Is $\frac{d}{dx} \ln(\phi(x))$ decreasing where $\phi$ is the Gaussian cumulative distribution function? I was plotting the function $x \mapsto \frac{d}{dx} \ln(\phi(x))$ and it seams like this is a decreasing function. But somehow I cannot prove this. Here, $\phi$ is the Gaussian cumulative distribution function, i.e.
$$
\phi(x) = \int_{-\infty}^x \gamma(t) \, dt
$$
and $\gamma$ is the Gaussian. Do you have any suggestion how to show this?

$(ln (\phi(x))'=\frac {\gamma (x)} {\phi(x)}$. In view of smoothness this function is decreasing iff its derivative is $\leq 0$. So we require $\phi \gamma' \leq \gamma^{2}$  or $\phi \leq \frac {\gamma^{2}} {\gamma'}$. Look at the behavior at $\infty$ to see that this is false.
