How is $\xi+2a\eta<0$ an "obvious necessary condition" for $y^3+2y^2(1-2a-\xi)+y(1-4\xi+8a\xi)-2\xi-4a\eta >0$ to be satisfied for positive $y$?

Question

How is $$\xi+2\alpha\eta<0$$ an 'obvious necessary condition' for the inequality

$$y^3+2y^2(1-2\alpha-\xi)+y(1-4\xi+8\alpha\xi)-2\xi-4\alpha\eta >0$$ to be satisfied for positive $y$ (as claimed in this passage of an article)?

Someone please help me understand. I want to use the author's method, but I just don't see how the last coefficient affects the inequality.

Answer 1

Well, the text says that the statement shall hold true for all positive $y$. If you let $y \to 0$ you get exactly the condition.

More formally, for some arbitrarily small $y = \epsilon > 0$ we have, to order $\cal{O}(\epsilon)$, that $$y^3+2y^2(1-2\alpha-\xi)+y(1-4\xi+8\alpha\xi)-2\xi-4\alpha\eta \\ \to \epsilon(1-4\xi+8\alpha\xi)-2(\xi+2\alpha\eta) $$.

For this to remain positive, since the first term gets arbitrarily small, the second term needs to remain positive, which is the case if and only if $\xi+2\alpha\eta < 0$.