[Math] Composition of a convex function and a convex decreasing function is quasi-concave

Question

Let $h$ be a convex decreasing function and $g$ a convex function. Is it true that $h(g(x))$ is a quasi-concave function?

Answer 1

Yes.

Since $g$ is convex, for $\lambda \in [0,1],$

$$g(\lambda x +(1-\lambda )y) \leq \lambda g(x) + (1-\lambda)g(y).$$

and

$$\lambda g(x) + (1-\lambda)g(y)\leq\max[g(x),g(y)]$$

Since h is decreasing,

$$h[g(\lambda x +(1-\lambda )y)] \geq h[\lambda g(x) + (1-\lambda)g(y)]\geq h(\max[g(x),g(y)])\geq \min[h(g(x)),h(g(y))].$$