Abstract
Abstract Let R ( x ) $R\left ( x\right ) $ be the Mills ratio of the standard Gaussian law. This paper investigates the higher order monotonicity of the function x ↦ Y a , b ( x ) = ( x 2 + a ) R 2 ( x ) + b x R ( x ) $$ x\mapsto Y_{a,b}\left ( x\right ) =\left ( x^{2}+a\right ) R^{2} \left ( x\right ) +bxR\left ( x\right ) $$ on ( 0 , ∞ ) $\left ( 0,\infty \right ) $ , where a , b ∈ R $a,b\in \mathbb{R}$ . In particular, we prove that the function x ↦ Y a , a − 2 ( x ) − a + 1 $x\mapsto Y_{a,a-2}\left ( x\right ) -a+1$ is completely monotonic on ( 0 , ∞ ) $\left ( 0,\infty \right ) $ if and only if a ≥ − 1 $a\geq -1$ . As a special case, x ↦ Y − 1 , − 3 ( x ) + 2 $x\mapsto Y_{-1,-3}\left ( x\right ) +2$ is completely monotonic on ( 0 , ∞ ) $\left ( 0,\infty \right ) $ , which implies the complete monotonicity of the functions R 3 ( 1 / R ) ″ $R^{3}\left ( 1/R\right ) ^{\prime \prime }$ and − R 3 ( ln R ) ‴ $-R^{3}\left ( \ln R\right ) ^{\prime \prime \prime }$ on the same interval. As applications, several functional inequalities are derived, and some known and new sharp bounds for R are established.