A gaussian tail estimation

Question

In the proof of minmax bound for multi-arm bandits, it refers an inequality $$\frac{\exp \left(-x^{2}\right)}{x+\sqrt{x^{2}+2}} \leq \int_{x}^{\infty} \exp \left(-t^{2}\right) d t \leq \frac{\exp \left(-x^{2}\right)}{x+\sqrt{x^{2}+4 / \pi}},x\ge0.$$ However, I've not found the proof in its reference and I didn't find it trivial. It is a tight bound I think. Can anyone share an idea?

Answer 1

It's from a less well-known (I think) inequality about Gaussian Mills' ratio $M(x)\equiv e^{x^{2}}\int_{x}^{\infty}e^{-t^2}dt$ that for $x\geq0$, $$\frac{1}{x+\sqrt{x^{2}+2}}<M(x)\leq\frac{1}{x+\sqrt{x^{2}+4/\pi}}. $$ The upper bound is tight while the lower bound is not. For the proof of a more general version of this inequality, see W. Gautschi (1959b). Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38(1), pp. 77–81.