Let $p:\mathbb{C} \longrightarrow \mathbb{C}$ be a polynomial with real coefficients and suppose that $p$ satisfies \begin{equation} \frac{p(y)}{y} \le \frac{p(x)}{x} \tag{*} \label{ratcond} \end{equation} whenever $y < 0$ and $x + y \ge 0$.

Is there anything known about such polynomials? Is \eqref{ratcond} related to the derivative of $p$ in some way or equivalent to another well-known condition?