Steck provides several approximations for the multivariate Mills' ratio. We first prove a new result for the univariate Mills' ratio and use it to give simple sufficient conditions for Steck's best approximation to be a lower bound.
"On a Lower Bound for the Multivariate Normal Mills' Ratio." Ann. Probab. 14 (4) 1399 - 1403, October, 1986. https://doi.org/10.1214/aop/1176992380