TITLE: Prove $F(x)=\frac{1}{x}\int_0^xf(t)$ is convex
QUESTION [6 upvotes]: Assume $f(x)$ is convex in $[0,\infty)$, Prove
$F(x)=\frac{1}{x}\int_0^xf(t)dt$ is convex

REPLY [5 votes]: A hint: Substitute $t:=\tau\, x$ $\ (0\leq\tau\leq 1)$ and look at the resulting integral.