Approximating Mills ratio

Consider the Mills ratio f ( x ) = ( 1 − Φ ( x ) ) / ϕ ( x ) , x ≥ 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 , ∞ ) 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.