Concavity of composition

Question

I have a question about the concavity of a composition of functions. Suppose $f$ is a pdf (if needed, a normal pdf). Suppose $g$ is a concave function. Is $g(f(x))$ concave? Thanks in advance!

Answer 1

It is not always true that the composition of concave functions is concave. For example, let $g(x) = -x^2$ with $f(x) = e^{-x^2}$. Then $g$ is concave but $h(x)=g(f(x))$ is not. We can easily see this by taking derivatives,

$$h'(x) = g'(f(x))f'(x) = 4xe^{-2x^2},\\ h''(x) = e^{-2x^2}(4-16x^2)$$

Since $h''(x)$ is not always negative (for example at $x=0$), we do not have concavity.

In general, $h''(x) = f''(x)g'(f(x))+f'(x)^2g''(f(x))$ will only be negative if $g$ is concave and also $g'(x) >0$ for all $x$.