A positive function $f: \mathbb{R} \to \mathbb{R}$ is said to be slowly varying if$$\lim \limits_{t \to \infty} \frac{f(tx)}{f(t)}=1$$ for all $x >0$. Assume that $f(x) \in [m,M]$ for all $x$. Is there a bound $$\left| \frac{f(tx)}{f(t)}-1 \right| \leq \mathcal{O}(t)$$ for all $x >0$, with $\mathcal{O}(t) \to 0$ as $t \to \infty$? Are there known regularity conditions for $f$ that would guarantee the existence of such bound?
I have tried to have a look at some results of Karamata, but not much help from that so far. Locally uniform convergence of the above ratio is a well known result in this topic (see Theorem A.1.). I'm interested in existence of a global analogue of this result under some additional regularity conditions on $f$ (also whether such result can be expected to hold at all).
EDIT: The original question also included the question whether $f(t)$ converges or not as $t \to \infty$. I have removed that part, since it has been previously answered (in negative) elsewhere.