Approximating Mills ratio

Consider the Mills ratio $f(x)=(1-\mathrm{\Phi }(x))\mathrm{/}\varphi (x),x\ge 0$, where $\varphi$ is the density function of the standard Gaussian law and $\mathrm{\Phi }$ its cumulative distribution.We introduce a general procedure to approximate $f$ on the whole $[0,\mathrm{\infty })$ which allows to prove interesting properties where $f$ is involved. As applications we present a new proof that $1\mathrm{/}f$ is strictly convex, and we give new sharp bounds of $f$ involving rational functions, functions with square roots or exponential terms. Also Chernoff type bounds for the Gaussian $Q$–function are studied.