Assume we have a function like this: $$ F(x) = \sum_{k=1}^x f(k) $$ And we have upper and lower bounds for it: $$ c_1 g(x) < F(x) < c_2h(x). $$ Can we find a lower bound for $f(x)$ with this information? I tried this but it didn't work.
But $c_1g(x) - c_2h(x-1)$ is always negative for particular $g,h$ functions I'm working on... I need to find a positive lower bound. Is it possible?
Let $\pi$(x) be the prime counting function.
$$g(x) = (\pi(x)-1)^2$$ $$h(x) = (\pi(2x-1)-1)^2$$