prove that The function h(x) = g(f(x)) is convex. given g(x) and f(x) are convex [duplicate]
SO, I have been reading on convex functions and came across the property that the function h(x) = g(f(x)) is convex. given g(x) and f(x) are convex, could someone give me the proof? given g is non-decresing
Solution 1:
This is not true. $g(x)=e^{-x}, f(x)=x^{2}$ is a counter-example. If add the condition that $g$ is also increasing then the result is true.
[ $g\circ f$ is concave on $(0,\frac 1 {\sqrt 2})$ and convex on $(\frac 1 {\sqrt 2}, \infty)$].