Suppose one has an entire complex analytic function $f(z)$ having no zeros on the real axis. Is it possible to find an analytic function $g(z)$ such that the coefficients of the power series expansion of $f(z)g(z)$ in $z$ are always real and positive (so that, e.g. "Descartes' rule of signs" can be used to "explain" the absence of zeros)? How can one find $g(z)$?
This is attempting be a kind-of generalization of the question Property of a polynomial with no positive real roots to complex analytic functions. The goal is to "explain" why $f(z)$ has no real roots, by studying the properties of $f(z)g(z)$ (or obtaining insights into $f(z)$ by studying the properties of $g(z)$). Are there (non-trivial) theorems that apply to complex analytic functions with no real zeros? Some Fredholm-alternative-like thingy?
Let $g(z)=\overline{f(\overline z)}$. Then, regardless of whether $f$ has any zeroes on the real axis, the function $fg$ is real on the real axis, so all its derivatives at the origin are real.
I'm mystified what it might mean to explain why a function has no zeroes on the real axis, or what sort of "explanation" you're hoping for. But the existence of $g$ so that all the coefficients of $fg$ are real has absolutely nothing to do with whether $f$ has any zeroes on the real axis.