Consider the rational function $R(z)=N(z)/D(z)$ where $N(z)$ and $D(z)$ are polynomials of $z$ with real coefficients. Furthermore, $N(0) \neq 0$, $D(0) \neq 0$, and $N(z)$ and $D(z)$ are relatively prime.
I am interested in the logarithmic derivatives of $N$ and $D$ evaluated at the origin, defined as $N_0 := N'(0)/N(0)$ and $D_0 := D'(0)/D(0)$. These can be written in terms of the zeros and poles of $R(z)$,
$$ N_0 = \sum_{r \in \mathbb{C}|N(r)=0}\frac{-1}{r}, $$ $$ D_0 = \sum_{p \in \mathbb{C}|D(p)=0}\frac{-1}{p}. $$
Note that because the polynomial coefficients are real, the roots and poles are either real or are in conjugate pairs, and so $N_0$ and $D_0$ must be real.
Now, suppose that I do not actually have a complete analytic functional representation for $R(z)$, but I do know some of its properties. What is the minimum that must I know about $R$ in order to evaluate (or bound) $N_0$ and $D_0$?
For example, I know that I can compute the difference between $N_0$ and $D_0$ directly from the local behavior of $R(z)$ near the origin: $$ N_0 - D_0 = R'(0)/R(0). $$
More generally, if I know the value $R$ takes along some curve (e.g. if I can evaluate $R$ on the imaginary axis), I could in theory apply analytic continuation to obtain the location of its zeros and poles. But it is not obvious to me how this might be done in closed form for arbitrary $R$. Could I construct a contour integral along the imaginary axis to compute $N_0$ and/or $D_0$ (or bound them)?