In trying to solve another the problem posed in the question <https://www.mathoverflow.net/q/385777/78539>, I'm led to consider the following problem.


Let $\mu\_\gamma$ be the Marchenko-Pastur distribution with parameter $\gamma \in (0,1)$. Note that $\mu\_\gamma$ is supported on $[t\_-,t\_+]$, where $t\_{\pm} = (1\pm \sqrt{\gamma})^2$. For $\lambda \ge 0$, define
$$
I\_\gamma(\lambda):=\int\_{t\_-}^{t\_+} \dfrac{t}{(t + \lambda)^2}d\mu\_\gamma(t).
$$ It is clear that $I\_\gamma$ is a decreasing function on $[0,\infty)$.



> 
> **Question 1.** What is an analytic formula for $I\_\gamma(\lambda)$ ?
> 
> 
> 


**Note.** I'm fine with good lower-bounds on $I\_\gamma(\lambda)$


Solution for the extreme cases: $\lambda \to 0^+$ and $\lambda \to \infty$
--------------------------------------------------------------------------


For any $t > 0$, one may write
$$
\frac{t}{(t + \lambda)^2} = \frac{1}{t+\lambda} - \frac{\lambda}{(t+\lambda)^2} = \begin{cases}\dfrac{1}{t},&\mbox{ if }\lambda \to 0^+,\\ \dfrac{1}{t+\lambda}+\mathcal O(\dfrac{1}{\lambda}),&\mbox{ if }\lambda \to \infty.\end{cases}
$$


Let $m\_\gamma(z) := \int \dfrac{1}{z-t}d\mu\_\gamma(t)$ be the Stieltjes transform of $\mu\_\gamma$. It is well-known that for all real $z>0$,
$$
m\_\gamma(-z)=\frac{-(1-\gamma + z) + \sqrt{(1-\gamma+z)^2 + 4\gamma z}}{2\gamma z}
$$
be the Cauchy transform of $\mu\_\gamma$.


* **Case 1:** $\lambda \to 0^+$.
One recognizes
$
I\_\gamma(0^+) = \lim\_{\lambda \to 0^+}\int\_{t\_-}^{t\_+}\dfrac{1}{t + \lambda}d\mu\_\gamma(t) = \lim\_{\lambda \to 0^+}m\_\gamma(-\lambda)=\dfrac{1}{1-\gamma}.
$
* **Case 2:** $\lambda \to \infty$. Likewise, one recognizes $I\_\gamma(\infty) = \lim\_{\lambda \to \infty}\int\_{t\_-}^{t\_+}\dfrac{1}{t + \lambda}d\mu\_\gamma(t) = \lim\_{\lambda \to \infty}m\_\gamma(-\lambda) = 0$.



For fixed $\lambda \ge 0$, Integrate the function $f_\lambda(x):=x/(x + \lambda)^2$ w.r.t. Marchenko-Pastur density
$$I(\lambda)=\int\_{t\_-}^{t\_+}\frac{\sqrt{\left(t\_+-t\right) \left(t-t\_-\right)}}{2 \pi {\gamma} (t+\lambda)^2}\,dt
=\frac{-\sqrt{ {\gamma}^2+2 {\gamma} ( {\lambda}-1)+( {\lambda}+1)^2}+ {\gamma}+ {\lambda}+1}{2 {\gamma} \sqrt{ {\gamma}^2+2 {\gamma} ( {\lambda}-1)+( {\lambda}+1)^2}}.$$


