Let $f$ is positive-valued concave function and $g$ is another positive-valued concave function, such that $f:\mathbb{R}_+ \mapsto \mathbb{R}_+$ and $g:\mathbb{R}_+ \mapsto \mathbb{R}_+$. Additionally, $f\ge g$ over all domain $\mathbb{R}_+$.
Do their difference $f-g$ is concave function too?
This question is different from this.
No, let $f(x) = 1-e^{-x}$ and $g(x) = 1 - e^{-x/2}$.
I was trying to find out necessary conditions for it to be true. I was wondering about $f>g$. In fact, a slight modification of the above counter-example is sufficient to get another counter-example for the case $f>g$.
I came up with another counter-example. In fact the difference between two concave functions such that $f>g$ can be convex. Let $f$ be defined over $\mathbb{R}$, $$ f(x)= \left\{ \begin{array}{l} -\vert x \vert^3 \quad\text{if}\quad x\in [-0.25,0.25] \\ -3/16\vert x\vert +1/32 \quad\text{else}, \end{array} \right. $$ and $g(x)= -x^2$. Then $f$ and $g$ are concave and $f>g$ but $h = f-g$ is convex. You can see an illustration below. I have also illustrated the counter-examples given by Erik M and GLay.
