(edited for more clarity)
For a given function $f$, which is continuous, and $a < b$ real numbers, I need to make an estimation of the type $ \Bigg| \frac{\int_a^b f(t) (-t)dt}{\int_a^b f(t)dt} \Bigg| \leq M $. That is to say, I want to find the minimal $M$ with that property.
(1) Are there any tricks for estimating this for such general $f$?
(2) I could further assume for my application that $f$ is positive... then my best result is that $a \leq M \leq b$, because $f(t)a \leq f(t)t \leq f(t) b \quad \forall t \in [a,b]$ Does somebody have an idea how to improve these bounds?
Well the estimation $$(b-a)\cdot \min_{x\in [a,b]}\{f(x)\}\leq \int_a^b f(x)dx \leq (b-a)\cdot \max_{x\in [a,b]}\{f(x)\}$$
is an exact estimation (i.e., we can find functions for which both inequalities become equalities), so with no other info about $f$, it's the best you can do...