Let $\phi(\cdot)$ and $\Phi(\cdot)$ be the standard normal density and cumulative distribution function, respectively and let $a>0$. Consider the function:
$$f_a(x) = \frac{\phi(a+x) - \phi(a-x)}{\Phi(a-x) - \Phi(-a-x)}$$
The function has a form somewhat similar to $g(x)=-x^3$ as the below plot shows for two values of $a$. I would like to check that this function is convex for $x<0$. I don't know for sure that this is true, but it seems to me that this is the case (just by looking at the graph). Computing the second derivative isn't very helpful and the other definition which involves analysing a cord that goes through arbitrary two points $x_1,x_2$ also didn't lead me anywhere. Could you maybe suggest something I could try to solve this doubt?
