Let $f:[0,\infty)\to[0,\infty)$ and $g:\mathbb{R}\to\mathbb{R}$ be two bounded continously differentiable functions. Do we have the existence of a constant $C$ such that $$\int_0^af(s)\int_0^b g(t)\left(g(t)-g(t-s) \right)dtds\leq C \int_0^af(s) \left| \int_0^b \left(g(t)-g(t-s) \right)dt \right|ds $$ for all $a,b\geq 0$?
I think of using the first mean value theorem for definite integrals, by this requires the function $g(t)-g(t-s)$ to have a constant sign.