Let $f:[0,\infty) \to \mathbb{R}$ obey the differential inequality $$f'' - 2\alpha f' + 2\alpha f \leq 0$$ where $0 < \alpha < 2$ is some constant. If $f(0) = 0$ and $f'(0) = 1$, can I say that $f(x) < e^x - 1$ for some $x$?

Note that the solution to the corresponding differential equation oscillates since the characteristic equation has complex roots (call this solution $g$). Thus we can certainly say $g(x) < e^x -1$ for infinitely many $x$. My first thought was to try to control $f$ by $g$ a la Gronwall's inequality. However, I was recently shown that the analogue to Gronwall for degree two differential equations doesn't hold.

Any ideas would be welcome. Also, any good references for differential inequalities that might help me solve this problem are equally welcome.