Counting complex solutions on a disk.

Question

I am looking for an example of an analytic map $f$ on the complex domain for which there exist $r>0$ such that for $\gamma=\{z:|z|=r\}$, a.s. for $\theta \in [0,2\pi)$ we have that $$\int_{f(\gamma)} \frac{1}{2 \pi i} \frac{dw}{w} < \left( f(\gamma)e^{-2\pi i \theta}\cap \mathbb{R}_{+} \right).$$

Answer 1

For $r=1$, the function $f(z) = (z+\sqrt2)^4$ works nicely. The path integral equals $0$, while the count on the right-hand side is always at least 1. (In fact, if you reduce $\sqrt2$ slightly, you can get the count to be at least 2 always.)

To deduce how I came up with this function, consider it as the composition of $z\mapsto z+\sqrt2$ with $z\mapsto z^4$, and think about what each map does to the complex plane.