Is there a way to solve the below complex polynomial equation analytically? Note $z \in \mathbb C^+ $ (upper half of $\mathbb C$-plane) $, n, p \in \mathbb Z^+, t_i \in \mathbb R, i=1,...,p$:
$$m(z) = \frac{1}{p} \sum_{i=1}^p \frac{1}{t_i \{1 - p/n - (p/n)zm(z)\} - z}$$
Some background: This is a discretized version of the Marcenko-Pastur equation. The $t_i$'s are the eigenvalues of the population covariance matrix $\Sigma_n \in \mathbb R^{p \times p}$. $m(z)$ represents the Stieltjes transform of any $z \in \mathbb C^+$. Assume we know $n, p, t_i$. $z \in \mathbb C^+$ is the input to the equation.
There is always a unique solution $m(z)$ in the set $\{m(z) \in \mathbb C : -\frac{n-p}{nz} + \frac{p}{n}m(z) \in \mathbb C^+ \}$, and it can be found numerically, but is there a way to solve for it analytically?
Thanks.