Suppose $f:\mathbb R \rightarrow \mathbb R$, $O(1)<f(x)<O(x)$ (i.e. has slower than linear growth but is unbounded), $\lim_{x\rightarrow \infty} f'(x)=0$ and $f'$ is monotonic. Is it possible to show that there exists a $\delta$ such that for $|x|>\delta$, $|x f'(x) |< |f(x)|$ for all such functions $f$?
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Certainly not as you've phrased the question at the moment. Take $f = -1/x$, for example. Did you want to take absolute values? – Rhys Aug 07 '15 at 19:23
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yes, thank you for the correction. – Jong Aug 07 '15 at 19:40