How to analyze the following complex number? $f(T) = \int_{0}^{1}e^{-iT\lambda(s)}ds$, where $\lambda(s)$ increases monotonously. I want to show that  $|f(T)|$ has an upper bound that decreases with $T$ for $T \gg 1$.
For example, if $\lambda(s) = s$ 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 which 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.

Remark: This answer was for the first version of the question which stated that $|f(T)|^2$ was monotonic.
The result is not true. If $\lambda(s) = s$, then
\begin{equation}
f(T) = \frac{1}{i T}(1 - e^{-i T})
\end{equation}
hence $f(2k\pi) = 0$ for $k\in {\mathbb{N}^*}$ but $f$ is not identically $0$, hence $|f(T)|^2$ is not monotonic.
