I am trying to show that $$ \int_0^\infty e^{-x^2/2}(x^2+s)^{-\beta}\, dx\leq 4s^{\frac12-\beta} $$ for $\beta>1/2$ and $s>0$. It appears in this article:
https://link.springer.com/article/10.1007/BF00534922,
page 250.
A simple attempt goes like this: $$ \int_0^\infty e^{-x^2/2}(x^2+s)^{-\beta}\, dx\leq s^{\frac12-\beta}\int_0^\infty e^{-x^2/2}(x^2+s)^{-\frac12}\, dx. $$ I don't believe the integral on the right converges for small enough $s$, so the bound must be tighter. Any help would be appreciated.