I am reading a paper in which the following argument is made. We have two positive real valued functions $f(x)$ and $g(x)$. We know that $$\int_0^x \int_0^y f(z) \ dz \ dy \leq g(x).$$

It is then stated "and easy computation shows that, except for negligibly small intervals," $$\log f(x) = O(x + \log g(x)).$$

I am at a loss on how this comes about. $g(x)$ is convex, if that helps. Can anyone explain?