I posted this question on MSE a few time ago, but it did not receive much attention. I thought there might be an elementary answer so didn't want to post it directly on MO. My apologies if this question is not a good fit here.

Let $f:\bar{\mathbb C}\to \bar{\mathbb C}$ be a rational function, and take a circle $C$ not crossing the zero- and polar-locus of $f$. The argument principle tells us the total variation $\Delta$ of $\arg f$ along $C$, obtained by performing the analytic continuation of $g : t\in[0,1]\mapsto \arg f(z(t))$ for some parameterization $z(\bullet)$ of $C$:$$ \Delta = g(1)-g(0)=2\pi(Z-P)$$ where $Z$ is the count of roots of $f$ encircled by $C$ and $P$ that of its poles (of course this value can also be obtained by integrating $\frac{f'}{if}$ along $C$).

What it does not tell is the size of $g([0,1])$. **Question:** how big can this interval get? Is there a known bound, *e.g.* in terms of the degree of $f$ and/or the zeroes and poles encircled by $C$?

Also, although the answer might not depend on the special shape of $C$, I don't want to assume $C$ to be «small enough» or to match too restrictive hypothesis of the like.