Let $f:\mathbb{R}\_+\rightarrow\mathbb{R}\_+$ be twice differentiable quasi-concave function satisfying $f(x)>0,\forall x \in \mathbb{R}\_+$. Let $g:\mathbb{R}\_+\rightarrow\mathbb{R}\_+$ be a positive, increasing and convex function (i.e. $g(x)>0,g'(x)>0,g''(x)>0$) satisfying
$$ g(x)\leq f(x), \forall x\in [a,b] \text{ with, } a<b\in \mathbb{R}\_+.$$
Is the sum
$$ h(x)=f(x)-g(x)$$
quasiconcave on the interval $[a,b]$? Can you recommend any references beyond standard books on convex optimization?



Establishing quasiconcavity
The answer is negative. Take $[a,b]=[1,3]$. Let
$f(x)=x+\epsilon(x-2)^2,$ where $\epsilon$ is small enough,
so that the function is increasing, thus quasiconcave.
Then take $g(x)=x+2\epsilon(x-2)^2-c$, where $c$ is such that
$g\leq f$ on $[a,b]$. Then the difference $f(x)-g(x)=-\epsilon(x-2)^2+c$ has a strict maximum at $x=2$ thus it is not quasiconcave.


