Let $R(x)$ be the residual function associated to the normal probability density, i.e.

$$R(x)~=~\int_x^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}}dy, \mbox{ for all } x\in R.$$

Define

$$\phi(s)~=~\int_0^1\log\Big(\frac{R(sx)}{R(s)}\Big)dx \mbox{ for all } s\ge 0.$$

Could we show the map $s\mapsto\phi(s)$ is increasing on $R_+$? Thanks for the reply!

Ps: Indeed, I have shown using Mathematica that the function $s\mapsto\phi(s)$ is increasing and moreover, for each $0\le x<1$, the maps $s\frac{R(sx)}{R(s)}$ is increasing, ex: $x=0, 0.1, 0.2, ..., 0.9$. Thus I strongly believe that it suffices to show the map $s\mapsto\frac{R(sx)}{R(x)}$ is increasing for $0\le x<1$. However, when I compute its derivative, it is still hard to determine its positivity. If someone has an idea to show that, many thanks!