$f(T) = \int_{0}^{1}e^{-iT\lambda(s)}ds$, where $\lambda(s)$ increases monotonously. I want to prove it satisfies $|f(T)|^{2}$ has a proper upper bound that decreases with $T$ for $T \gg 1$.
For example, $\lambda(s) = s$ then $|f(T)| \le 2/T$.
This question comes from numerical analysis of my research. It's the expression of error analysis in adiabatic evolution. More specifically, the digital error has expression : $\sum_{k=1}^{L}e^{-iT\sum_{j<k}\lambda_{j}}/L$, where $\lambda_{j}$ equals $\lambda_{j} = \lambda(j/L)$. From numerical result I am sure it has an upper bound decays with $T$. However, I don't know how to prove that. I wonder if there is related formulas in complex analysis and fourier series.