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?