Let $c>0$, $f\colon \mathbb{R} \to [c,\infty)$ be differentiable and convex.
Do we have
$$ \left\|\frac{f'}{f}\right\|_{\infty} < \infty ?$$
This seems to be true in simple examples, but I am not sure whether this is true in general, so I would appreciate some hint or a counterexample.
I think this is false : take $f : x \mapsto e^{x^2}$. This is convex, but $f'(x) \gg f(x)$ as $x$ tends to infinity.