Find a function $\alpha: \mathbb R_+ \to \mathbb R_+$ so that for $\epsilon > 0$ it is $\frac{\log(\alpha(\epsilon))}{\alpha(\epsilon)} < \epsilon$

Question

Ideally, $\alpha$ should be bounded by a polynomial in $\frac 1 \epsilon$.

I feel like intuitively, this is obvious, because the logarithm grows much slower than the identity function. However, I fail to find an actual function $\alpha$ that works for all values of $\epsilon$.

(The reason I'm asking this question is because I'm trying to prove a (fully-polynomial) approximation scheme.)

Answer 1

Here's an answer to your question, but don't think it'll be useful for approximation (in which case you may need to edit your question to add additional properties to $\alpha(\epsilon)$). $$\alpha(\epsilon) = \epsilon + 1$$

$\alpha(\epsilon)$ is bound by polynomial $p(\frac{1}{\epsilon}) = .17$ and $\frac{\log \left(\epsilon+1\right)}{\epsilon+1} < \epsilon$ for all $\epsilon > 0$