I have a seemingly simple question that I haven't been able to resolve:
Question. Suppose that $f:(0,\infty)\to(0,\infty)$ is nondecreasing. Suppose that $f\leq L$ for a slowly varying function $L$. Does this imply that $f$ is also slowly varying?
For example, If we have some arbitrary function $f$ such that $0\leq f\leq L$ where $L$ is slowly varying, then obviously this doesn't imply that $g$ is also slowly varying. To see this, we can take $f(x)=\sin(x)+2$.
However, if we also assume that $f$ is nondecreasing, the question strikes me as nontrivial. For instance, a bounded nondecreasing function is convergent, hence slowly varying, and so the claim is at least true in this limited context.
Edit. I mean slowly varying in the standard sense of the term, i.e., $L$ is slowly varying if and only if $$\lim_{x\to\infty}\frac{L(ax)}{L(x)}=1$$ for every $a>0$.