Can we construct a function that is monotone increasing and nonnegative, such that $f'(x)^2 \ge \alpha f(x)f''(x)$ for each $x\in \mathbb R$, where $\alpha$ is greater than $1$. If not, how can we give a proof?
Note: we say $f(x)$ is monotone increasing, iff $f(x)<f(y)$ for all $x<y$.
I have tried a lot of examples but havn't found a solution. For example, consider $f(x)=b\exp(ax)$, then $f'(x)^2=f(x)f''(x)=b^2a^2\exp(ax)$, so the constraint "$\alpha$ is greater than 1" is not true.