Higher order monotonicity involving the Mills ratio with applications

Let R ( x ) be the Mills ratio of the standard Gaussian law. This paper investigates the higher order monotonicity of the function x↦ Ya,b ( x ) = ( x²+a ) R² ( x ) +bxR ( x )on ( 0,∞ ), where a,b∈ R. In particular, we prove that the function x↦ Ya,a-2 ( x ) -a+1 is completely monotonic on ( 0,∞ ) if and only if a≥ -1. As a special case, x↦ Y-1,-3 ( x ) +2 is completely monotonic on ( 0,∞ ), which implies the complete monotonicity of the functions R³ ( 1/R ) and -R³ ( ln R ) on the same interval. As applications, several functional inequalities are derived, and some known and new sharp bounds for R are established.