# A certain ratio condition for polynomials with real coefficients

Let $$p:\mathbb{C} \longrightarrow \mathbb{C}$$ be a polynomial with real coefficients and suppose that $$p$$ satisfies $$$$\frac{p(y)}{y} \le \frac{p(x)}{x} \tag{*} \label{ratcond}$$$$ whenever $$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?

• Why do you say that SOS implies your condition? Your condition fails, for example for $p(x)=(x-1)^2$. – Alexandre Eremenko Jan 30 at 13:47
• @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. – Pietro Paparella Jan 30 at 15:00