For $r,s\in\mathbb{N}$, let $$L(z):=\sum_{j=-r}^{s}a_{j}z^{j}$$ be a Laurent polynomial with real coefficients such that there exists a closed curve $\gamma$ encircling the origin, i.e., $0\in\mbox{Int }\gamma$ (interior of $\gamma$), and $L$ is real valued if restricted to $\gamma$.

Although it is not obvious, there are many of such Laurent polynomials. The most simple example is $L(z)=1/z+z$. Then $\gamma$ is just the unit circle. More involved example is, for instance, $L(z)=2/z+6z+z^{2}$. However, to show the existence of the curve in this case is a more difficult task (but doable).

My **conjecture** is that, if $L$ has the property as above, then
$$\mbox{Int }\gamma \textbf{ is a convex set}.$$
Can you prove/disprove it?