Approximating Mills ratio

Consider the Mills ratio $f(x)=\big(1-Φ(x)\big)/ϕ(x), \, x\ge 0$, where $ϕ$ is the density function of the standard Gaussian law and $Φ$ its cumulative distribution.We introduce a general procedure to approximate $f$ on the whole $[0,\infty)$ which allows to prove interesting properties where $f$ is involved. As applications we present a new proof that $1/f$ 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.

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