The Stieltjes transform method is a technique to understand the spectrum of a random matrix. On the way to proving a local semicircle law with this technique, one encounters the stieltjes transform of the semicircle law, which is defined as $$ m(z) := \int_\mathbb{R} \frac{\rho}{x - z} dx $$ where $\rho = \frac{1}{2 \pi} \sqrt{(4 - x^2)_+}$. One can show that $m(z)$ satisfies the equation $$ m(z) + 1/ m(z) + z = 0 $$ with Im(z) > 0. One can solve for $m(z)$ to find that $$ m(z) = \frac{-z + \sqrt{z^2 - 4}}{2}. $$ Now it is claimed that for $x \in [-20, 20]$ and $y \in (0, 20]$ that for $z = x + i y$,
1) There is a constant $c > 0$ such that $$ c \leq |m(z)| \leq 1 - cy $$
2) There exist constants $C', c' >0$ such that $$ c' \sqrt{\kappa + y} \leq |1 - m^2(z)| \leq C' \sqrt{ \kappa + y} $$ where $\kappa := ||x| - 2|$.
3) Also for $|x| \leq 2$, there exist constants $C'', c''>0$ such that $$ c'' \sqrt{\kappa + y} \leq Im(m(z)) \leq C'' \sqrt{\kappa + y}. $$
4) For $|x| \geq 2$, $$ c'' \frac{y}{\sqrt{\kappa + y}} \leq Im(m(z)) \leq C'' \frac{y}{\sqrt{\kappa + y}}. $$
I can show the explicit form of $m(z)$ , but I'm having difficulty demonstrating any of the 4 properties. Any suggestions or explicit calculations would be helpful.
I'll give you my thoughts on 4). Let $z = x + i y$ then $$ Im(\frac{-z + \sqrt{z^2 - 4}}{2}) = -y/2 + Im(\sqrt{x^2 - y^2 + 2xy i})/2. $$ This should just be a brute force calculation and there is an explicit formula for the imaginary part of a square root. I can't guarantee this is the correct approach.