Let $f:[1,\infty)\to\mathbb{R}$ be a measurable function with $\lim_{t\to\infty} f(t)=0$. I want to show that the function $x\mapsto \int_1^x \frac{f(t)}{t} dt$ is asymptotically sublogarithmic, i.e. $\lim_{x\to\infty}\frac{1}{\log x} \int_1^x \frac{f(t)}{t} dt = 0.$ Although I think I should be able to prove this, I am not.
Integral of $1/x$ times a decaying function
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2Hint: Suppose |f(t)|<\delta for all t>t_0. Show that \limsup_{x\to\infty}\frac{1}{\log x} \int_1^x \frac{|f(t)|}{t} dt < \delta. Do it by decomposing the interval of integration in to two parts: below $t_0$ and above $t_0$. – 2012-06-29
2 Answers
Since $\lim_{t\to\infty} f(t) = 0$ you have that for every $\epsilon > 0$ there exists $M > 0$ such that $|f(t)| < \epsilon$ for all $t > M$. For $y > M$ you then have
$ \frac{1}{\log y} \int_1^y \frac{f(t)}{t}\mathrm{d}t = \frac{1}{\log y} \int_1^M \frac{f(t)}{t} \mathrm{d}t + \frac{1}{\log y} \int_M^y \frac{f(t)}{t}\mathrm{d}t $
The first integral contributes a constant term (only depending on $M$). The second integral can be bounded by
$ \lvert\int_M^y \frac{f(t)}{t} \mathrm{d}t\rvert \leq \epsilon \log \frac{y}{M} $
Hence we have that
$ \left\lvert \frac{1}{\log y}\int_1^y \frac{f(t)}{t} \mathrm{d}t\right\rvert \leq \frac{C_M}{\log y} + \epsilon $
By choosing $y > M$ large enough we can make the first term also $< \epsilon$ using that $\log y$ grows unboundedly. This means that for every $\epsilon$ you can choose $Y_0$ sufficienly large such that for every $y > Y_0$,
$ \left\lvert\frac{1}{\log y}\int_0^y \frac{f}{t} \mathrm{d}t \right\rvert < 2\epsilon $
as desired.
For every $\epsilon>0$, $\frac{1}{\log x}\int_1^x \frac{f(t)}{t}\,dt-\epsilon = \frac{1}{\log x}\int_1^x \frac{f(t)-\epsilon}{t}\,dt$ and since $f(t)-\epsilon<0$ for all large $t$, you should be able to show that $\limsup_{t\to\infty}\frac{1}{\log x}\int_1^x \frac{f(t)-\epsilon}{t}\,dt\le 0$ Hence $\limsup_{t\to\infty}\frac{1}{\log x}\int_1^x \frac{f(t)}{t}\,dt\le \epsilon$ and replacing $f$ with $-f$ gives you a bound from the other side..