  Consider the Mills ratio $f(x)=\big(1-\Phi(x)\big)/\phi(x), \, x\ge 0$, where
$\phi$ is the density function of the standard Gaussian law and $\Phi$ 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.
