny simple proofs.
Am I missing something? Is there a really simple proof. It honestly seems like there should be, using only the Hermite-Bielher and Kermite-Kakeya theorem, but I can't seem to find it. Maybe I only proved a special case and the other cases have counter-examples.
Thanks for reading. Simply having people to argue with might be beneficial.