Composition of a convex function

Question

If $f:[a,b]\rightarrow R$ is convex function and $f'(x)\geq 0$ for all $x\in [a,b]$ and $g:U\rightarrow [a,b]$ is convex function, how to show that $f(g(u)), u\in U$ is convex function?

Answer 1

Note that $f$ is increasing. Let $x,y\in U, \lambda\in(0,1)$. From Jensen's inequality follows $$f(g(\lambda x + (1-\lambda)y))\leq f(\lambda g(x) + (1-\lambda)g(y))\leq \lambda f(g(x)) + (1-\lambda)f(g(y)),$$ which means the convexity of $f(g(u))$.