For a (random) hermitian matrix $M$ and a complex $z$, it is well known that
$$ \left| \int_{\mathbb{R}} \frac{1}{z-x} \text{d}\mu_M(x) \right| = \left| \frac{1}{n} \text{Tr} (z-M)^{-1} \right| \leq \frac{1}{\left|\Im(z)\right|} $$
but I am looking for a majoration involving the real part of $z$.
Could you help me, please ?
Thank you for your consideration on this matter,
Nawak