Positiveness of the double integral

How it can be proved that the following double integral (which emerged in a physics project) $$\int\limits_{-1}^1d\tau\;\tau \int\limits_0^\infty dk\frac{k\sin{(kr\tau)}}{(1+\beta k^\alpha)^{2/(1-\alpha)}}$$ is positive for all $r>0,\,\beta>0,\,0<\alpha<1$? For $\alpha\le\frac{1}{2}$, the inner integral should be considered as a generalized function.

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It makes sense to integrate against the characteristic function of any interval. I think that means you can reduce your question to whether $\int_{-1}^1\int_a^b\frac{ksin(kr\tau)}{(1+\beta k^{\alpha})^{2/(1-\alpha)}}dkd\tau$ is a positive number. You can right some code to approximate this pretty well, so I would start by doing numerical experiments to see if it is true. The numbers you get will lead you to the correct estimates to prove it if it's true. –  Charlie Frohman Sep 20 at 14:06
For $\alpha \ge 1/2$, the inner integral is absolutely convergent and can be replaced by a sum over integrals over the smaller intervals $[\frac{n\pi}{r\tau},\frac{(n+1)\pi}{r\tau}]$. Then, for $r$ sufficiently small, this sum can be shown to be alternating with monotonically decreasing terms. For $\tau>0$, the first term is positive, hence the whole sum is positive. This of course doesn't cover all the cases that you want. –  Igor Khavkine Sep 20 at 14:51