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

Let $h$ be a convex decreasing function and $g$ a convex function. Is it true that $h\left(g\left(x\right)\right)$ is a quasi-concave function?
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periasemdy
Yes.Since $g$ is convex, for $\lambda \in \left[0,1\right],$,
$g\left(\lambda x+\left(1-\lambda \right)y\right)\le \lambda g\left(x\right)+\left(1-\lambda \right)g\left(y\right).$
and
$\lambda g\left(x\right)+\left(1-\lambda \right)g\left(y\right)\le max\left[g\left(x\right),g\left(y\right)\right]$
Since $h$ is decreasing,
$h\left[g\left(\lambda x+\left(1-\lambda \right)y\right)\right]\ge h\left[\lambda g\left(x\right)+\left(1-\lambda \right)g\left(y\right)\right]\ge h\left(max\left[g\left(x\right),g\left(y\right)\right]\right)\ge min\left[h\left(g\left(x\right)\right),h\left(g\left(y\right)\right)\right].$