Given a real symmetric $n$ dimensional matrix $A$, with eigenvalues $\lambda_i$ I am defining its Cauchy transform as the function, $f_A(z) = \sum_i \frac{1}{z-\lambda_i}\,$

- Is there any information about the $\lambda_i (A) $s that can be inferred from $f_A(z)$?

For example I would want an analysis of the following kind :

Say one can estimate the value of $z (A,\epsilon) \gt \max \{ \lambda_i \}$ such that for all $z \geq z (A,\epsilon)$, $f_A(z) \leq \epsilon$. Now suppose one is given a vector $v$ of the same dimension as $A$.

- Is there any connection between $\max \{\lambda(A + vv^T) \} - \max\{ \lambda(A) \}$ and $z(A + vv^T,\epsilon) - z(A,\epsilon)$? (at least for any restricted class of $A$ and $v$?)

Here I picked on $z(A,\epsilon)$ as an example of a property of the Cauchy transform which I feel might be sensitive to how the maximum eigenvalue of $A$ changes under rank-$1$ updates to $A$. But may be this $z(A,\epsilon)$ is not the right quantity - but there is may be some other property of the Cauchy transform which knows about, $\max \{\lambda(A + vv^T) \} - \max\{ \lambda(A) \}$ ?

If necessary assume that $v = (e_i \pm e_j)$ for some $j > n/2$ and $i<= n/2$.

If necessary assume that $A$ is constructed as follows : First take the matrix $D - Ad$ where $D$ is the diagonal matrix of degrees of some bi-partite graph and $Ad$ is its adjacency matrix. Then flip some of the off-diagonal $-1$ entries of $D-A$ to $1$ keeping the entire thing symmetric.

Some possibly related things I saw while searching,