In a question asked by Bobby Ocean, the following theorem is cited:
Hermite-Kakeya Theorem(for polynomials) - 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. (see Rahman & Schmeisser, page 197-199).
Question Is there a similar theorem for entire functions as stated below:
Hermite-Kakeya (for entire function) - Given two entire functions, $f$ and $g$, and $f(z)$ and $g(z)$ are real when $z\in\mathbb{R}$, then $f(z)+g(z) r$ has only real zeros for every $r\in\mathbb{R}$, if and only if, $f$ and $g$ have real interlacing zeros.
For example $$f(x)=\cos(\sqrt{x})=\prod_{k=1}^{\infty}\left(1-\frac{x}{((k-1/2)\pi)^2}\right)\tag{1}$$
$$g(x)=\frac{\sin(\sqrt{x})}{\sqrt{x}}=\prod_{k=1}^{\infty}\left(1-\frac{x}{(k\pi)^2}\right)\tag{1}$$
Thanks- mike