Compute an integral explicit (or explicit upperbound? Possible?)

Question

I would just like to know if there is an explicit formula/computation for this integral:

$$\int_{a}^1 \left(|x-a|^{\alpha} - |x-b|^{\alpha} \right)^2 dx$$ where $\alpha\in (-1/2,0)$ so the integral is convergent and $0<a<b<1$ are fixed.

Is it possible to find an explicit expression? If not, any other upperbound of this in terms of $a,b$ and $\alpha$? Any ideas? I tried everything that is naiv, i.e. change of variables, trivial upperbounds... try to find a simpler majorizing function but I am not so good at these things.

Thanks a lot for any help!

Answer 1

By shifting the integral, you can set $a$ to zero while changing the upper limit of integration. Then you can rescale $x$ to set $b=1$, say, to get $$ (b-a)^{2\alpha+1}\int_0^\frac{1-a}{b-a} [z^\alpha-|1-z|^\alpha]^2 dz $$

In the special case $b=1$, the integral $$ \int_0^1 [z^\alpha-(1-z)^\alpha]^2 dz $$ can be evaluated explicitly since it is essentially a beta function, to get $$\frac{2}{2 \alpha +1}-\frac{2 \Gamma (\alpha +1)^2}{\Gamma (2 \alpha +2)}$$ which will straightforwardly give you an upper bound if $b<1$.