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A monotonicity property related to Laurent polynomials

Let $L$ be a Laurent polynomial with real coefficients, i.e., $$L(z)=\sum_{j=-r}^{s}a_{j}z^{j},$$ where $r,s\in\mathbb{N}$ and $a_{j}\in\mathbb{R}$. Assume further that the set $L^{-1}(\mathbb{R})\...