exponentially decaying weighted integral lower bound

For $f\in L^2$ for example, can we achieve sharpest lower/upper bound for the following integral for small $\epsilon>0$

$$\int_{0}^{T}e^{-\dfrac{t}{\epsilon}} |f(t)|dt $$ andunder what conditions on $f$ or error terms, we can make the following hold$$\int_{0}^{T}|f(t)|dt + error\quad \leq \int_{0}^{T}e^{-\dfrac{t}{\epsilon}} |f(t)|dt$$