Bounds on the distance between probability distributions in terms characteristic functions

Question

I am looking for the bounds on the distance between probability distributions in terms characteristic functions.

For example, I am aware of the following bound \begin{align} d(P,Q) \le \frac{1}{T} \int_{-T}^T \frac{|\phi_P(t)-\phi_Q(t)|}{t} dt+ \frac{q_{max}}{ \pi T} \end{align} where the $d(P,Q)$ is the Kolmogorov-Smirnov distance and $q_{max}$ the maximum value of the pdf of $Q$.

Specifically, am looking for the bonds that would depend on the following difference

\begin{align} | t \phi_P^{\prime}(t)+\phi_P(t)-(t \phi_Q^{\prime}(t)+\phi_Q(t)) | \end{align}