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(Alexandre Eremenko gave the answer to the question I posted earlier. Then I realised the question didn't give what I wanted. Here is the updated question.)

Suppose that $f:[a,\infty)\to \mathbb{R}$ is smooth at infinity. I would like to have some result like if $f'$ is monotone and does not grow too fast, then something like the following holds

(1) $\lim_{x\to\infty} |f'(x)|/|x f(x)| = 0$.

(2) $\lim_{x\to\infty} |f'(x)|/|f(x)| = 0$.

For (2), it is easy to show that if there exists $\alpha,\beta$ with $\alpha\leq \beta<\alpha+1$ such that $x^\alpha \leq |f'(x)| \leq x^\beta$ then (2) holds. Can this be relaxed?

For (1), my feeling is that it should hold for $f'(x)$ up to $f(x) = e^x$ or even up to $e^{x^2}$. But I do not have a proof...

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    $\begingroup$ what do you exactly mean by "smooth at infinity"? $\endgroup$
    – user35593
    Commented Oct 27, 2015 at 16:16
  • $\begingroup$ If $f(x)=e^xsin(x)$ your ratio does not converge. $\endgroup$
    – user35593
    Commented Oct 27, 2015 at 16:19
  • $\begingroup$ I mean, $f$ is $C^\infty[a,\infty)$ for some $a$ $\endgroup$
    – user58955
    Commented Oct 27, 2015 at 16:48
  • $\begingroup$ @user35593 In that case, (1) still holds and the limit exists, (2) doesn't hold. Well I don't expect (2) to hold for that, I only expect that (2) holds for $f$ with polynomial growth. Moreover, the limits are $\liminf$. $\endgroup$
    – user58955
    Commented Oct 27, 2015 at 16:51

1 Answer 1

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Suppose that your function is non-negative and increasing. Negation of (1) is $f'(x)>cxf(x)$ for $x$ large enough. This implies $f'(x)/f(x)>cx$ and integration leads to $f(x)>\exp(cx^2/2)$. Similarly (2) would imply $f(x)>\exp(cx)$.

If the function is not increasing for $x>a$ with some $a$, there is nothing to prove, because then $f'$ will have an unbounded sequence of zeros. If it is increasing and tends to infinity, then it is of course non-negative for large $x$.

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  • $\begingroup$ Thanks! This is indeed an answer. Somehow I realised I do need limit instead of lim inf in my current research and have modified the questions. I want to exclude the oscillatory case and imposed that $f'$ is monotone. Do you see if it remains true now? $\endgroup$
    – user58955
    Commented Oct 27, 2015 at 21:32
  • $\begingroup$ The limit of course is not granted under any growth conditions on $f$. $\endgroup$
    – Alexandre Eremenko
    Commented Oct 28, 2015 at 2:02
  • $\begingroup$ But there's constraint that $f'$ is monotonically increasing? $\endgroup$
    – user58955
    Commented Oct 28, 2015 at 2:35
  • $\begingroup$ Also I am looking at growth condition on $f'$ now $\endgroup$
    – user58955
    Commented Oct 28, 2015 at 3:07

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