Bounds for the derivative of inverse Mills ratio of standard normal distribution

Question

I'm trying to find bounds for the derivative of the inverse Mills ratio $\lambda(x)=\dfrac{\phi(x)}{\Phi(x)}$:

$\lambda^{\prime}(x)=-\lambda(x) (x+\lambda(x))$

While my matlab numerical results suggest that $\lambda^{\prime}(x)\in(-1,0)$, I could not prove that $\lambda^{\prime}(x)>-1$ for any $x$. Any ideas on how to prove it?

Answer 1

Proof from Sampford (1953):

Observe that $\dfrac{e^{-\frac{1}{2} u^2}}{\int^{x}_{-\infty}e^{-\frac{1}{2} u^2}\, \rm du}$ is a pdf defined on $(-\infty, x]$ and its variance is given by $1-\lambda(x)x -\lambda^2(x)$

Thus $1-\lambda(x)x -\lambda^2(x) = 1 + \lambda^{\prime}(x)>0$.