Small-$r$ asymptotics of an integral of $1/log ^alpha r$

Question

Let $f(r)=frac{1}{(-log r)^{alpha}}$ and let $xi_r$ be the unique value in $(0,r)$ such that$f(xi_r)=frac{1}{r}int_{0}^rf(t)dt$, where $alphain (0,1)$ and $r>0$. My question is about the order of $xi_r$?
Can we have $xi_r=o(r)$ as $r$ tends to $0$? moreover, can we get $xi_r=o(frac{r}{log^2(1/r)})$?
Is there any reference about this question?

Carlo Beenakker · Answer

Starting from the integral (in terms of an incomplete Gamma function)
$$g(r)=frac{1}{r}int_0^r (-ln t)^{-alpha},dt=frac{1}{r}Gamma(1-alpha,-ln r),$$
we expand for $rrightarrow 0$, or $yequiv -ln rrightarrowinfty$,
$$g(r)=y^{-1-alpha}bigl(y-alpha+{cal O}(1/y)bigr).$$
We then wish to solve $g(r)=(-ln xi)^{-alpha}$ for $xi$, or for $zequiv-lnxi$,
$$Rightarrow z=y^{1/alpha+1}bigl(y-alpha+{cal O}(1/y)bigr)^{-1/alpha}=y+1+{cal O}(1/y).$$
The resulting small-$r$ asymptotics is
$$xi=frac{r}{e}bigl(1+{cal O}(1/ln r)bigr).$$

Small-$r$ asymptotics of an integral of $1/log ^alpha r$

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