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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?

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  • $\begingroup$ Why do you say that SOS implies your condition? Your condition fails, for example for $p(x)=(x-1)^2$. $\endgroup$ – Alexandre Eremenko Jan 30 at 13:47
  • $\begingroup$ @AlexandreEremenko I was looking for a quick example but should’ve known that not all SOS polynomials would work; thank you for pointing this out. $\endgroup$ – Pietro Paparella Jan 30 at 15:00

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