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 $\alpha\in (0,1)$ 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?



Small-$r$ asymptotics of an integral of $1/\log ^\alpha r$
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 $r\rightarrow 0$, or $y\equiv -\ln r\rightarrow\infty$,
$$g(r)=y^{-1-\alpha}\bigl(y-\alpha+{\cal O}(1/y)\bigr).$$
We then wish to solve $g(r)=(-\ln \xi)^{-\alpha}$ for $\xi$, or for $z\equiv-\ln\xi$,
$$\Rightarrow z=y^{1/\alpha+1}\bigl(y-\alpha+{\cal O}(1/y)\bigr)^{-1/\alpha}=y+1+{\cal O}(1/y).$$
The resulting small-$r$ asymptotics is
$$\xi=\frac{r}{e}\bigl(1+{\cal O}(1/\ln r)\bigr).$$


