One general class of functions I have just found, perhaps a hint as to an even more general class, where the sort-and-scan approach works, (sorting pairs $(a,b)$ decreasing by $a/b$) is $F(x) = x$, and any positive differentiable $G(x)$ which satisfies $G'(x)/G(x) \leq \beta /x$ for some $\beta < 1$, for all $x \geq \min_i b_i$.  Example non-trivial $G(x)$ include $G(x) = x^\gamma$ for any power $0 < \gamma < 1$ (subsuming one of the given examples when $\gamma = 1/2$).  Also, at least if all $b_i > e$, then $G(x) = \log x$ also works.  If anyone is interested I can also post a proof here, it's pretty simple.  Hopefully someone else can come up with an even more general class with a proof though.