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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